Backward SLE and the symmetry of the welding

Abstract

The backward chordal Schramm–Loewner evolution naturally defines a conformal welding homeomorphism of the real line. We show that this homeomorphism is invariant under the automorphism $x\mapsto -1/x$, and conclude that the associated solution to the welding problem (which is a natural renormalized limit of the finite time Loewner traces) is reversible. The proofs rely on an analysis of the action of analytic circle diffeomorphisms on the space of hulls, and on the coupling techniques of the second author.

1 Introduction

1.1 Introduction and results

The Schramm–Loewner evolution ${{\mathrm{SLE}}}_{\kappa }$, first introduced in [16], is a stochastic process of random conformal maps that has received a lot of attention over the last decade. We refer to the introductory text [7] for basic facts and definitions. In this paper we are largely concerned with chordal ${{\mathrm{SLE}}}_{\kappa }$, which can be viewed as a family of random curves $\gamma$ that join $0$ and $\infty$ in the closure of the upper half plane $\overline{\mathbb {H}}.$ A fundamental property of chordal SLE is reversibility: The law of $\gamma$ is invariant under the automorphism $z\mapsto -1/z$ of $\mathbb {H},$ modulo time parametrization. This has first been proved by the second author in [20] for $\kappa \le 4$, and recently by Miller and Sheffield for $4<\kappa \le 8$ in [11]. It is known to be false for $\kappa >8$ [15, 21].

In the early years of SLE, Oded Schramm, Wendelin Werner and the first author made an attempt to prove reversibility along the following lines: The “backward” flow

$$\begin{aligned} \partial _t f_t(z)=\frac{-2}{f_t(z)-\sqrt{\kappa }B_t}, \quad f_0(z)=z,\quad 0\le t\le T, \end{aligned}$$

generates curves $\beta _T=\beta [0,T]$ whose law is that of the chordal SLE trace $\gamma [0,T]$ (up to translation by $\sqrt{\kappa }B_T$). When $\kappa \le 4,$ these curves are simple, and each point of $\beta$ (with the exception of the endpoints) corresponds to two points on the real line under the conformal map $f_t$. The conformal welding homeomorphism $\phi$ of $\beta _T$ is the auto-homeomorphism of the interval $f_T^{-1}(\beta _T)$ that interchanges these two points. In other words, it is the rule that describes which points on the real line are to be identified (laminated) in order to form the curve $\beta _T.$ It is known [15] that, for $\kappa <4$, the welding almost surely uniquely determines the curve. The welding homeomorphism can be obtained by restricting the backward flow to the real line: Two points $x\ne y$ on the real line are to be welded if and only if their swallowing times coincide, $\phi (x)=y$ if and only if $\tau _x=\tau _y$, see Sect. 3.5. An idea to prove reversibility was to prove the invariance of $\phi$ under $x\mapsto -1/x$, and to relate this to reversibility of a suitable limit of the curves $\beta _T$. But the attempts to prove invariance of $\phi$ failed, and this program of proving reversibility was never completed successfully.

Instead, other methods of proving reversibility became available. In this paper, we turn the above strategy around: We use the coupling techniques of the second author, introduced in [20] for his proof of reversibility of (forward) SLE traces, to prove the invariance of the welding. The main result of this paper is the following:

Theorem 1.1

Let $\kappa \in (0,4]$, and $\phi$ be a backward chordal SLE$_\kappa$ welding. Let $h(z)=-1/z$. Then $h\circ \phi \circ h$ has the same distribution as $\phi$.

As a consequence, in the range $\kappa \in (0,4)$ where the SLE trace is conformally removable, we obtain the reversibility of suitably normalized limits of the $\beta _T$ (see Sect. 6 for details):

Theorem 1.2

Let $\kappa \in (0,4)$, and $\beta$ be a normalized global backward chordal SLE$_\kappa$ trace. Let $h(z)=-1/z$. Then $h(\beta {\setminus }\{0\})$ has the same distribution as $\beta {\setminus }\{0\}$ as random sets.

In the important paper [18], Sheffield obtains a representation of the SLE welding in terms of a quantum gravity boundary length measure, and also relates it to a simple Jordan arc, which differs from our $\beta$ only through normalization. However, Theorems 1.1 and 1.2 do not follow easily from his work. A similar random welding homeomorphism is constructed in [3], where the main point is the very difficult existence of a curve solving the welding problem. Our approach to the welding is different: In order to prove Theorem 1.1, in Sect. 2 we develop a framework to study the effect of analytic perturbations of weldings on the corresponding hulls. We show in Sect. 4 that a Möbius image of a backward chordal SLE$_\kappa$ process is a backward radial SLE($\kappa , -\kappa -6$) process, and the welding is preserved under this conformal transformation. In Sect. 5 we apply the coupling technique to show that backward radial SLE($\kappa , -\kappa -6$) started from an ordered pair of points $(a,b)$ commutes with backward radial SLE($\kappa , -\kappa -6$) started from $(b,a)$, and use this in Sect. 6 to prove Theorem 1.1.

In a subsequent paper [25] of the second author, Theorem 1.1 is used to study the ergodic properties of a forward SLE$_\kappa$ trace near the tip at a fixed capacity time.

1.2 Notation

Let $\widehat{\mathbb {C}}=\mathbb {C}\cup \{\infty \}$, $\mathbb {D}=\{z\in \mathbb {C}:|z|<1\}$, $\mathbb {D}^*=\widehat{\mathbb {C}}{\setminus }\overline{\mathbb {D}}$, $\mathbb {T}=\{z\in \mathbb {C}:|z|=1\}$, and $\mathbb {H}=\{z\in \mathbb {C}:{{\mathrm{Im}}}z>0\}$. Let $I_\mathbb {R}(z)=\overline{z}$ and $I_\mathbb {T}(z)=1/\overline{z}$ be the reflections about $\mathbb {R}$ and $\mathbb {T}$, respectively. Let $e^i$ denote the map $z\mapsto e^{iz}$. Let $\cot _2(z)=\cot (z/2)$ and $\sin _2(z)=\sin (z/2)$. For a real interval $J$, let $C(J)$ denote the space of real valued continuous functions on $J$. An increasing or decreasing function in this paper is assumed to be strictly monotonic. We use $B(t)$ to denote a standard real Brownian motion. By $f:D\mathop {\twoheadrightarrow }\limits ^\mathrm{Conf}E$ we mean that $f$ maps $D$ conformally onto $E$. By $f_n\mathop {\longrightarrow }\limits ^\mathrm{l.u.}f$ in $U$ we mean that $f_n$ converges to $f$ uniformly on every compact subset of $U$. We will frequently use the notation $D_n\mathop {\longrightarrow }\limits ^\mathrm{Cara}D$ as in Definition 7.1.

The outline of this paper is the following. In Sect. 2, we derive some fundamental results in Complex Analysis, which are interesting on their own. In Sect. 3, we review the properties of forward Loewner processes, and derive some properties of backward Loewner processes. In Sect. 4, we discuss how are backward Loewner processes transformed by conformal maps. In Sect. 5, we present and prove certain commutation relations between backward SLE$(\kappa ;\mathbf {\rho })$ processes. In the last section, we prove the reversibility of backward chordal SLE$_\kappa$ processes for $\kappa \in (0,4]$ and propose questions in other cases. In the appendix, we discuss some results on the topology of domains and hulls.

2 Extension of conformal maps

2.1 Interior hulls in $\mathbb {C}$

An interior hull (in $\mathbb {C}$) is a nonempty compact connected set $K\subset \mathbb {C}$ such that $\mathbb {C}{\setminus } K$ is also connected. For every interior hull $K$ in $\mathbb {C}$, there are a unique $r\ge 0$ and a unique $\phi _K:\widehat{\mathbb {C}}{\setminus } K\mathop {\twoheadrightarrow }\limits ^\mathrm{Conf}\widehat{\mathbb {C}}{\setminus } r\overline{\mathbb {D}}$ such that $\phi _K(\infty )=\infty$ and $\phi _K'(\infty ):=\lim _{z\rightarrow \infty }z/\phi _K(z)=1$. We call ${{\mathrm{rad}}}(K):=r$ the radius of $K$ and ${{\mathrm{cap}}}(K):=\ln (r)$ the capacity of $K$. The radius is $0$ iff $K$ contains only one point. In general, we have ${{\mathrm{rad}}}(K)\le {{\mathrm{diam}}}(K)\le 4{{\mathrm{rad}}}(K)$. We call $K$ nondegenerate if it contains more than one point. For such $K$, there is a unique $\varphi _K:\widehat{\mathbb {C}}{\setminus } K\mathop {\twoheadrightarrow }\limits ^\mathrm{Conf}\mathbb {D}^*$ such that $\varphi _K(\infty )=\infty$ and $\varphi _K'(\infty )>0$. In fact, $\varphi _K=\phi _K/{{\mathrm{rad}}}(K)$. Let $\psi _K=\varphi _K^{-1}$ for such $K$.

For any Jordan curve $J$ in $\mathbb {C}$, let $D_J$ denote the Jordan domain bounded by $J$, and let $D_J^*=\widehat{\mathbb {C}}{\setminus }(D_J\cup J)$. Suppose $f_J:\mathbb {D}\mathop {\twoheadrightarrow }\limits ^\mathrm{Conf}D_J$ and $f_J^*=\psi _{\overline{D_J}}:\mathbb {D}^*\mathop {\twoheadrightarrow }\limits ^\mathrm{Conf}D_J^*$. Then both $f_J$ and $f_J^*$ extend continuously to a homeomorphism from $\mathbb {T}$ onto $J$. Let $h=(f_J^*)^{-1}\circ f_J$. Then $h$ is an orientation-preserving automorphism of $\mathbb {T}$. We call such $h$ a conformal welding. Not every homeomorphism of $\mathbb {T}$ is a conformal welding, but it is well-known (and an easy consequence of the uniformization theorem) that every analytic automorphism is a conformal welding, and that the associated Jordan curve is analytic (c.f. [2, Chapter II, Section 1, 3D]). Also see [10] for the quasiconformal theory of conformal welding, and [4] for deep generalizations and further references.

Lemma 2.1

Let $\beta$ be an analytic Jordan curve. Let $\Omega \subset \mathbb {C}$ be a neighborhood of $\mathbb {T}$. Suppose $W$ is a conformal map defined in $\Omega$, maps $\mathbb {T}$ onto $\mathbb {T}$, and preserves the orientation of $\mathbb {T}$. Let $\Omega ^\beta =\beta \cup D_{\beta }\cup \psi _{\overline{D_\beta }}(\Omega \cap \mathbb {D}^*)$. Then there is a conformal map $V$ defined in $\Omega ^\beta$ such that $V\circ \psi _{\overline{D_\beta }}=\psi _{\overline{D_{V(\beta )}}}\circ W$ in $\Omega \cap \mathbb {D}^*$ (Fig. 1).

Fig. 1

The situation of Theorem 2.2 and Lemma 2.1. Given $H$ and $W$, $V$ can be constructed to be analytic in $H$. In Lemma 2.1, the boundary of $H$ is assumed to be an analytic Jordan curve, while in Theorem 2.2, no regularity assumption is made

Proof

Fix a conformal map $f_{\beta }:\mathbb {D}\mathop {\twoheadrightarrow }\limits ^\mathrm{Conf}D_{\beta }$ and let $h_{\beta }=\varphi _{\overline{D_\beta }}\circ f_{\beta }$ be the associated conformal welding homeomorphism. Define $h=W\circ h_{\beta }.$ Since $\beta$ is analytic, $h$ is analytic and there is an analytic Jordan curve $\gamma$ and a conformal map $f_{\gamma }:\mathbb {D}\mathop {\twoheadrightarrow }\limits ^\mathrm{Conf}D_{\gamma }$ such that $h=h_{\gamma }=\varphi _{\overline{D_\gamma }}\circ f_{\gamma }$. Define $V=f_{\gamma }\circ f_{\beta }^{-1}$ on $D_{\beta }.$ Since ${\beta }$ and $\gamma$ are analytic curves, $V$ extends conformally to a neighborhood of $\beta$ with $V(\beta )=\gamma$. On $\beta$, this extension (still denoted $V$) satisfies $V=(\psi _{\overline{D_\gamma }}\circ h_\gamma )\circ (h_{\beta }^{-1}\circ \psi _{\overline{D_\beta }}^{-1}) = \psi _{\overline{D_\gamma }}\circ W \circ \psi _{\overline{D_\beta }}^{-1}$. Therefore $V$ extends conformally to all of $\Omega ^\beta$ and satisfies the desired property. $\square$

Theorem 2.2

Let $H$ be a nondegenerate interior hull. Let $\Omega \subset \mathbb {C}$ be a neighborhood of $\mathbb {T}$. Suppose $W$ is a conformal map defined in $\Omega$, maps $\mathbb {T}$ onto $\mathbb {T}$, and preserves the orientation of $\mathbb {T}$. Let $\Omega ^H=H\cup \psi _H(\Omega \cap \mathbb {D}^*)$. Then there is a conformal map $V$ defined in $\Omega ^H$ such that $V\circ \psi _H=\psi _{V(H)}\circ W$ in $\Omega \cap \mathbb {D}^*$. If another conformal map $\widetilde{V}$ satisfies the properties of $V$, then $\widetilde{V}=aV+b$ for some $a>0$ and $b\in \mathbb {C}$ (Fig. 1).

Proof

First, define a sequence of analytic Jordan curves $(\beta _n)$ by

$$\begin{aligned} \beta _n=\psi _H(\{e^{\frac{1}{n} +i\theta }:0\le \theta \le 2\pi \}),\quad n\in \mathbb {N}. \end{aligned}$$

Then $\beta _n\cup D_{\beta _n}\rightarrow H$ in $d_\mathcal{H}$ (see Appendix B). From Lemma 2.1, for each $n\in \mathbb {N}$, there is a conformal map $V_n$ defined in $\Omega ^{\beta _n}:=\beta _n\cup D_{\beta _n}\cup \psi _{\beta _n}(\Omega \cap \mathbb {D}^*)$ such that $V_n\circ \psi _{\beta _n}=\psi _{V_n(\beta _n)}\circ W$ in $\Omega \cap \mathbb {D}^*$. Note that for any $a_n>0$ and $b_n\in \mathbb {C}$, $a_nV_n+b_n$ satisfies the same property as $V_n$. Thus, we may assume that $0\in V_n(\beta _n)\subset \overline{\mathbb {D}}$ and $V_n(\beta _n)\cap \mathbb {T}\ne \emptyset$. Let $\gamma _n=V_n(\beta _n)$, $n\in \mathbb {N}$. Then each $\gamma _n$ is an interior hull contained in the interior hull $\overline{\mathbb {D}}$, and ${{\mathrm{diam}}}(\gamma _n)\ge 1$. So ${{\mathrm{rad}}}(\gamma _n)\ge 1/4$. From Corollary 8.2, $(\gamma _n)$ contains a subsequence which converges to some interior hull $K$ contained in $\overline{\mathbb {D}}$ with radius at least $1/4$. So $K$ is nondegenerate. By passing to a subsequence, we may assume that $\gamma _n\rightarrow K$. From $\beta _n\rightarrow H$ and $\gamma _n\rightarrow K$ we get $\psi _{\beta _n}\mathop {\longrightarrow }\limits ^\mathrm{l.u.}\psi _H$ in $\Omega \cap \mathbb {D}^*$ and $\psi _{\gamma _n}\mathop {\longrightarrow }\limits ^\mathrm{l.u.}\psi _K$ in $W(\Omega \cap \mathbb {D}^*)$. Thus, $\psi _{\beta _n}(\Omega \cap \mathbb {D}^*)\mathop {\longrightarrow }\limits ^\mathrm{Cara}\psi _H(\Omega \cap \mathbb {D}^*)$ by Lemma 7.2.

Since $V_n\circ \psi _{\beta _n}=\psi _{\gamma _n}\circ W$ in $\Omega \cap \mathbb {D}^*$, we find that $V_n=\psi _{\gamma _n}\circ W\circ \psi _{\beta _n}^{-1}$ in $\psi _{\beta _n}(\Omega \cap \mathbb {D}^*)$. Let $V=\psi _K\circ W\circ \psi _H^{-1}$ in $\psi _H(\Omega \cap \mathbb {D}^*)$. Then $V_n\mathop {\longrightarrow }\limits ^\mathrm{l.u.}V$ in $\psi _H(\Omega \cap \mathbb {D}^*)$. We may find $r>1$ such that for any $s\in (1,r]$, $s\mathbb {T}\subset \Omega \cap \mathbb {D}^*$. Then $\psi _{H}(r\mathbb {T})$ is a Jordan curve in $\psi _H(\Omega \cap \mathbb {D}^*)$ surrounding $H$, and the Jordan domain bounded by $\psi _{H}(r\mathbb {T})$ is contained in $\Omega ^H=H\cup \psi _H(\Omega \cap \mathbb {D}^*)$. Since $\psi _{H}(r\mathbb {T})$ is a compact subset of $\psi _H(\Omega \cap \mathbb {D}^*)$, we have $V_n\rightarrow V$ uniformly on $\psi _{H}(r\mathbb {T})$. It is easy to see that $\Omega ^{\beta _n}\mathop {\longrightarrow }\limits ^\mathrm{Cara}\Omega ^H$. For $n$ big enough, $\psi _{H}(r\mathbb {T})$ together with its interior is contained in $\Omega ^{\beta _n}$. From the maximum principle, $V_n$ converges uniformly in the interior of $\psi _{H}(r\mathbb {T})$ to a conformal map which extends $V$. We still use $V$ to denote the extended conformal map. Then $V$ is a conformal map defined in $\Omega ^H$, and $V_n\mathop {\longrightarrow }\limits ^\mathrm{l.u.}V$ in $\Omega ^H$. Letting $n\rightarrow \infty$ in the equality $V_n\circ \psi _{\beta _n}=\psi _{\gamma _n}\circ W$ in $\Omega \cap \mathbb {D}^*$, we conclude that $V\circ \psi _H=\psi _{V(H)}\circ W$ in $\Omega \cap \mathbb {D}^*$. So the existence part is proved.

If $\widetilde{V}=aV+b$ for some $a>0$ and $b\in \mathbb {C}$, then $\psi _{\widetilde{V}(H)}=a\psi _{V(H)}+b$, which implies $\widetilde{V}\circ \psi _H=\psi _{\widetilde{V}(H)}\circ W$. Finally, suppose $\widetilde{V}$ satisfies the properties of $V$. Then $\widetilde{V}\circ V^{-1}$ is a conformal map in $V(\Omega ^H)$. Since $V\circ \psi _H=\psi _{V(H)}\circ W$ and $\widetilde{V}\circ \psi _H=\psi _{\widetilde{V}(H)}\circ W$ in $\Omega \cap \mathbb {D}^*$, we find that $\widetilde{V}\circ V^{-1}=\psi _{\widetilde{V}(H)}\circ \psi _{V(H)}^{-1}$ in $\psi _{V(H)}(W(\Omega \cap \mathbb {D}^*))\!=\!V(\Omega ^H){\setminus } V( H)$. Note that $\psi _{\widetilde{V}(H)}\circ \psi _{V(H)}^{-1}$ is a conformal map defined in $\widehat{\mathbb {C}}{\setminus } V(H)$. Since $V(\Omega ^H)\cup (\widehat{\mathbb {C}}{\setminus } V(H))=\widehat{\mathbb {C}}$, we may define an analytic function $h$ in $\mathbb {C}$ such that $h=\widetilde{V}\circ V^{-1}$ in $V(\Omega ^H)$ and $h=\psi _{\widetilde{V}(H)}\circ \psi _{V(H)}^{-1}$ in $\mathbb {C}{\setminus } V(H)$. From the properties of $\psi _{\widetilde{V}(H)}$ and $\psi _{V(H)}$, we have $h(\infty )=\infty$ and $h'(\infty )>0$. Thus, $h(z)=az+b$ for some $a>0$ and $b\in \mathbb {C}$, which implies that $\widetilde{V}=aV+b$. $\square$

Now we obtain a new proof of the following well-known result about conformal welding.

Corollary 2.3

Let $W$ be conformal in a neighborhood of $\mathbb {T}$, maps $\mathbb {T}$ onto $\mathbb {T}$, and preserves the orientation of $\mathbb {T}$. If $h$ is a conformal welding, then $W\circ h$ and $h\circ W$ are also conformal weldings.

Proof

Apply Theorem 2.2 to $H=\overline{D_J}$, where $J$ is the Jordan curve for the conformal welding $h$. We find a conformal map $V$ defined in $\Omega ^H=D_J\cup f_J^*(\Omega \cap \mathbb {D}^*)$ such that $V\circ f_J^*=\psi _{V(H)}\circ W$ in $\Omega \cap \mathbb {D}^*$. Let $J'=V(J)$. Then $J'$ is also a Jordan curve, $V(H)=\overline{D_{J'}}$, and $\psi _{V(H)}=f_{J'}^*$. Let $f_{J'}=V\circ f_J$. Then $f_{J'}:\mathbb {D}\mathop {\twoheadrightarrow }\limits ^\mathrm{Conf}D_J$. Thus,

$$\begin{aligned} W\circ h=W\circ (f_J^*)^{-1}\circ f_J= \psi _{H(H)}^{-1}\circ V\circ f_J=(f_{J'}^*)^{-1}\circ f_{J'}, \end{aligned}$$

which implies that $W\circ h$ is a conformal welding. As for $h\circ W$, note that $(h\circ W)^{-1}=W^{-1}\circ h^{-1}$ and that $h$ is a conformal welding if and only if $h^{-1}$ is a conformal welding. $\square$

2.2 Hulls in the upper half plane

Let $\mathbb {H}=\{z\in \mathbb {C}:{{\mathrm{Im}}}z>0\}$. A subset $K$ of $\mathbb {H}$ is called an $\mathbb {H}$-hull if it is bounded, relatively closed in $\mathbb {H}$, and $\mathbb {H}{\setminus } K$ is simply connected. For every $\mathbb {H}$-hull $K$, there are a unique $c\ge 0$ and a unique $g_K:\mathbb {H}{\setminus } K\mathop {\twoheadrightarrow }\limits ^\mathrm{Conf}\mathbb {H}$ such that $g_K(z)=z+\frac{c}{z}+O(\frac{1}{z^2})$ as $z\rightarrow \infty$. The number $c$ is called the $\mathbb {H}$-capacity of $K$, and is denoted by ${{\mathrm{hcap}}}(K)$. Let $f_K=g_K^{-1}$. The empty set is an $\mathbb {H}$-hull with ${{\mathrm{hcap}}}(\emptyset )=0$ and $g_\emptyset =f_\emptyset ={{\mathrm{id}}}_{\mathbb {H}}$.

Definition 2.4

Let $K_1$ and $K_2$ be $\mathbb {H}$-hulls.

• 1. If $K_1\subset K_2$, define $K_2/K_1=g_{K_1}(K_2{\setminus } K_1)$. We call $K_2/K_1$ a quotient hull of $K_2$, and write $K_2/K_1\prec K_2$.

• 2. The product $K_1\cdot K_2$ is defined to be $K_1\cup f_{K_1}(K_2)$.

The following facts are easy to check.

• 1. $K_2/K_1$ and $K_1\cdot K_2$ in the definition are also $\mathbb {H}$-hulls.

• 2. For any two $\mathbb {H}$-hulls $K_1$ and $K_2$, $K_1\subset K_1\cdot K_2$ and $K_2=(K_1\cdot K_2)/K_1\prec K_1\cdot K_2$. If $K_1\subset K_2$, then $K_1\cdot (K_2/K_1)=K_2$.

• 3. The space of $\mathbb {H}$-hulls with the product “$\cdot$” is a semigroup with identity element $\emptyset$, and “$\prec$” is a transitive relation of this space.

• 4. $f_{K_1\cdot K_2}=f_{K_1}\circ f_{K_2}$ in $\mathbb {H}$; $g_{K_1\cdot K_2}=g_{K_2}\circ g_{K_1}$ in $\mathbb {H}{\setminus }(K_1\cdot K_2)$.

• 5. ${{\mathrm{hcap}}}(K_1\cdot K_2)={{\mathrm{hcap}}}(K_1)+{{\mathrm{hcap}}}(K_2)$. If $K_1\subset K_2$ or $K_1\prec K_2$, then ${{\mathrm{hcap}}}(K_1)\le {{\mathrm{hcap}}}(K_2)$.

From $f_{K_1\cdot K_2}=f_{K_1}\circ f_{K_2}$ in $\mathbb {H}$ we can conclude that $f_{K_1}=f_{K_1\cdot K_2}\circ g_{K_2}$ in $\mathbb {H}{\setminus } K_2$. So $f_{K_1}$ is an analytic extension of $f_{K_1\cdot K_2}\circ g_{K_2}$, which means that $K_1$ is uniquely determined by $K_1\cdot K_2$ and $K_2$. So the following definition makes sense.

Definition 2.5

Let $K_1$ and $K_2$ be $\mathbb {H}$-hulls such that $K_1\prec K_2$. We use $K_2:K_1$ to denote the unique $\mathbb {H}$-hull $K\subset K_2$ such that $K_2/K=K_1$.

For an $\mathbb {H}$-hull $K$, the base of $K$ is the set $B_K=\overline{K}\cap \mathbb {R}$. Let the double of $K$ be defined by $K^{{{\mathrm{db}}}}=K\cup I_{\mathbb {R}}(K)\cup B_K$, where $I_\mathbb {R}(z):=\overline{z}$. Then $g_K$ extends to a conformal map (still denoted by $g_K$) in $\widehat{\mathbb {C}}{\setminus } K^{{{\mathrm{db}}}}$, which satisfies $g_K(\infty )=\infty$, $g_K'(\infty )=1$, and $g_K\circ I_\mathbb {R}=I_\mathbb {R}\circ g_K$. Moreover, $g_K(\widehat{\mathbb {C}}{\setminus } K^{{{\mathrm{db}}}})=\widehat{\mathbb {C}}{\setminus } S_K$ for some compact $S_K\subset \mathbb {R}$, which is called the support of $K$. So $f_K$ extends to a conformal map from $\widehat{\mathbb {C}}{\setminus } S_K$ onto $\widehat{\mathbb {C}}{\setminus } K^{{{\mathrm{db}}}}$.

Lemma 2.6

$f_K$ can not be extended analytically beyond $\widehat{\mathbb {C}}{\setminus }S_K$.

Proof

Suppose $f_K$ can be extended analytically near $x_0\in \mathbb {R}$, then the image of $f_K$ contains a neighborhood of $f_K(x_0)\in \mathbb {R}$. So $f_K(\mathbb {H})=\mathbb {H}{\setminus } K$ contains a neighborhood of $f_K(x_0)$ in $\mathbb {H}$. This then implies that $f_K(x_0)\in \mathbb {R}{\setminus } B_K$. Thus, there is $y_0\in \mathbb {R}{\setminus } S_K$ such that $f_K(y_0)=f_K(x_0)$. Since $f_K$ is conformal in $\mathbb {H}$, we must have $x_0=y_0\in \mathbb {R}{\setminus }$ $S_K$. $\square$

Lemma 2.7

If $K_1=K_2/K_0\prec K_2$, then $S_{K_1}\subset S_{K_2}$, $f_{K_2}=f_{K_0}\circ f_{K_1}$ in $\widehat{\mathbb {C}}{\setminus } S_{K_2}$, and $g_{K_2}=g_{K_1}\circ g_{K_0}$ in $\widehat{\mathbb {C}}{\setminus } K^{{{\mathrm{db}}}}_2$.

Proof

Since $K_2=K_0\cdot K_1$, we have $f_{K_2}=f_{K_0}\circ f_{K_1}$ in $\mathbb {H}$, which implies that $g_{K_0}\circ f_{K_2}=f_{K_1}$ in $\mathbb {H}$. Since $f_{K_2}$ maps $\widehat{\mathbb {C}}{\setminus } S_{K_2}$ conformally onto $\widehat{\mathbb {C}}{\setminus } K^{{{\mathrm{db}}}}_2\subset \widehat{\mathbb {C}}{\setminus } K^{{{\mathrm{db}}}}_0$, and $g_{K_0}$ is analytic in $\widehat{\mathbb {C}}{\setminus } K^{{{\mathrm{db}}}}_2$, we see that $g_{K_0}\circ f_{K_2}$ is analytic in $\widehat{\mathbb {C}}{\setminus } S_{K_2}$. Since $g_{K_0}\circ f_{K_2}=f_{K_1}$ in $\mathbb {H}$, from Lemma 2.6 we have $S_{K_1}\subset S_{K_2}$, and $g_{K_0}\circ f_{K_2}=f_{K_1}$ in $\widehat{\mathbb {C}}{\setminus } S_{K_2}$. Composing $f_{K_0}$ to the left of both sides, we get $f_{K_2}=f_{K_0}\circ f_{K_1}$ in $\widehat{\mathbb {C}}{\setminus } S_{K_2}$. Taking inverse, we obtain the equality for $g_K$’s. $\square$

Definition 2.8

$S\subset \widehat{\mathbb {C}}$ is called $\mathbb {R}$-symmetric if $I_\mathbb {R}(S)=S$. An $\mathbb {R}$-symmetric map $W$ is a function defined in an $\mathbb {R}$-symmetric domain $\Omega$, which commutes with $I_\mathbb {R}$, and maps $\Omega \cap \mathbb {H}$ into $\mathbb {H}$.

Remarks

• 1. For any $\mathbb {H}$-hull $K$, $g_K$ and $f_K$ are $\mathbb {R}$-symmetric conformal maps.

• 2. Let $W$ be an $R$-symmetric conformal map defined in $\Omega$. If an $\mathbb {H}$-hull $K$ satisfies $K^{{{\mathrm{db}}}}\subset \Omega$ and $\infty \not \in W( K^{{{\mathrm{db}}}})$, then $W(K)$ is also an $\mathbb {H}$-hull and ${W(K)}^{{{\mathrm{db}}}}=W( K^{{{\mathrm{db}}}})$.

Definition 2.9

Let $\Omega$ be an $\mathbb {R}$-symmetric domain and $K$ be an $\mathbb {H}$-hull. If $K^{{{\mathrm{db}}}}\subset \Omega$, we write $\Omega _K$ or $(\Omega )_K$ for $S_K\cup g_K(\Omega {\setminus } K^{{{\mathrm{db}}}})$, and call it the collapse of $\Omega$ via $K$. If $S_K\subset \Omega$, we write $\Omega ^K$ or $(\Omega ^K)$ for $K^{{{\mathrm{db}}}}\cup f_K(\Omega {\setminus } S_K)$, and call it the lift of $\Omega$ via $K$.

Remarks

• 1. In the definition, $\Omega _K$ is an $\mathbb {R}$-symmetric domain containing $S_K$; $\Omega ^K$ is an $\mathbb {R}$-symmetric domain containing $K^{{{\mathrm{db}}}}$.

• 2. $(\Omega _K)^K=\Omega$ and $(\Omega ^K)_K=\Omega$ if the lefthand sides are well defined.

• 3. $\Omega _{K_1\cdot K_2}=(\Omega _{K_1})_{K_2 }$ and $\Omega ^{K_1\cdot K_2}=(\Omega ^{K_2 })^{K_1}$ if either sides are well defined.

Definition 2.10

Let $W$ be an $\mathbb {R}$-symmetric conformal map with domain $\Omega$. Let $K$ be an $\mathbb {H}$-hull such that $K^{{{\mathrm{db}}}}\subset \Omega$ and $\infty \not \in W( K^{{{\mathrm{db}}}})$. We write $W_K$ or $(W)_K$ for the conformal extension of $g_{W(K)}\circ W\circ f_K$ to $\Omega _K$, and call it the collapse of $W$ via $K$.

Remarks

• 1. Since $g_{W(K)}\circ W\circ f_K:\Omega _K{\setminus } S_K\mathop {\twoheadrightarrow }\limits ^\mathrm{Conf}W(\Omega ){\setminus } S_{W(K)}$, the existence of $W_K$ follows from the Schwarz reflection principle. $W_K$ is an $\mathbb {R}$-symmetric conformal map, and $W_K(S_K)=S_{W(K)}$.

• 2. The $g_K$ and $f_K$ defined at the beginning of this section should not be understood as the collapse of $g$ and $f$ via $K$.

• 3. $W_{K_1\cdot K_2}=(W_{K_1})_{K_2 }$ if either side is well defined.

• 4. $V_{W(K)}\circ W_K=(V\circ W)_K$ if either side is well defined. In particular, $(W^{-1})_{W(K)}=(W_K)^{-1}$.

Let $B_K^{{{\mathrm{cv}}}}$ and $S_K^{{{\mathrm{cv}}}}$ be the convex hulls of $B_K$ and $S_K$, respectively. Let $K^{{{\mathrm{db}}},{{\mathrm{cv}}}}= K^{{{\mathrm{db}}}}\cup B_K^{{{\mathrm{cv}}}}$. Then $g_K:\widehat{\mathbb {C}}{\setminus } K^{{{\mathrm{db}}},{{\mathrm{cv}}}}\mathop {\twoheadrightarrow }\limits ^\mathrm{Conf}\widehat{\mathbb {C}}{\setminus } S_K^{{{\mathrm{cv}}}}$. If $K\ne \emptyset$, then $S_K^{{{\mathrm{cv}}}}$ is a bounded closed interval, $K^{{{\mathrm{db}}},{{\mathrm{cv}}}}$ is a nondegenerate interior hull, and $\psi _{K^{{{\mathrm{db}}},{{\mathrm{cv}}}}}=f_K\circ \psi _{S_K^{{{\mathrm{cv}}}}}$. If $S_K^{{{\mathrm{cv}}}}\subset \Omega$, then $\Omega ^K=K^{{{\mathrm{db}}},{{\mathrm{cv}}}}\cup f_K(\Omega {\setminus } S_K^{{{\mathrm{cv}}}})$. The lemma below is a part of Lemma 5.3 in [19], where $S_K^{{{\mathrm{cv}}}}$ was denoted by $[c_K,d_K]$.

Lemma 2.11

If $K_1\subset K_2$, then $S_{K_1}^{{{\mathrm{cv}}}}\subset S_{K_2}^{{{\mathrm{cv}}}}$.

Theorem 2.12

Let $W$ be an $\mathbb {R}$-symmetric conformal map with domain $\Omega$. Let $K$ be an $\mathbb {H}$-hull such that $S_K\subset \Omega$ and $\infty \not \in W(S_K)$. Then there is a unique $\mathbb {R}$-symmetric conformal map $V$ defined in $\Omega ^K$ such that $V_K=W$ (Fig. 2).

Fig. 2

The situation of Theorem 2.12. Given $K$ and $W$, there is a unique $V$, also denoted $W^K$, which is analytic across $K$ and its reflection $I_{\mathbb {R}}(K)$, see Definition 2.13

Proof

We first consider the existence. If $K=\emptyset$, since $f_\emptyset ={{\mathrm{id}}}$ and $\Omega ^\emptyset =\Omega$, $V=W$ is what we need. Now suppose $K\ne \emptyset$ and $S_K^{{{\mathrm{cv}}}}\subset \Omega$. Note that $S_K^{{{\mathrm{cv}}}}$ is a bounded closed interval, and so is $W(S_K^{{{\mathrm{cv}}}})$. Let $\Omega _{\mathbb {T}}=\psi _{S_K^{{{\mathrm{cv}}}}}^{-1}(\Omega {\setminus } S_K^{{{\mathrm{cv}}}})$. Define a conformal map $W_{\mathbb {T}}$ in $\Omega _{\mathbb {T}}$ by $W_{\mathbb {T}}=\psi _{W(S_K^{{{\mathrm{cv}}}})}^{-1}\circ W\circ \psi _{S_K^{{{\mathrm{cv}}}}}$. Then $W_\mathbb {T}(z)\rightarrow \mathbb {T}$ as $\Omega _\mathbb {T}\ni z\rightarrow \mathbb {T}$. Thus, $W_\mathbb {T}$ extends conformally across $\mathbb {T}$, maps $\mathbb {T}$ onto $\mathbb {T}$, and preserves the orientation of $\mathbb {T}$. Apply Theorem 2.2 to $W_\mathbb {T}$ and $K^{{{\mathrm{db}}},{{\mathrm{cv}}}}$. We find a conformal map $\widehat{V}$ defined in

$$\begin{aligned} K^{{{\mathrm{db}}},{{\mathrm{cv}}}}\cup \psi _{K^{{{\mathrm{db}}},{{\mathrm{cv}}}}}(\Omega _\mathbb {T})=K^{{{\mathrm{db}}},{{\mathrm{cv}}}}\cup f_K(\Omega {\setminus } S_K^{{{\mathrm{cv}}}}) =\Omega ^K \end{aligned}$$

such that $\widehat{V}\circ \psi _{K^{{{\mathrm{db}}},{{\mathrm{cv}}}}}=\psi _{\widehat{V}(K^{{{\mathrm{db}}},{{\mathrm{cv}}}})}\circ W_\mathbb {T}$ in $\Omega _\mathbb {T}$. Let $\widetilde{V}=I_\mathbb {R}\circ \widehat{V}\circ I_\mathbb {R}$. Then $\widetilde{V}(K^{{{\mathrm{db}}},{{\mathrm{cv}}}})=I_\mathbb {R}\circ \widehat{V}(K^{{{\mathrm{db}}},{{\mathrm{cv}}}})$. So $\psi _{\widetilde{V}(K^{{{\mathrm{db}}},{{\mathrm{cv}}}})}=I_\mathbb {R}\circ \psi _{\widehat{V}(K^{{{\mathrm{db}}},{{\mathrm{cv}}}})}\circ I_\mathbb {R}$. Since $I_\mathbb {R}$ commutes with $\psi _{K^{{{\mathrm{db}}},{{\mathrm{cv}}}}}$ and $W_\mathbb {T}$, we see that $\widetilde{V}$ also satisfies the properties of $\widehat{V}$. So $\widetilde{V}=a\widehat{V}+b$ for some $a>0$ and $b\in \mathbb {C}$. Thus, $I_\mathbb {R}\circ \widehat{V}\circ I_\mathbb {R}=a\widehat{V}+b$. Considering the values of $\widehat{V}$ on $\Omega ^K\cap \mathbb {R}$, we find that $a=1$ and ${{\mathrm{Re}}}b=0$. Note that $\widehat{V}-\frac{b}{2}$ satisfies the property of $\widehat{V}$, and commutes with $I_\mathbb {R}$. By replacing $\widehat{V}$ with $\widehat{V}-\frac{b}{2}$, we may assume that $\widehat{V}$ is an $\mathbb {R}$-symmetric conformal map.

Since $\widehat{V}\circ \psi _{K^{{{\mathrm{db}}},{{\mathrm{cv}}}}}=\psi _{\widehat{V}(K^{{{\mathrm{db}}},{{\mathrm{cv}}}})}\circ W_\mathbb {T}$ in $\Omega _\mathbb {T}$, from $\psi _{K^{{{\mathrm{db}}},{{\mathrm{cv}}}}}=f_K\circ \psi _{S_K^{{{\mathrm{cv}}}}}$, $\psi _{\widehat{V}(K^{{{\mathrm{db}}},{{\mathrm{cv}}}})}=f_{\widehat{V}(K)}\circ \psi _{S_{\widehat{V}(K)}^{{{\mathrm{cv}}}}}$, and the definitions of $W_\mathbb {T}$ and $\Omega _\mathbb {T}$, we have

$$\begin{aligned} \widehat{V}\circ f_K=f_{\widehat{V}(K)}\circ \psi _{S_{\widehat{V}(K)}^{{{\mathrm{cv}}}}}\circ \psi _{W(S_K^{{{\mathrm{cv}}}})}^{-1}\circ W \end{aligned}$$

(2.1)

on $\Omega {\setminus } S_K^{{{\mathrm{cv}}}}$. Let $h=\psi _{S_{\widehat{V}(K)}^{{{\mathrm{cv}}}}}\circ \psi _{W(S_K^{{{\mathrm{cv}}}})}^{-1}$. Since $S_{\widehat{V}(K)}^{{{\mathrm{cv}}}}$ and $W(S_K^{{{\mathrm{cv}}}})$ are both bounded closed intervals, we have $h(z)=az+b$ for some $a>0$ and $b\in \mathbb {R}$. Let $V=h^{-1}\circ \widehat{V}$. Then $V$ is also an $\mathbb {R}$-symmetric conformal map defined on $\Omega ^K$, and $f_{V(K)}=h^{-1}\circ f_{\widehat{V}(K)}\circ h$. From (2.1) we have

$$\begin{aligned} f_{V(K)}\circ W=h^{-1}\circ f_{\widehat{V}(K)}\circ h\circ W=h^{-1}\circ \widehat{V}\circ f_K=V\circ f_K. \end{aligned}$$

This finishes the existence part in the case that $K\ne \emptyset$ and $S_K^{{{\mathrm{cv}}}}\subset \Omega$.

Now we still assume that $K\ne \emptyset$ but do not assume that $S_K^{{{\mathrm{cv}}}}\subset \Omega$. Let $\Omega _0=\Omega$ and $W_0=W$. We will construct $\mathbb {H}$-hulls $K_1,\ldots ,K_n$ and $\mathbb {R}$-symmetric domains $\Omega _1,\ldots ,\Omega _n$ such that $K_n\cdot K_{n-1}\cdots K_1=K$, $\Omega _j=\Omega _{j-1}^{K_j}$, and $S_{K_j}^{{{\mathrm{cv}}}}\subset \Omega _{j-1}$, $1\le j\le n$. When they are constructed, using the above result, we can obtain $\mathbb {R}$-symmetric conformal maps $W_j$ defined on $\Omega _j$, $1\le j\le n$, such that $(W_j)_{K_j}=W_{j-1}$, $1\le j\le n$. Let $V=W_n$. Then $V$ is defined in $\Omega _n=\Omega _0^{K_n\cdots K_1}=\Omega ^K$, and $V_K=(W_n)_{K_n\cdots K_1}=W_0=W$. So $V$ is what we need.

It remains to construct $K_j$ and $\Omega _j$ with the desired properties. Since $\Omega \cap \mathbb {R}$ is a disjoint union of open intervals, and $S_K$ is a compact subset of $\Omega \cap \mathbb {R}$, we may find finitely many components of $\Omega \cap \mathbb {R}$ which cover $S_K$. There exist mutually disjoint $\mathbb {R}$-symmetric Jordan curves $J_1,\ldots ,J_n$ in $\Omega$ such that their interiors $D_{J_1},\ldots ,D_{J_n}$ are mutually disjoint and contained in $\Omega$, and $S_K\subset \bigcup _{k=1}^n D_{J_k}$. Then $J^K_j:=f_K(J_j)$, $1\le j\le n$ are $\mathbb {R}$-symmetric Jordan curves, which together with their interiors are mutually disjoint, and $K^{{{\mathrm{db}}}}\subset \bigcup _{k=1}^n D_{f_K(J_k)}$. Let $H_j=K\cap \bigcup _{k=j}^n D_{J^K_j}$, $1\le j\le n$. Then each $H_j$ is an $\mathbb {H}$-hull, and $K=H_1\supset H_2\supset \cdots \supset H_n$. Let $K_j=H_j/H_{j+1}$, $1\le j\le n-1$, and $K_n=H_n$. Then we have $K_n\cdots K_1=H_1=K$.

Construct $\Omega _j$, $1\le j\le n$, such that $\Omega _j=\Omega _{j-1}^{K_j}$, $1\le j\le n$. Then

$$\begin{aligned} \Omega _{j-1}=(\Omega _0)^{K_{j-1}\cdots K_1}=(\Omega ^{K_n\cdots K_1})_{K_n\cdots K_j}=(\Omega ^K)_{H_j},\quad 1\le j\le n. \end{aligned}$$

It suffices to show that $S_{K_j}^{{{\mathrm{cv}}}}\subset \Omega _{j-1}$. We have

$$\begin{aligned} K_j=H_j/H_{j+1}=g_{H_{j+1}}(H_j{\setminus } H_{j+1})=g_{H_{j+1}}(K\cap D_{J^K_j}). \end{aligned}$$

Thus, $K^{{{\mathrm{db}}}}_j\subset D_{g_{H_{j+1}}(J^K_j)}$, which implies that $S_{K_j}\subset D_{g_{H_{j}}(J^K_j)}$. Since $\mathbb {R}\cap D_{g_{H_{j}}(J^K_j)}$ is an interval, we have $S_{K_j}^{{{\mathrm{cv}}}}\subset D_{g_{H_{j}}(J^K_j)}$. Since $\overline{D_{J^K_j}}\subset \Omega ^K$, and $J^K_j$ has positive distance from $H_j$, we have $D_{g_{H_{j}}(J^K_j)}\subset (\Omega ^K)_{H_j}=\Omega _{j-1}$. So $K_j$ and $\Omega _j$ satisfy the properties we need. This finishes the proof of the existence part.

Now we prove the uniqueness. Suppose $\widetilde{V}$ is another $\mathbb {R}$-symmetric conformal map defined on $\Omega ^K$ such that $\widetilde{V}_K=W$. Then

$$\begin{aligned} g_{V(K)}\circ V=W\circ g_K=g_{\widetilde{V}(K)}\circ \widetilde{V} \end{aligned}$$

on $\Omega {\setminus } K^{{{\mathrm{db}}}}$. Thus, $\widetilde{V}\circ V^{-1}=f_{\widetilde{V}(K)}\circ g_{V(K)}$ on $V(\Omega {\setminus } K^{{{\mathrm{db}}}})=V(\Omega ){\setminus } V( K^{{{\mathrm{db}}}})$. We know that $\widetilde{V}\circ V^{-1}$ is a conformal map defined on $V(\Omega )$, while $f_{\widetilde{V}(K)}\circ g_{V(K)}$ is a conformal map defined on $\widehat{\mathbb {C}}{\setminus } {V(K)}^{{{\mathrm{db}}}}=\widehat{\mathbb {C}}{\setminus } V( K^{{{\mathrm{db}}}})$. Since $V(\Omega )$ and $\widehat{\mathbb {C}}{\setminus } V( K^{{{\mathrm{db}}}})$ cover $\widehat{\mathbb {C}}$, we may define an analytic function $h$ on $\mathbb {C}$ such that $h=\widetilde{V}\circ V^{-1}$ on $V(\Omega )$ and $h=f_{\widetilde{V}(K)}\circ g_{V(K)}$ on $\widehat{\mathbb {C}}{\setminus } V( K^{{{\mathrm{db}}}})$. From the properties of $f_{\widetilde{V}(K)}$ and $g_{V(K)}$, we see that $h(z)-z\rightarrow 0$ as $z\rightarrow \infty$. So $h={{\mathrm{id}}}$, which implies that $\widetilde{V}=V$. So the uniqueness is proved. $\square$

Definition 2.13

We use $W^K$ to denote the unique $V$ in Theorem 2.12, and call it the lift of $W$ via $K$. Let $W^\mathcal{H}$ be the map defined by $W^\mathcal{H}(K)=W^K(K)$.

Remarks

• 1. $(W_K)^K=W$ and $(W^K)_K=W$.

• 2. The range of $W^K$ is $W^K(\Omega ^K)=(W(\Omega ))^{W^\mathcal{H}(K)}$.

• 3. $W^{K_1\cdot K_2}=(W^{K_2 })^{K_1}$, $V^{W^K(K)}\circ W^K=(V\circ W)^K$, and $(W^K)^{-1}=(W^{-1})^{W(K)}$.

• 4. The domain (resp. range) of $W^\mathcal{H}$ is the set of $\mathbb {H}$-hulls whose supports are contained in the domain (resp. range) of $W$; and $S_{W^\mathcal{H}(K)}=W(S_K)$.

• 5. $V^\mathcal{H}\circ W^\mathcal{H}=(V\circ W)^\mathcal{H}$; $(W^\mathcal{H})^{-1}=(W^{-1})^\mathcal{H}$.

Lemma 2.14

Suppose $K_1\prec K_2$, $S_{K_2}$ lies in the domain of an $\mathbb {R}$-symmetric conformal map $W$, and $\infty \not \in W(S_{K_2})$. Then $W^\mathcal{H}(K_1)\prec W^\mathcal{H}(K_2)$, and

$$\begin{aligned} W^\mathcal{H}(K_2):W^\mathcal{H}(K_1)=W^{K_2}(K_2:K_1). \end{aligned}$$

(2.2)

Proof

From Lemma 2.7, $S_{K_1}\subset S_{K_2}$. So $W^{K_1}$ and $W^{K_2}$ exist. Let $K_0=K_2:K_1\subset K_2$. Then $W^{K_2}(K_0)\subset W^{K_2}(K_2)$ and

$$\begin{aligned} W^{K_2}(K_2)/W^{K_2}(K_0)&= g_{W^{K_2}(K_0)}\circ W^{K_2}(K_2{\setminus } K_0)\\&= g_{W^{K_2}(K_0)}\circ W^{K_2}\circ f_{K_0}(K_2/K_0)=(W^{K_2})_{K_0}(K_2/K_0)\\&= (W^{K_0\cdot K_1})_{K_0}(K_1)=W^{K_1}(K_1). \end{aligned}$$

Thus, $W^{K_1}(K_1)\prec W^{K_2}(K_2)$ and $W^{K_2}(K_2):W^{K_1}(K_1)=W^{K_2}(K_0)$. $\square$

Definition 2.15

Let $\mathcal{P}^*$ denote the set of pair of $\mathbb {H}$-hulls $(H_1,H_2)$ such that $H^{{{\mathrm{db}}}}_1\cap H^{{{\mathrm{db}}}}_2=\emptyset$. Let $\mathcal{P}_*$ denote the set of pair of $\mathbb {H}$-hulls $(K_1,K_2)$ such that $S_{K_1}\cap S_{K_2}=\emptyset$. Define $g_\mathcal{P}$ on $\mathcal{P}^*$ by $g_\mathcal{P}(H_1,H_2)=(g_{H_2}(H_1),g_{H_1}(H_2))$. Define $f^\mathcal{P}$ on $\mathcal{P}_*$ by $f^\mathcal{P}(K_1,K_2)=(f_{K_2}^\mathcal{H}(K_1),f_{K_1}^\mathcal{H}(K_2))$ (Fig. 3).

Fig. 3

The pair $(H_1,H_2)$ uniquely determines the pair $(K_1,K_2),$ and vice versa, see Definition 2.15 and Theorem 2.16

Remarks

• 1. $g_\mathcal{P}$ is well defined on $\mathcal{P}^*$ because for $j=1,2$, $K^{{{\mathrm{db}}}}_{3-j}$ is contained in the domain of $g_{K_j}$: $\widehat{\mathbb {C}}{\setminus } K^{{{\mathrm{db}}}}_j$. The value of $g_\mathcal{P}$ is a pair of $\mathbb {H}$-hulls.

• 2. $f^\mathcal{P}$ is well defined on $\mathcal{P}_*$ because for $j=1,2$, $S_{K_{3-j}}$ is contained in the domain of $f_{K_j}$: $\widehat{\mathbb {C}}{\setminus } S_{K_j}$. The value of $f^\mathcal{P}$ is a pair of $\mathbb {H}$-hulls.

Theorem 2.16

$g_\mathcal{P}$ and $f^\mathcal{P}$ are bijections between $\mathcal{P}^*$ and $\mathcal{P}_*$, and are inverse of each other. Moreover, if $(H_1,H_2)=f^\mathcal{P}(K_1,K_2)$, then

• (i) $H_1\cdot K_2=H_2\cdot K_1=H_1\cup H_2$;

• (ii) $f_{K_2}(S_{K_1})=S_{H_1}$ and $f_{K_1}(S_{K_2})=S_{H_2}$;

• (iii) $S_{H_1\cup H_2}=S_{K_1}\cup S_{K_2}$.

Proof

Let $(H_1,H_2)\in \mathcal{P}^*$ and $(K_1,K_2)=g_\mathcal{P}(H_1,H_2)$. Then $(\widehat{\mathbb {C}}{\setminus } H^{{{\mathrm{db}}}}_1)_{H_2}=\widehat{\mathbb {C}}{\setminus } K^{{{\mathrm{db}}}}_1$, $S_{H_1}\subset \widehat{\mathbb {C}}{\setminus } K^{{{\mathrm{db}}}}_2$, and $(\widehat{\mathbb {C}}{\setminus } S_{H_1})_{K_2}=\widehat{\mathbb {C}}{\setminus } g_{K_2}(S_{H_1})$. Since $g_{H_1}:\widehat{\mathbb {C}}{\setminus } H^{{{\mathrm{db}}}}_1\mathop {\twoheadrightarrow }\limits ^\mathrm{Conf}\widehat{\mathbb {C}}{\setminus } S_{H_1}$ and $g_{H_1}(H_2)=K_2$, we get $(g_{H_1})_{H_2}:\widehat{\mathbb {C}}{\setminus } K^{{{\mathrm{db}}}}_1\mathop {\twoheadrightarrow }\limits ^\mathrm{Conf}\widehat{\mathbb {C}}{\setminus } g_{K_2}(S_{H_1})$. From the normalization of $g_{H_1},g_{H_2},g_{K_2}$ at $\infty$, we conclude that

$$\begin{aligned} (g_{H_1})_{H_2}=g_{K_1},\quad g_{K_2}(S_{H_1})=S_{K_1}. \end{aligned}$$

(2.3)

From $S_{H_1}\subset \widehat{\mathbb {C}}{\setminus } K^{{{\mathrm{db}}}}_2$ and $g_{K_2}(S_{H_1})=S_{K_1}$, we see that $S_{K_1}\cap S_{K_2}=\emptyset$, i.e., $(K_1,K_2)\in \mathcal{P}_*$. Since $f_{H_1}=g_{H_1}^{-1}$, $f_{K_1}=g_{K_1}^{-1}$, and $g_{H_1}(H_2)=K_2$, from (2.3) we get $(f_{H_1})_{K_2}=f_{K_1}$, which implies that $(f_{K_1})^{K_2}=f_{H_1}$. Thus, $f_{K_1}^\mathcal{H}(K_2)=f_{H_1}(K_2)=H_2$. Similarly, $f_{K_2}^\mathcal{H}(K_1)=H_1$. Thus, $f^\mathcal{P}(K_1,K_2)=(H_1,H_2)$. So $f^\mathcal{P}\circ g_\mathcal{P}={{\mathrm{id}}}_{\mathcal{P}^*}$.

Let $(K_1,K_2)\in \mathcal{P}_*$ and $H_1=f_{K_2}^\mathcal{H}(K_1)$. Then $S_{H_1}=f_{K_2}(S_{K_1})$ is disjoint from $K^{{{\mathrm{db}}}}_2$. Thus, we may define another $\mathbb {H}$-hull $H_2:=f_{H_1}(K_2)$. Then $H^{{{\mathrm{db}}}}_2\subset \widehat{\mathbb {C}}{\setminus } H^{{{\mathrm{db}}}}_1$. So $(H_1,H_2)\in \mathcal{P}^*$. We have $(\widehat{\mathbb {C}}{\setminus } S_{K_2})^{K_1}=\widehat{\mathbb {C}}{\setminus } f_{K_1}(S_{K_2})$ and $(\widehat{\mathbb {C}}{\setminus } K^{{{\mathrm{db}}}}_2)^{H_1}=\widehat{\mathbb {C}}{\setminus } H^{{{\mathrm{db}}}}_2$. Since $f_{K_2}:\widehat{\mathbb {C}}{\setminus } S_{K_2}\mathop {\twoheadrightarrow }\limits ^\mathrm{Conf}\widehat{\mathbb {C}}{\setminus } K^{{{\mathrm{db}}}}_2$ and $f_{K_2}^\mathcal{H}(K_1)=H_1$, we see that $(f_{K_2})^{K_1}:\widehat{\mathbb {C}}{\setminus } f_{K_1}(S_{K_2})\mathop {\twoheadrightarrow }\limits ^\mathrm{Conf}\widehat{\mathbb {C}}{\setminus } H^{{{\mathrm{db}}}}_2$. From the normalization of $f_{K_1},f_{H_1},f_{K_2}$ at $\infty$, we conclude that

$$\begin{aligned} (f_{K_2})^{K_1}=f_{H_2},\quad f_{K_1}(S_{K_2})=S_{H_2}. \end{aligned}$$

(2.4)

Since $H_1=f_{K_2}^\mathcal{H}(K_1)$, we get $f_{K_2}=g_{H_1}\circ f_{H_2}\circ f_{K_1}$ on $(\widehat{\mathbb {C}}{\setminus } S_{K_2}){\setminus } S_{K_1}$, which implies that $f_{H_1}\circ f_{K_2}= f_{H_2}\circ f_{K_1}$ on $\widehat{\mathbb {C}}{\setminus } (S_{K_1}\cup S_{K_2})$. So

$$\begin{aligned} H_2\cdot K_1=H_1\cdot K_2=H_1\cup f_{H_1}(K_2)=H_1\cup H_2. \end{aligned}$$

(2.5)

Thus, $K_1=g_{H_2}(H_1)$ and $K_2=g_{H_1}(H_2)$, i.e., $(K_1,K_2)=g_\mathcal{P}(H_1,H_2)$. This shows that the range of $g_\mathcal{P}$ is $\mathcal{P}_*$, which combining with $f^\mathcal{P}\circ g_\mathcal{P}={{\mathrm{id}}}_{\mathcal{P}^*}$ shows that $f^\mathcal{P}=(g_\mathcal{P})^{-1}$ and $g_\mathcal{P}=(f^\mathcal{P})^{-1}$.

In the previous paragraph, since $(K_1,K_2)=g_\mathcal{P}(H_1,H_2)$, $f^\mathcal{P}(K_1,K_2)=(H_1,H_2)$. Thus, (i) follows from (2.5); the second parts of (ii) follow from (2.4), and the first part follows from symmetry. Finally, since $g_{K_2}\circ g_{H_1}=g_{H_1\cdot K_2}=g_{H_1\cup H_2}$, from $g_{H_1}:\widehat{\mathbb {C}}{\setminus } ( H^{{{\mathrm{db}}}}_1\cup H^{{{\mathrm{db}}}}_2)\mathop {\twoheadrightarrow }\limits ^\mathrm{Conf}\widehat{\mathbb {C}}{\setminus }(S_{H_1}\cup K^{{{\mathrm{db}}}}_2)$, $g_{K_2}:\widehat{\mathbb {C}}{\setminus }(S_{H_1}\cup K^{{{\mathrm{db}}}}_2)\mathop {\twoheadrightarrow }\limits ^\mathrm{Conf}\widehat{\mathbb {C}}{\setminus } (g_{K_2}(S_{H_1})\cup S_{K_2})$, and (2.3), we get (iii). $\square$

Definition 2.17

For $(K_1,K_2)\in \mathcal{P}_*$, we define the quotient union of $K_1$ and $K_2$ to be $K_1\vee K_2=H_1\cup H_2$, where $(H_1,H_2)=f^\mathcal{P}(K_1,K_2)$.

Remark   From Theorem 2.16, $K_1,K_2\prec K_1\vee K_2$ and $S_{K_1\vee K_2}=S_{K_1}\cup S_{K_2}$.

The space of $\mathbb {H}$-hulls has a natural metric $d_\mathcal{H}$ described in Appendix C. Let $\mathcal{H}_S$ denote the set of $\mathbb {H}$-hulls whose supports are contained in $S$. From Lemma 9.2, if $F$ is compact, $(\mathcal{H}_F,d_\mathcal{H})$ is compact, and $H_n\rightarrow H$ in $\mathcal{H}_F$ implies that $f_{H_n}\mathop {\longrightarrow }\limits ^\mathrm{l.u.}f_H$ in $\mathbb {C}{\setminus } F$.

Theorem 2.18

• (i) Let $F\subset \mathbb {R}$ be compact. Let $W$ be an $\mathbb {R}$-symmetric conformal map whose domain contains $F$. Then $W^\mathcal{H}:\mathcal{H}_F\rightarrow \mathcal{H}_{W(F)}$ is continuous.

• (ii) Let $E$ and $F$ be two nonempty compact subsets of $\mathbb {R}$ with $E\cap F=\emptyset$. Then $f^\mathcal{P}$ and $(K_1,K_2)\mapsto K_1\vee K_2$ are continuous on $\mathcal{H}_{E}\times \mathcal{H}_{F}$.

Proof

(i) First, $W^\mathcal{H}$ is well defined on $\mathcal{H}_F$, and the range of $W^\mathcal{H}$ is $\mathcal{H}_{W(F)}$. Suppose $(H_n)$ is a sequence in $\mathcal{H}_F$ and $H_n\rightarrow H_0\in \mathcal{H}_F$. To prove the continuity of $W^\mathcal{H}$, we need to show that $W^\mathcal{H}(H_n)\rightarrow W^\mathcal{H}(H_0)$. Suppose this is not true. Since $\mathcal{H}_{W(F)}$ is compact, by passing to a subsequence, we may assume that $W^\mathcal{H}(H_n)\rightarrow K_0 \ne W^\mathcal{H}(H_0)$. For each $n_k$, $W^{H_{n_k}}=f_{W^\mathcal{H}(H_{n_k})}\circ W\circ g_{H_{n_k}}$ on $f_{H_{n_k}}(\Omega {\setminus } F)$. We have $g_{H_{n_k}}\mathop {\longrightarrow }\limits ^\mathrm{l.u.}g_{H_0}$ in $f_{H_{0}}(\Omega {\setminus } F)$ and $f_{W^\mathcal{H}(H_{n_k})}\mathop {\longrightarrow }\limits ^\mathrm{l.u.}f_{K_0}$ in $W(\Omega ){\setminus } W(F)$. Thus, $W^{H_{n_k}}\mathop {\longrightarrow }\limits ^\mathrm{l.u.}f_{K_0}\circ W\circ g_{H_0}=:V$ in $f_{H_{0}}(\Omega {\setminus } F)$. The domain of $W^{H_{n_k}}$ is $\Omega ^{H_{n_k}}= H^{{{\mathrm{db}}}}_{n_k}\cup f_{H_{n_k}}(\Omega {\setminus } S_{H_{n_k}})$, which converges to $\Omega ^{H_{0}}= H^{{{\mathrm{db}}}}_{0}\cup f_{H_{0}}(\Omega {\setminus } S_{H_{0}})\supset f_{H_{0}}(\Omega {\setminus } F)$. It is clear that $\Omega ^{H_0}{\setminus } f_{H_0}(\Omega {\setminus } F)$ is compact. Since $W^{H_{n_k}}\mathop {\longrightarrow }\limits ^\mathrm{l.u.}V$ in $f_{H_{0}}(\Omega {\setminus } F)$, from the maximum principle, $W^{H_{n_k}}$ converges locally uniformly in $\Omega ^{H_0}$. We still let $V$ denote the limit function. Since $H_{n_k}\rightarrow H_0$ and $W^{H_{n_k}}(H_{n_k})\rightarrow K_0$, we have $V(H_0)=K_0$. Since $f_{K_0}\circ W\circ g_{H_0}=V$ in $f_{H_{0}}(\Omega {\setminus } F)$, we see that $f_{V(H_0)}\circ W\circ g_{H_0}=V$ in $f_{H_0}(\Omega {\setminus } S_{H_0})$. Thus, $V=W^{H_0}$. So $K_0=W^{H_0}(H_0)=W^\mathcal{H}(H_0)$. This is the contradiction we need.

(ii) To show $f^\mathcal{P}$ is continuous, it suffices to show that, if $(K_1^{n},K_2^{n})$ is a sequence in $\mathcal{H}_{E}\times \mathcal{H}_{F}$ which converges to $(K_1^{0},K_2^{0})\in \mathcal{H}_{E}\times \mathcal{H}_{F}$, then it has a subsequence $(K_1^{(n_k)},K_2^{(n_k)})$ such that $f^\mathcal{P}(K_1^{(n_k)},K_2^{(n_k)})\rightarrow f^\mathcal{P}(K_1^{0},K_2^{0})$. Let $(H_1^{n},H_2^{n})=f^\mathcal{P}(K_1^{n},K_2^{n})$, $n\in \mathbb {N}$. From Theorem 2.16 (iii), $S_{H_1^{n}\cup H_2^{n}}=S_{K_1^{n}}\cup S_{K_2^{n}}\subset E\cup F$. From Lemma 9.2, $(H_1^{n}\cup H_2^{n})$ has a convergent subsequence with limit in $\mathcal{H}_{E\cup F}$. From Lemma 2.11, $S_{H_1^{n}}\subset S_{H_1^{n}\cup H_2^{n}}^{{{\mathrm{cv}}}}\subset A$, where $A$ is the convex hull of $E\cup F$. From Lemma 9.2, $(H_1^{n})$ has a convergent subsequence. For the same reason, $(H_2^{n})$ also has a convergent subsequence. By passing to subsequences, we may assume that $H_1^{n}\cup H_2^{n}\rightarrow M^{0}\in \mathcal{H}_{E\cup F}$ and $H_j^{n}\rightarrow H_j^{0}$, $j=1,2$.

From Theorem 2.16 (i) and the continuity of the dot product, we get $H_1^{0}\cdot K_2^{0}=H_2^{0}\cdot K_1^{0}=M^{0}$. This implies that $M^{0}=H_1^{0}\cup f_{H_1^{0}}(K_2^{0})$. The measures $(\mu _{H_1^{n}})$ (see Appendix C) converges to $\mu _{H_1^{0}}$ weakly. Each $\mu _{H_1^{n}}$ is supported by $S_{H_1^{n}}$. From Theorem 2.16 (ii), $S_{H_1^{n}}\!=\!f_{{K_2^{n}}}(S_{K_1^{n}})\subset f_{K_2^{n}}(E)$. Since $E$ is a compact subset of $\mathbb {C}{\setminus } F$, we have $f_{K_2^{n}}\rightarrow f_{K_2^{0}}$ uniformly on $E$. Thus, $f_{K_2^{n}}(E)\rightarrow f_{K_2^{0}}(E)$ in the Hausdorff metric. So $\mu _{H_1^{0}}$ is supported by $f_{K_2^{0}}(E)$, which implies that $S_{H_1^{0}}\subset f_{K_2^{0}}(E)$. Hence $f_{H_1^{0}}(K_2^{0})$ is another $\mathbb {H}$-hull, which is bounded away from $H_1^{0}$. From $K_2^{n}\rightarrow K_2^{0}$ we have $\mathbb {H}{\setminus } K_2^{n}\mathop {\longrightarrow }\limits ^\mathrm{Cara}\mathbb {H}{\setminus } K_2^{0}$. From (9.1) we get $f_{H_1^{n}}\mathop {\longrightarrow }\limits ^\mathrm{l.u.}f_{H_1^{0}}$ in $\mathbb {C}{\setminus } S_{H_1^{0}}$. Thus, $\mathbb {H}{\setminus } f_{H_1^{n}}(K_2^{n})\mathop {\longrightarrow }\limits ^\mathrm{Cara}\mathbb {H}{\setminus } f_{H_1^{0}}(K_2^{0})$. Since $H_2^{n}=f_{H_1^{n}}(K_2^{n})$, we have $\mathbb {H}{\setminus } H_2^{n}\mathop {\longrightarrow }\limits ^\mathrm{Cara}\mathbb {H}{\setminus } f_{H_1^{0}}(K_2^{0})$. On the other hand, $\mathbb {H}{\setminus } H_2^{n}\mathop {\longrightarrow }\limits ^\mathrm{Cara}\mathbb {H}{\setminus } H_2^{0}$. Since $\mathbb {H}{\setminus } H_2^{0}$ and $\mathbb {H}{\setminus } f_{H_1^{0}}(K_2^{0})$ both contain a neighborhood of $\infty$ in $\mathbb {H}$, they must be the same domain. Thus, $H_2^{0}=f_{H_1^{0}}(K_2^{0})$ is bounded away from $H_1^{0}$, i.e., $(H_1^{n},H_2^{n})\in \mathcal{P}^*$. For the same reason, $H_1^{0}=f_{H_2^{0}}(K_1^{0})$. Thus, $(H_1^{n},H_2^{n})\rightarrow (H_1^{0},H_2^{0})=f^\mathcal{P}(K_1^{0},K_2^{0})$. This shows that $f^\mathcal{P}$ is continuous. Finally, since $K_1\vee K_2=H_1\cdot K_2$ if $(H_1,H_2)=f^\mathcal{P}(K_1,K_2)$, we see that $(K_1,K_2)\mapsto K_1\vee K_2$ is also continuous. $\square$

Corollary 2.19

• (i) Let $W$ be an $\mathbb {R}$-symmetric conformal map with domain $\Omega$. Then $W^\mathcal{H}$ is measurable on $\mathcal{H}_{\Omega \cap \mathbb {R}}$.

• (ii) $f^\mathcal{P}$ and $(K_1,K_2)\mapsto K_1\vee K_2$ are measurable on $\mathcal{P}_*$.

Proof

(i) We may find an increasing sequence of compact subsets $(F_n)$ of $\Omega \cap \mathbb {R}$ such that $\mathcal{H}_{\Omega \cap \mathbb {R}}=\bigcup _{n=1}^\infty \mathcal{H}_{F_n}$. From Theorem 2.18 (i), $W^\mathcal{H}$ is continuous on each $\mathcal{H}_{F_n}$. Thus, $W^\mathcal{H}$ is measurable on $\mathcal{H}_{\Omega \cap \mathbb {R}}$.

(ii) We may find a sequence of pairs of disjoint bounded closed intervals of $\mathbb {R}$: $(E_n,F_n)$, $n\in \mathbb {N}$, such that $\mathcal{P}_*=\bigcup _{n=1}^\infty \mathcal{H}_{E_n}\times \mathcal{H}_{F_n}$. From Theorem 2.18 (ii), $f^\mathcal{P}$ and $(K_1,K_2)\mapsto K_1\vee K_2$ are continuous on each $\mathcal{H}_{E_n}\times \mathcal{H}_{F_n}$, and so they are measurable on $\mathcal{P}_*$. $\square$

2.3 Hulls in the unit disc

A subset $K$ of $\mathbb {D}=\{|z|<1\}$ is called a $\mathbb {D}$-hull if $\mathbb {D}{\setminus } K$ is a simply connected domain containing $0$. For every $\mathbb {D}$-hull $K$, there is a unique $g_K:\mathbb {D}{\setminus } K\mathop {\twoheadrightarrow }\limits ^\mathrm{Conf}\mathbb {D}$ such that $g_K(0)=0$ and $g_K'(0)>0$. Then $\ln g_K'(0)\ge 0$ is called the $\mathbb {D}$-capacity of $K$, and is denoted by ${{\mathrm{dcap}}}(K)$. Let $f_K=g_K^{-1}$.

We may define $K_1\cdot K_2$, $K_2/K_1$ (when $K_1\subset K_2$), and $K_1\prec K_2$ on the space of $\mathbb {D}$-hulls as in Definition 2.4. Then the remarks after Definition 2.4 still hold if $\mathbb {H}$ is replaced by $\mathbb {D}$ and ${{\mathrm{hcap}}}$ is replaced by ${{\mathrm{dcap}}}$. Then we may define $K_2:K_1$ (when $K_1\prec K_2$) as in Definition 2.5. For a $\mathbb {D}$-hull $K$, the base $B_K$ of $K$ is $\overline{K}\cap \mathbb {T}$, and the double of $K$ is $K^{{{\mathrm{db}}}}=K\cup I_\mathbb {T}(K)\cup B_K$, where $I_\mathbb {T}(z):=1/\overline{z}$. Then $g_K$ extends to a conformal map (still denoted by $g_K$) on $\widehat{\mathbb {C}}{\setminus } K^{{{\mathrm{db}}}}$, which commutes with $I_\mathbb {T}$. Moreover, $g_K(\widehat{\mathbb {C}}{\setminus } K^{{{\mathrm{db}}}})=\widehat{\mathbb {C}}{\setminus } S_K$ for some compact $S_K\subset \mathbb {T}$, which is called the support of $K$. So $f_K$ extends to a conformal map from $\widehat{\mathbb {C}}{\setminus } S_K$ onto $\widehat{\mathbb {C}}{\setminus } K^{{{\mathrm{db}}}}$, which commutes with $I_\mathbb {T}$. Then Lemmas 2.6 and 2.7 still hold here.

We may define $\mathbb {T}$-symmetric sets and $\mathbb {T}$-symmetric conformal maps using Definition 2.8 with $\mathbb {R}$ and $\mathbb {H}$ replaced by $\mathbb {T}$ and $\mathbb {D}$, respectively. For a $\mathbb {T}$-symmetric domain $\Omega$ and a $\mathbb {D}$-hull $K$, we may define domains $\Omega _K$ (when $K^{{{\mathrm{db}}}}\subset \Omega$) and $\Omega ^K$ (when $S_K\subset \Omega$) using Definition 2.9. If $W$ is a $\mathbb {T}$-symmetric conformal map with domain $\Omega$, and if $\Omega _K$ is defined, we may then define $W_K$ using Definition 2.10, which is a $\mathbb {T}$-symmetric conformal map on $\Omega _K$. The remarks after Definitions 2.8, 2.9, and 2.10 hold here with minor modifications. We claim that Theorem 2.12 holds here with modifications. We need several lemmas.

The theorem below relates the $\mathbb {H}$-hulls with $\mathbb {D}$-hulls. To distinguish the two set of symbols, we use $f^{\mathbb {H}}_K$, $g^\mathbb {H}_K$, $B_K^\mathbb {R}$, $S_K^\mathbb {R}$, and $K^{\mathbb {R}{{\mathrm{db}}}}$ for $\mathbb {H}$-hulls, and $f^{\mathbb {D}}_K$, $g^\mathbb {D}_K$, $B_K^\mathbb {T}$, $S_K^\mathbb {T}$, and $K^{\mathbb {T}{{\mathrm{db}}}}$ for $\mathbb {D}$-hulls.

Theorem 2.20

• (i) Let $W$ be a Möbius transformation that maps $\mathbb {D}$ onto $\mathbb {H}$, and $K$ be a $\mathbb {D}$-hull such that $W^{-1}(\infty )\not \in S_K^\mathbb {T}$. Then there is a unique Möbius transformation $W^K$ that maps $\mathbb {D}$ onto $\mathbb {H}$ such that $W^K(K)$ is an $\mathbb {H}$-hull, $g^\mathbb {H}_{W^K(K)}\circ W^K\circ f^\mathbb {D}_K=W$ in $\widehat{\mathbb {C}}{\setminus } S_K^\mathbb {T}$, and $S^\mathbb {R}_{W^K(K)}=W(S^\mathbb {T}_K)$.

• (ii) Let $W$ be a Möbius transformation that maps $\mathbb {H}$ onto $\mathbb {D}$, and $K$ be an $\mathbb {H}$-hull. Then there is a unique Möbius transformation $W^K$ that maps $\mathbb {H}$ onto $\mathbb {D}$ such that $W^K(K)$ is a $\mathbb {D}$-hull, $g^\mathbb {D}_{W^K(K)}\circ W^K\circ f^\mathbb {H}_K=W$ in $\widehat{\mathbb {C}}{\setminus } S_K^\mathbb {R}$, and $S^\mathbb {T}_{W^K(K)}=W(S^\mathbb {R}_K)$.

Proof

(i) Let $z_0=W^{-1}(\infty )\in \mathbb {T}{\setminus } S_K^\mathbb {T}$. Then $w_0:=f_K^\mathbb {D}(z_0)\in \mathbb {T}{\setminus } B_K^\mathbb {T}$ is well defined. Let $W^K_0(z)=i\frac{w_0+z}{w_0-z}$. Then $W^K_0$ is a Möbius transformation that maps $\mathbb {D}$ onto $\mathbb {H}$ and takes $w_0$ to $\infty$. Let $L_0=W^K_0(K)$. Since $w_0$ is bounded away from $K$, we see that $L_0$ is an $\mathbb {H}$-hull. We have $W^K_0:\widehat{\mathbb {C}}{\setminus } K^{\mathbb {T}{{\mathrm{db}}}}\mathop {\twoheadrightarrow }\limits ^\mathrm{Conf}\widehat{\mathbb {C}}{\setminus } L_0^{\mathbb {R}{{\mathrm{db}}}}$. Define $G=g^\mathbb {H}_{L_0}\circ W^K_0\circ f^\mathbb {D}_K\circ W^{-1}$ on $\widehat{\mathbb {C}}{\setminus } W(S_K^\mathbb {T})$. Then $G:\widehat{\mathbb {C}}{\setminus } W(S_K^\mathbb {T})\mathop {\twoheadrightarrow }\limits ^\mathrm{Conf}\widehat{\mathbb {C}}{\setminus } S_{L_0}^\mathbb {R}$, fixes $\infty$, and maps $\mathbb {H}$ onto $\mathbb {H}$. So $G(z)=az+b$ for some $a>0$ and $b\in \mathbb {R}$. Let $W^K=G^{-1}\circ W^K_0$. Then $W^K$ is also a Möbius transformation that maps $\mathbb {D}$ onto $\mathbb {H}$, and $W^K(K)$ is also an $\mathbb {H}$-hull with $S_{W^K(K)}^\mathbb {R}=G^{-1}(S_{L_0}^\mathbb {R})=W(S_K^\mathbb {T})$ and $g^\mathbb {H}_{W^K(K)}=G^{-1}\circ g^\mathbb {H}_{L_0}\circ G$. Thus,

$$\begin{aligned} g^\mathbb {H}_{W^K(K)}\circ W^K\circ f^\mathbb {D}_K\circ W^{-1}&= G^{-1}\circ g^\mathbb {H}_{L_0}\circ G\circ G^{-1}\circ W^K_0\circ f^\mathbb {D}_K\circ W^{-1}\\&= G^{-1}\circ g^\mathbb {H}_{L_0}\circ W^K_0\circ f^\mathbb {D}_K\circ W^{-1}=G^{-1}\circ G\\ {}&= {{\mathrm{id}}}_{\widehat{\mathbb {C}}{\setminus } W(S_K)}. \end{aligned}$$

This implies that $g^\mathbb {H}_{L}\circ W^K\circ f^\mathbb {D}_K=W$ in $\widehat{\mathbb {C}} {\setminus } S_K$. So we proved the existence. On the other hand, if $W^K$ satisfies the desired property, then from $W^K=f^\mathbb {H}_{L}\circ W\circ g^\mathbb {D}_K$ we get $W^K(w_0)=\infty$. So $W^K=G_0\circ W^K_0$, where $G_0(z)=az+b$ for some $a>0$ and $b\in \mathbb {R}$. The above argument shows that $G_0=G^{-1}$. So we get the uniqueness.

(ii) We may use the proof of (i) with slight modifications: replace $\infty$ by $0$, swap $\mathbb {H}$ and $\mathbb {D}$, swap $\mathbb {R}$ and $\mathbb {T}$, and define $W^K_0(z)=\frac{z-w_0}{z-\overline{w_0}}$. $\square$

We also use $W^\mathcal{H}(K)$ to denote the hull $W^K(K)$ in the above lemma. The following lemma is similar to Lemma 2.14.

Lemma 2.21

Let $K_1$ and $K_2$ be two $\mathbb {H}$-(resp. $\mathbb {D}$-)hulls such that $K_1\prec K_2$. Let $W$ be a Möbius transformation that maps $\mathbb {H}$ onto $\mathbb {D}$ (resp. maps $\mathbb {D}$ onto $\mathbb {H}$) such that $\infty \not \in W(S_{K_2})$. Then $W^\mathcal{H}(K_1)\prec W^\mathcal{H}(K_2)$ and (2.2) still holds.

The following lemma is used to treat the case $S_K=\mathbb {T}$ in Theorem 2.23.

Lemma 2.22

Let $W$ be a $\mathbb {T}$-symmetric conformal map with domain $\Omega \supset \mathbb {T}$. Let $(K_n)$ be a sequence of $\mathbb {D}$-hulls which converges to $K$. Suppose that for each $n$, there is a $\mathbb {T}$-symmetric conformal map $V^{\langle n\rangle }$ defined on $\Omega ^{K_n}$ such that $V^{\langle n\rangle }_{K_n}=W$. Then there is a $\mathbb {T}$-symmetric conformal map $V$ defined on $\Omega ^{K}$ such that $V_K=W$. Moreover, $V(K)$ is a subsequential limit of $(V^{\langle n\rangle }(K_n))$.

Proof

Since $K_n\rightarrow K$, $\Omega ^{K_n}\mathop {\longrightarrow }\limits ^\mathrm{Cara}\Omega ^K$. Since $V^{\langle n\rangle }$ maps $\Omega ^{K_n}\cap \mathbb {D}$ into $\mathbb {D}$, the family $(V^{\langle n\rangle }|_{\Omega ^{K_n}\cap \mathbb {D}})$ is uniformly bounded. Thus, $(V^{\langle n\rangle })$ contains a subsequence, which convergence locally uniformly in $\Omega ^K\cap \mathbb {D}$. To save the symbols, we assume that $(V^{\langle n\rangle })$ itself converges locally uniformly in $\Omega ^K\cap \mathbb {D}$. Since each $V^{\langle n\rangle }$ is $\mathbb {T}$-symmetric, the sequence also converges locally uniformly in $\Omega ^K\cap \mathbb {D}^*$. From the maximum principle, $(V^{\langle n\rangle })$ converges locally uniformly in $\Omega ^K$. Let $V$ be the limit function. Since each $V^{\langle n\rangle }$ maps $\mathbb {T}$ onto $\mathbb {T}$, and $V^{\langle n\rangle }\rightarrow V$ uniformly on $\mathbb {T}$, $V$ can not be constant. From Lemma 7.2, $V$ is a conformal map. It is $\mathbb {T}$-symmetric because each $V^{\langle n\rangle }$ is $\mathbb {T}$-symmetric. Since $K_n\rightarrow K$, we have $V^{\langle n\rangle }(K_n)\rightarrow V(K)$. From $V^{\langle n\rangle }_{K_n}=W$ we have $g_{V^{\langle n\rangle }(K_n)}\circ V^{\langle n\rangle }\circ f_{K_n}=W$ in $\Omega {\setminus } \mathbb {T}$. Letting $n\rightarrow \infty$ we get $g_{V(K)}\circ V\circ f_K=W$ in $\Omega {\setminus }\mathbb {T}$. By continuation, this equality also holds on $\Omega {\setminus } S_K$. Thus, $V_K=W$. $\square$

Theorem 2.23

Let $W$ be a $\mathbb {T}$-symmetric conformal map with domain $\Omega$. Let $K$ be a $\mathbb {D}$-hull such that $S_K\subset \Omega$. Then there is a unique $\mathbb {T}$-symmetric conformal map $V$ defined on $\Omega ^K$ such that $V_K=W$.

Proof

We first consider the existence. Case 1. $S_K^\mathbb {T}\ne \mathbb {T}$. We will apply Theorems 2.12 and 2.20 for this case. Pick $z_0\in \mathbb {T}{\setminus } S_K^\mathbb {T}$ and let $h(z)=i\frac{z_0+z}{z_0-z}$. From Theorem 2.20 (i), there is a Möbius transformation $h^K$ that maps $\mathbb {D}$ onto $\mathbb {H}$ such that $L:=h^K(K)$ is an $\mathbb {H}$-hull, and $g^\mathbb {H}_{L}\circ h^K\circ f^\mathbb {D}_K=h$ in $\widehat{\mathbb {C}}{\setminus } S_K^\mathbb {T}$. Since $W$ is a homeomorphism on $S_K$, $W(S_K)\ne \mathbb {T}$. So there is $z_W\in \mathbb {T}{\setminus } W(S_K)$. Let $h_W(z)=z_W\cdot \frac{z-i}{z+i}$. Then $h_W$ is a Möbius transformation that maps $\mathbb {H}$ onto $\mathbb {D}$ and takes $\infty$ to $z_W$. Let $\widetilde{W}=h_W^{-1}\circ W\circ h^{-1}$. Then $\widetilde{W}$ is an $\mathbb {R}$-symmetric conformal map with domain $h(\Omega )$, and $\widetilde{W}(S^\mathbb {R}_L)=h_W^{-1}\circ W(S_K^\mathbb {T}) \not \ni \infty$. From Theorem 2.12, there is an $\mathbb {R}$-symmetric conformal map $\widetilde{V}$ with domain $L^{\mathbb {R}{{\mathrm{db}}}}\cup f_L^\mathbb {R}(h(\Omega ){\setminus } S_L^\mathbb {R})$ such that $L^*:=\widetilde{V}(L)$ is an $\mathbb {H}$-hull, and $\widetilde{V}=f^\mathbb {H}_{L^*}\circ \widetilde{W}\circ g^\mathbb {H}_L$ in $\widehat{\mathbb {C}}{\setminus } L^{\mathbb {R}{{\mathrm{db}}}}$. From Theorem 2.20 (ii), there is a Möbius transformation $h_W^{L^*}$ that maps $\mathbb {H}$ onto $\mathbb {D}$ such that $K^*:=h_W^{L^*}(L^*)$ is a $\mathbb {D}$-hull, and $g^\mathbb {D}_{K^*}\circ h_W^{L^*}\circ f^\mathbb {H}_{L^*}=h_W$ in $\widehat{\mathbb {C}}{\setminus } S_{L^*}^\mathbb {R}$. Finally, let $V=h_W^{L^*}\circ \widetilde{V}\circ h^K$. Then

$$\begin{aligned} V(K)=h_W^{L^*}\circ \widetilde{V}(L)=h_W^{L^*}(L^*)=K^*, \end{aligned}$$

and

$$\begin{aligned} g_{K^*}\circ V\circ f_K&= g_{K^*}\circ h_W^{L^*}\circ \widetilde{V}\circ h^K\circ f_K\\&= g_{K^*}\circ h_W^{L^*}\circ f^\mathbb {H}_{L^*}\circ \widetilde{W}\circ g^\mathbb {H}_L\circ h^K\circ f_K\\&= h_W\circ \widetilde{W}\circ h=W \end{aligned}$$

in $\widehat{\mathbb {C}}{\setminus } K^{\mathbb {T}{{\mathrm{db}}}}$. This finishes the existence part for Case 1.

Case 2. $S_K=\mathbb {T}$. First, we may approximate $K$ using $\mathbb {D}$-hulls bounded by $\mathbb {T}$ and a Jordan curve in $\mathbb {D}$. For example, let $J_n=f_{K}(\{|z|=1-1/(2n)\})$, and let $K_n=\mathbb {D}{\setminus } D_{J_n}$. Then each $K_n$ is a $\mathbb {D}$-hull, and $K_n\rightarrow K$. Second, if $K'$ has the form of $\mathbb {D}{\setminus } D_J$ for some Jordan curve $J$, then we may define a curve $\beta$, which starts from $\beta (0)=z_0\in \mathbb {T}$, then follows a simple curve in $\mathbb {D}\cap D_J^*$ to a point on $J$, and then follows $J$ in the clockwise direction, and ends when it finishes one round. Suppose the domain of $\beta$ is $[0,1]$. Then $\beta$ is simple on $[0,1-\varepsilon ]$ for any $\varepsilon >0$. Let $K_n=\beta ((0,1-1/n])$, $n\in \mathbb {N}$. Then each $K_n$ is a $\mathbb {D}$-hull with $S_{K_n}\ne \mathbb {T}$, and $K_n\rightarrow K'$. Thus, $K$ can be approximated by a sequence of $\mathbb {D}$-hulls $(K_n)$ such that $S_{K_n}\ne \mathbb {T}$ for each $K_n$. Then the existence of $V$ follows from Case 1 and Lemma 2.22.

Now we prove the uniqueness. Suppose $\widetilde{V}$ is another $\mathbb {T}$-symmetric conformal map defined on $\Omega ^K$ such that $\widetilde{V}_K=W$. We may use the argument in the proof of Theorem 2.12 to construct an analytic function $h$ on $\mathbb {C}$ such that $h=\widetilde{V}\circ V^{-1}$ on $V(\Omega )$ and $h=f_{\widetilde{V}(K)}\circ g_{V(K)}$ on $\mathbb {C}{\setminus } V( K^{{{\mathrm{db}}}})$. Then $h$ is $\mathbb {T}$-symmetric. From the properties of $f_{\widetilde{V}(K)}$ and $g_{V(K)}$, we see that $h(0)=0$ and $h'(0)>0$. So $h={{\mathrm{id}}}$, which implies that $\widetilde{V}=V$. $\square$

We may then define $W^K$ and $W^\mathcal{H}$ using Definition 2.13 with Theorem 2.23 in place of Theorem 2.12 and $\mathbb {D}$ in place of $\mathbb {H}$. The remarks after Definition 2.13 hold here with minor modifications, and so does Lemma 2.14. Then we define $\mathcal{P}^*$, $\mathcal{P}_*$, $g_\mathcal{P}$, and $f^\mathcal{P}$ using Definition 2.15 with $\mathbb {H}$ replaced by $\mathbb {D}$. Then Theorem 2.16 still holds here, and we may define the quotient union $K_1\vee K_2$ for $(K_1,K_2)\in \mathcal{P}_*$.

The space of $\mathbb {D}$-hulls has a natural metric $d_\mathcal{H}$ described in Appendix D. Let $\mathcal{H}_S$ denote the set of $\mathbb {D}$-hulls whose supports are contained in $S$. We claim that Theorem 2.18 still holds here if every $\mathbb {R}$ is replaced by $\mathbb {T}$. For part (i), if $F\ne \mathbb {T}$, then the proof of Theorem 2.18 (i) still goes through with Lemma 10.2 in place of Lemma 9.2; if $F=\mathbb {T}$, then the continuity of $W^\mathcal{H}$ follows from Lemma 2.22. For part (ii), the proof of Theorem 2.18 still goes through with some modifications. The relatively compactness of $(H_n\cup J_n)$ follows from Lemma 10.2 instead of Lemma 9.2 because $S_{H_n\cup J_n}\subset E\cup F\subsetneqq \mathbb {T}$. To show the relatively compactness of $(H_n)$ and $(J_n)$, instead of applying Lemma 2.11, we now apply Lemma 10.1, and use the relatively compactness of $(H_n\cup J_n)$ and the inequalities ${{\mathrm{dcap}}}(H_n),{{\mathrm{dcap}}}(J_n)\le {{\mathrm{dcap}}}(H_n\cup J_n)$. In addition, (10.2) will be used in place of (9.1). This finishes the proof of Theorem 2.18 in the radial case. Then Corollary 2.19 in the radial case immediately follows.

The proof of Theorem 2.18 (i) may also be used to show that the map $K\mapsto W^K(K)$ in Theorem 2.20 (i) (resp. (ii)) is continuous if restricted to $\mathcal{H}_F^\mathbb {D}$ (resp. $\mathcal{H}_F^\mathbb {H}$), where $F$ is a compact subset of $\mathbb {T}{\setminus } W^{-1}(\infty )$ (resp. $\mathbb {R}$). We then can conclude that the maps $K\mapsto W^K(K)$ in Theorem 2.20 (i) and (ii) are both measurable.

3 Loewner equations and Loewner chains

3.1 Forward Loewner equations

We review the definitions and basic facts about (forward) Loewner equations. The reader is referred to [7] for details. Let $\lambda \in C([0,T))$, where $T\in (0,\infty ]$. The chordal Loewner equation driven by $\lambda$ is

$$\begin{aligned} \partial _t g_t(z)=\frac{2}{g_t(z)-\lambda (t)}, \quad g_0(z)=z. \end{aligned}$$

We assume that $g_t(\infty )=\infty$ for $0\le t<\infty$. For $z\in \mathbb {C}$, suppose that the maximal interval for $t\mapsto g_t(z)$ is $[0,\tau _z)$. Let $K_t=\{z\in \mathbb {H}:\tau _z\le t\}$, i.e., the set of $z\in \mathbb {H}$ such that $g_t(z)$ is not defined. Then $g_t$ and $K_t$, $0\le t<T$, are called the chordal Loewner maps and hulls driven by $\lambda$. It is known that each $K_t$ is an $\mathbb {H}$-hull with ${{\mathrm{hcap}}}(K_t)=2t$, and for $0<t<T$, $g_t=g_{K_t}$ with exactly the same domain: $\widehat{\mathbb {C}}{\setminus } K^{{{\mathrm{db}}}}_t$. At $t=0$, $K_0=\emptyset$ and $g_0={{\mathrm{id}}}_{\widehat{\mathbb {C}}{\setminus }\{\lambda (0)\}}$.

We say that $\lambda$ generates a chordal trace $\beta$ if

$$\begin{aligned} \beta (t):=\lim _{\mathbb {H}\ni z\rightarrow \lambda (t)} g_t^{-1}(z)\in \overline{\mathbb {H}} \end{aligned}$$

exists for $0\le t<T$, and $\beta$ is a continuous curve. We call such $\beta$ the chordal trace driven by $\lambda$. If the chordal trace $\beta$ exists, then for each $t$, $\mathbb {H}{\setminus } K_t$ is the unbounded component of $\mathbb {H}{\setminus } \beta ((0,t])$, and $f_t$ extends continuously from $\mathbb {H}$ to $\mathbb {H}\cup \mathbb {R}$. The trace $\beta$ is called simple if it is a simple curve and $\beta (t)\in \mathbb {H}$ for $0<t<T$, in which case $K_t=\beta ((0,t])$ for $0\le t<T$.

The radial Loewner equation driven by $\lambda$ is

$$\begin{aligned} \partial _t g_t(z)=g_t(z)\frac{e^{i\lambda (t)}+g_t(z)}{e^{i\lambda (t)}-g_t(z)},\quad 0\le t<T;\quad g_0(z)=z. \end{aligned}$$

We assume that $g_t(\infty )=\infty$ for $0\le t<\infty$. For each $t\in [0,T)$, let $K_t$ be the set of $z\in \mathbb {D}:=\{|z|<1\}$ at which $g_t$ is not defined. Then $g_t$ and $K_t$, $0\le t<T$, are called the radial Loewner maps and hulls driven by $\lambda$. It is known that, each $K_t$ is a $\mathbb {D}$-hull with ${{\mathrm{dcap}}}(K_t)=t$, and for $0<t<T$, $g_t=g_{K_t}$ with exactly the same domain: $\widehat{\mathbb {C}}{\setminus } K^{{{\mathrm{db}}}}_t$. At $t=0$, $K_0=\emptyset$ and $g_0={{\mathrm{id}}}_{\widehat{\mathbb {C}}{\setminus } \{e^{i\lambda (0)}\}}$.

We say that $\lambda$ generates a radial trace $\beta$ if

$$\begin{aligned} \beta (t):=\lim _{\mathbb {D}\ni z\rightarrow e^{i\lambda (t)}} g_t^{-1}(z)\in \overline{\mathbb {D}}\end{aligned}$$

exists for $0\le t<T$, and $\beta$ is a continuous curve. We call such $\beta$ the radial trace driven by $\lambda$. If the radial trace $\beta$ exists, then for each $t$, $\mathbb {D}{\setminus } K_t$ is the component of $\mathbb {D}{\setminus } \beta ((0,t])$ that contains $0$. The trace $\beta$ is called simple if it is a simple curve and $\beta (t)\in \mathbb {D}$ for $0<t<T$, in which case $K_t=\beta ((0,t])$ for $0\le t<T$.

Let $\cot _2(z)=\cot (z/2)$. The covering radial Loewner equation driven by $\lambda$ is

$$\begin{aligned} \partial _t \widetilde{g}_t(z)=\cot _2(\widetilde{g}_t(z)-\lambda (t)),\quad 0\le t<T,\quad \widetilde{g}_0(z)=z. \end{aligned}$$

For each $t\in [0,T)$, let $\widetilde{K}_t$ be the set of all $z\in \mathbb {H}$ at which $\widetilde{g}_t$ is not defined. Then $\widetilde{g}_t$ and $\widetilde{K}_t$, $0\le t<T$, are called the covering radial Loewner maps and hulls driven by $\lambda$. We have $\widetilde{g}_t:\mathbb {H}{\setminus }\widetilde{K}_t\mathop {\twoheadrightarrow }\limits ^\mathrm{Conf}\mathbb {H}$. If $g_t$ and $K_t$, $0\le t<T$, are the radial Loewner maps and hulls driven by $\lambda$, then $\widetilde{K}_t=(e^i)^{-1}(K_t)$ and $e^i\circ \widetilde{g}_t=g_t\circ e^i$, where $e^i$ denotes the map $z\mapsto e^{iz}$.

For $\kappa >0$, chordal (resp. radial) SLE$_\kappa$ is defined by solving the chordal (resp. radial) Loewner equation with $\lambda (t)=\sqrt{\kappa }B(t)$. Such driving function a.s. generates a chordal (resp. radial) trace, which is simple if $\kappa \in (0,4]$.

3.2 Backward Loewner equations

Let $\lambda \in C([0,T))$. The backward chordal Loewner equation driven by $\lambda$ is

$$\begin{aligned} \partial _t f_t(z)=\frac{-2}{f_t(z)-\lambda (t)}, \quad f_0(z)=z. \end{aligned}$$

(3.1)

We assume that $f_t(\infty )=\infty$ for $0\le t<T$. Let $L_t=\mathbb {H}{\setminus } f_t(\mathbb {H})$. We call $f_t$ and $L_t$, $0\le t<T$, the backward chordal Loewner maps and hulls driven by $\lambda$.

Define a family of maps $f_{t_2,t_1}$, $t_1,t_2\in [0,T)$, such that, for any fixed $t_1\in [0,T)$ and $z\in \widehat{\mathbb {C}}{\setminus }\{\lambda (t_1)\}$, the function $t_2\mapsto f_{t_2,t_1}(z)$ is the maximal solution of the ODE

$$\begin{aligned} \partial _{t_2} f_{t_2,t_1}(z)=\frac{-2}{f_{t_2,t_1}(z)-\lambda (t_2)},\quad f_{t_1,t_1}(z)=z. \end{aligned}$$

Note that $f_{t,0}=f_{t}$ and $f_{t,t}={{\mathrm{id}}}_{\widehat{\mathbb {C}}{\setminus }\{\lambda (t)\}}$, $0\le t<T$. If $t_1\in (0,T)$, then $t_2$ could be bigger or smaller than $t_1$. Some simple observations give the following lemma.

Lemma 3.1

• (i) For any $t_1,t_2,t_3\in [0,T)$, $f_{t_3,t_2}\circ f_{t_2,t_1}$ is a restriction of $f_{t_3,t_1}$. In particular, this implies that $f_{t_1,t_2}=f_{t_2,t_1}^{-1}$.

• (ii) For any fixed $t_0\in [0,T)$, $f_{t_0+t,t_0}$, $0\le t<T-t_0$, are the backward chordal Loewner maps driven by $\lambda (t_0+t)$, $0\le t<T-t_0$.

• (iii) For any fixed $t_0\in [0,T)$, $f_{t_0-t,t_0}$, $0\le t\le t_0$, are the (forward) chordal Loewner maps driven by $\lambda (t_0-t)$, $0\le t\le t_0$.

Let $L_{t_2,t_1}=\mathbb {H}{\setminus } f_{t_2,t_1}(\mathbb {H})$ for $0\le t_1\le t_2<T$. From (i), (iii), and the properties of forward chordal Loewner maps, we see that, if $0\le t_1< t_2<T$, then $L_{t_2,t_1}$ is an $\mathbb {H}$-hull with ${{\mathrm{hcap}}}(L_{t_2,t_1})=2(t_2-t_1)$, and $f_{t_2,t_1}=f_{L_{t_2,t_1}}$. If $t_1=t_2$, this is almost still true except that $f_{t_1,t_1}={{\mathrm{id}}}_{\widehat{\mathbb {C}}{\setminus }\{\lambda (t_1)\}}$ and $f_{L_{t_1,t_1}}=f_\emptyset ={{\mathrm{id}}}_{\widehat{\mathbb {C}}}$. Since $L_{t,0}=L_t$, and $\lambda (t)\in \mathbb {R}$ does not lie in the range of $f_t$, which is $\widehat{\mathbb {C}}{\setminus } L^{{{\mathrm{db}}}}_t$ for $t>0$, we get the following lemma.

Lemma 3.2

For $0\le t<T$, $L_t$ is an $\mathbb {H}$-hull with ${{\mathrm{hcap}}}(L_t)=2t$. If $t\in (0,T)$, then $f_t=f_{L_t}$ with the same domain: $\widehat{\mathbb {C}}{\setminus } S_{L_t}$, and $\lambda (t)\in B_{L_t}$.

If $t_2\ge t_1\ge t_0$, from $f_{t_2,t_1}\circ f_{t_1,t_0}=f_{t_2,t_0}$ we get $L_{t_2,t_0}=L_{t_2,t_1}\cdot L_{t_1,t_0}$. From Lemmas 2.7 and 3.1, we obtain the following lemma.

Lemma 3.3

For any $0\le t_1<t_2<T$, $L_{t_1}\prec L_{t_2}$ and $S_{L_{t_1}}\subset S_{L_2(t_2)}$. For any fixed $t_0\in [0,T)$, the family $L_{t_0}:L_{t_0-t}=L_{t_0,t_0-t}$, $0\le t\le t_0$, are the chordal Loewner hulls driven by $\lambda (t_0-t)$, $0\le t\le t_0$.

Note that $S_{L_0}=S_\emptyset =\emptyset$, and its is easy to see that, for $0<t_0<T$, $S_{L_{t_0}}$ is the set of $x\in \mathbb {R}$ such that the solution $f_t(x)$ to (3.1) blows up before or at $t_0$, i.e., $S_{L_{t_0}}=\{x\in \mathbb {R}:\tau _x\le t_0\}$. So every $S_{L_t}$, $0<t<T$, is a real interval, and $\bigcap _{0<t<T} S_{L_t}=\{\lambda (0)\}$.

If for every $t_0\in [0,T)$, $\lambda (t_0-t)$, $0\le t\le t_0$, generates a (forward) chordal trace, which we denote by $\beta _{t_0}(t_0-t)$, $0\le t\le t_0$, then we say that $\lambda$ generates backward chordal traces $\beta _{t_0}$, $0\le t_0<T$. If this happens, then for any $0\le t_1\le t_2<T$, $\mathbb {H}{\setminus } L_{t_2,t_1}$ is the unbounded component of $\mathbb {H}{\setminus } \beta _{t_2}([t_1,t_2))$, and $f_{t_2,t_1}$ extends continuously from $\mathbb {H}$ to $\overline{\mathbb {H}}$ such that

$$\begin{aligned} f_{t_2,t_1}(\lambda (t_1))=\beta _{t_2}(t_1),\quad 0\le t_1\le t_2<T. \end{aligned}$$

(3.2)

Here we still use $f_{t_2,t_1}$ to denote the continuation if there is no confusion. For $0\le t_0\le t_1\le t_2<T$, the equality $f_{t_2,t_0}=f_{t_2,t_1}\circ f_{t_1,t_0}$ still holds after continuation, which together with (3.2) implies that

$$\begin{aligned} f_{t_2,t_1}(\beta _{t_1}(t))=\beta _{t_2}(t),\quad 0\le t\le t_1\le t_2<T. \end{aligned}$$

(3.3)

Remark One should keep in mind that each $\beta _t$ is a continuous function defined on $[0,t]$, $\beta _t(0)$ is the tip of $\beta _t$, and $\beta _t(t)$ is the root of $\beta _t$, which lies on $\mathbb {R}$. The parametrization is different from a forward chordal trace $\beta$, of which $\beta (0)$ is the root.

The backward radial Loewner equations and the backward covering radial Loewner equation driven by $\lambda \in C([0,T))$ are the following two equations respectively:

$$\begin{aligned} \partial _t f_t(z)&= -f_t(z)\frac{e^{i\lambda (t)}+f_t(z)}{e^{i\lambda (t)}-f_t(z)},\quad f_0(z)=z;\\ \partial _{t} \widetilde{f}_{t}(z)&= -\cot _2({\widetilde{f}_{t}(z)-\lambda (t)}),\quad \widetilde{f}_{0}(z)=z. \end{aligned}$$

We have $f_t\circ e^i=e^i\circ \widetilde{f}_t$. Let $L_t=\mathbb {D}{\setminus } f_t(\mathbb {D})$. We call $f_t$ and $L_t$, $0\le t<T$, the backward radial Loewner maps and hulls driven by $\lambda$, and call $\widetilde{f}_t$, $0\le t<T$, the backward covering radial Loewner maps driven by $\lambda$.

By introducing $f_{t_2,t_1}$ in the radial setting, we find that Lemma 3.1 holds if the word “chordal” is replaced by “radial”. The following lemma is similar to Lemma 3.2.

Lemma 3.4

For $0\le t<T$, $L_t$ is a $\mathbb {D}$-hull with ${{\mathrm{dcap}}}(L_t)=t$. If $t\in (0,T)$, then $f_t=f_{L_t}$ with the same domain: $\widehat{\mathbb {C}}{\setminus } S_{L_t}$, and $e^{i\lambda (t)}\in B_{L_t}$.

We find that Lemma 3.3 holds here if the word “chordal” is replaced by “radial”. So we may define backward radial traces $\beta _t$, $0\le t<T$, in a similar manner.

The following lemma holds only in the radial case.

Lemma 3.5

If $T=\infty$, then $\mathbb {T}{\setminus } \bigcup _{0<t<\infty } S_{L_t}$ contains at most one point.

Proof

Let $S_\infty =\bigcup _{0<t<\infty } S_{L_t}$. From Koebe’s $1/4$ theorem, as $t\rightarrow \infty$, ${{\mathrm{dist}}}(0,L_t)\rightarrow 0$, which implies that the harmonic measure of $\mathbb {T}{\setminus } B_{L_t}$ in $\mathbb {D}{\setminus } L_t$ seen from $0$ tends to $0$. Since $f_t:\mathbb {D}\mathop {\twoheadrightarrow }\limits ^\mathrm{Conf}\mathbb {D}{\setminus } L_t$, $f_t(0)=0$, and $f_t(\mathbb {T}{\setminus } S_{L_t})=\mathbb {T}{\setminus } B_{L_t}$, the above harmonic measure at time $t$ equals to $|\mathbb {T}{\setminus } S_{L_t}|/(2\pi )$. Thus, $|\mathbb {T}{\setminus } S_\infty |=\lim _{t\rightarrow \infty }|\mathbb {T}{\setminus } S_{L_t}|= 0$. $\square$

For $\kappa >0$, the backward chordal (resp. radial) SLE$_\kappa$ is defined by solving backward chordal (resp. radial) Loewner equation with $\lambda (t)=\sqrt{\kappa }B(t)$, $0\le t<\infty$. Since for any fixed $t_0>0$, $(\lambda (t_0-t)-\lambda (t_0),0\le t\le t_0)$ has the distribution of $(\sqrt{\kappa }B(t),0\le t\le t_0)$, using the existence of forward chordal (resp. radial) SLE$_\kappa$ traces, we conclude that $\lambda$ a.s. generates a family of backward chordal (resp. radial) traces.

3.3 Normalized global backward trace

First we consider a backward chordal Loewner process generated by $\lambda (t)$, $0\le t<T$. Let $S_t=S_{L_t}$, $0\le t<T$, and $S_T=\bigcup _{0\le t<T} S_t$. Then $(S_t)$ is an increasing family, and $S_T$ is an interval. The following Lemma is similar in spirit to Proposition 5.1 in [18].

Lemma 3.6

There exists a family of conformal maps $F_{T,t}$, $0\le t<T$, on $\mathbb {H}$ such that $F_{T,t_1}=F_{T, t_2}\circ f_{t_2,t_1}$ in $\mathbb {H}$ if $0\le t_1\le t_2<T$. Let $D_t=F_{T,t}(\mathbb {H})$, $0\le t<T$, and $D_T=\bigcup _{t<T} D_t$. If $(\widehat{F}_{T,t})$ satisfies the same property as $(F_{T,t})$, then there is a conformal map $h_T$ defined on $D_T$ such that $\widehat{F}_{T,t}=h_T\circ F_{T,t}$, $0\le t<T$. If there is $z_0\in \mathbb {H}$ such that

$$\begin{aligned} \lim _{t\rightarrow T} \frac{{{\mathrm{Im}}}f_t(z_0)}{|f_t'(z_0)|}=\infty , \end{aligned}$$

(3.4)

then we may construct $(F_{T,t})$ such that $D_T=\mathbb {C}$, and we have $S_T=\mathbb {R}$.

Proof

Fix $z_0\in \mathbb {H}$. Let $z_{t}=f_{t}(z_0)$ and $u_t=f_t'(z_0)$, $0\le t<T$. For $t\in [0,T)$, let $M_t(z)=\frac{z-z_t}{u_t}$ and $F_t=M_t\circ f_t$. Then $F_t$ maps $z_0$ to $0$ and has derivative $1$ at $z_0$. For $0\le t_1\le t_2<T$, define $F_{t_2,t_1}=M_{t_2}\circ f_{t_2,t_1}$. Then $F_{t_2,t_1}\circ f_{t_1,t_0}=F_{t_2,t_0}$ if $t_0\le t_1\le t_2$. Setting $t_0=0$ we get $F_{t_2,t_1}\circ f_{t_1}=F_{t_2}$. Thus, $F_{t_2,t_1}$ is a conformal map on $\mathbb {H}$ with $F_{t_2,t_1}(z_{t_1})=0$ and $F_{t_2,t_1}'(z_{t_1})=1/{u_{t_1}}$. By Koebe’s distortion theorem, for any $t_1\in [0,T)$, $\{F_{t_2,t_1}:t_2\in [t_1,T)\}$ is uniformly bounded on each compact subset of $\mathbb {H}$. This implies that every sequence in this family contains a subsequence which converges locally uniformly, and the limit function is also conformal on $\mathbb {H}$, maps $z_{t_1}$ to $0$, and has derivative $1/{u_{t_1}}$ at $z_{t_1}$.

From a diagonal argument, we can find a sequence $(t_n)$ in $[0,T)$ such that $t_n\rightarrow T$ and for any $q\in \mathbb {Q}\cap [0,T)$, $(F_{t_n,q})$ converges locally uniformly on $\mathbb {H}$. Let $F_{T,q}$, $q\in \mathbb {Q}\cap [0,T)$, denote the limit functions, which are conformal on $\mathbb {H}$. Since $F_{t_n,q_2}\circ f_{q_2,q_1}=F_{t_n,q_1}$ for each $n$, we have $F_{T,q_2}\circ f_{q_2,q_1}=F_{T,q_1}$. For $t\in [0,T)$, choose $q\in \mathbb {Q}\cap [t,T)$ and define the conformal map $F_{T,t}=F_{T,q}\circ f_{q,t}$ on $\mathbb {H}$. If $q_1\le q_2\in \mathbb {Q}\cap [t,T)$, then $F_{T,q_1}\circ f_{q_1,t}=F_{T,q_2}\circ f_{q_2,q_1}\circ f_{q_1,t}=F_{T,q_2}\circ f_{q_2,t}$. Thus, the definition of $F_{T,t}$ does not depend on the choice of $q$. If $0\le t_1\le t_2<T$, by choosing $q\in \mathbb {Q}\cap [0,T)$ with $q\ge t_1\vee t_2$, we get $F_{T,t_2}\circ f_{t_2,t_1}=F_{T,q}\circ f_{q,t_2}\circ f_{t_2,t_1}=F_{T,q}\circ f_{q,t_1}=F_{T,t_1}$.

If (3.4) holds, then we start the construction of $(F_{T,t})$ with such $z_0$. Since $F_{T,t}:(\mathbb {H};z_t)\mathop {\twoheadrightarrow }\limits ^\mathrm{Conf}(D_t;0)$ and $F_{T,t}'(z_t)=1/u_t$, Koebe’s $1/4$ theorem implies that ${{\mathrm{dist}}}(0,\partial D_t)\ge \frac{1}{4}{{{\mathrm{Im}}}z_t}/{|u_t|}=\frac{1}{4}\frac{{{\mathrm{Im}}}f_t(z_0)}{|f_t'(z_0)|}$, which tends to $\infty$ as $t\rightarrow T$. So $D_T$ has to be $\mathbb {C}$.

Suppose $\widehat{F}_{T,t}$, $0\le t<T$, satisfies the same property as $F_{T,t}$, $0\le t<T$. Let $h_t=\widehat{F}_{T,t}\circ F_{T,t}^{-1}$, $0\le t<T$. Then each $h_t$ is a conformal map defined on $D_t$. If $0\le t_1<t_2<T$, then

$$\begin{aligned} h_{t_1}\circ F_{T,t_1}=\widehat{F}_{T,t_1}(z)=\widehat{F}_{T,t_2}\circ f_{t_2,t_1}=h_{t_2}\circ F_{T,t_2}\circ f_{t_2,t_1}=h_{t_2}\circ F_{T,t_1} \end{aligned}$$

in $\mathbb {H}$, which implies that $h_{t_1}=h_{t_2}|_{D_{t_1}}$. So we may define a conformal map $h_T$ on $D_T$ such that $h_t=h_T|_{D_t}$ for $0\le t<T$. Such $h_T$ is what we need.

Suppose that (3.4) holds but $S_T\ne \mathbb {R}$. Since $S_T$ is an interval, $\overline{S_T}\ne \mathbb {R}$. Choose $\widehat{z}_0\in \mathbb {R}{\setminus } \overline{S_T}$, and start the construction with $\widehat{z}_0$ in place of $z_0$ at the beginning of this proof. Let $\widehat{F}_{T,t}$, $0\le t<T$, denote the family of maps constructed in this way. Then each $\widehat{F}_{T,t}$ is an $\mathbb {R}$-symmetric conformal map, which implies that $\widehat{D}_T\subset \mathbb {H}$. However, now $D_T=\mathbb {C}$ and $h_T:D_T\mathop {\twoheadrightarrow }\limits ^\mathrm{Conf}\widehat{D}_T$, which is impossible. Thus, $S_T= \mathbb {R}$ when (3.4) holds. $\square$

Let $(F_{T,t})$, $D_t$, and $D_T$ be as in Lemma 3.6. Let $F_T=F_{T,0}$. Suppose $\lambda$ generates backward chordal traces $\beta _{t_0}$, $0\le t_0<T$, which satisfy

$$\begin{aligned} \forall t_0\in [0,T),\quad \exists t_1\in (t_0,T),\quad \beta _{t_1}([0,t_0])\subset \mathbb {H}. \end{aligned}$$

(3.5)

We may define $\beta (t)$, $0\le t<T$, as follows. For every $t\in [0,T)$, pick $t_0\in (t,T)$ such that $\beta _{t_0}(t)\in \mathbb {H}$, which is possible by (3.5), and define

$$\begin{aligned} \beta (t)=F_{T,t_0}\beta _{t_0}(t)\in D_{t_0}\subset D_T. \end{aligned}$$

(3.6)

Since $F_{T,t_1}=F_{T, t_2}\circ f_{t_2,t_1}$ in $\mathbb {H}$, from (3.3) we see that the definition of $\beta$ does not depend on the choice of $t_0$. Let $t_0\in [0,T)$. From (3.5), there is $t_1>t_0$ such that $\beta _{t_1}([0,t])\in \mathbb {H}$. Since $\beta (t)=F_{T,t_0}(\beta _{t_0}(t))$, $0\le t\le t_0$, we see that $\beta$ is continuous on $[0,t_0]$. Thus, $\beta (t)$, $0\le t<T$, is a continuous curve in $D_T$.

Fix any $x\in S_T$. Then $x\in S_{t_0}$ for some $t_0\in (0,T)$. So $f_{t_0}(x)$ lies on the outer boundary of $L_{t_0}$, which implies that $f_{t_0}(x)\in \beta _{t_0}(t)$ for some $t\in [0,t_0]$. From (3.5), there is $t_1\in (t_0,T)$ such that $\beta _{t_1}([0,t_0])\subset \mathbb {H}$. Then $f_{t_1}(x)=f_{t_1,t_0}(\beta _{t_0}(t))= \beta _{t_1}(t)\in \mathbb {H}$. From the continuity of $f_{t_1}$ on $\mathbb {H}\cup \mathbb {R}$, there is a neighborhood $U$ of $x$ in $\mathbb {H}\cup \mathbb {R}$ such that $f_{t_1}(U)\subset \mathbb {H}$. This shows that $U\cap \mathbb {R}\subset S_{t_1}\subset S_T$. Since $F_T=F_{T,t_1}\circ f_{t_1}$ in $\mathbb {H}$, we find that $F_T$ has continuation on $U$. Since $x\in S_T$ is arbitrary, we conclude that $S_T$ is an open interval, and $F_T$ has continuation to $\mathbb {H}\cup S_T$.

Now we assume that $\lambda$ generates backward chordal traces, and both (3.4) and (3.5) hold. Then $D_T=\mathbb {C}$, $S_T=\mathbb {R}$, a continuous curve $\beta (t)$, $0\le t<T$, is well defined, and $F_T$ extends continuously to $\mathbb {H}\cup \mathbb {R}$. Moreover, $F_T$ is unique up to a map $z\mapsto az+b$ for some $a,b\in \mathbb {C}$, $a\ne 0$. With some suitable normalization condition, the family $F_{T,t}$ and the curve $\beta$ will be determined by $\lambda$. We will use the following normalization:

$$\begin{aligned} F_T(\lambda (0))=\lambda (0),\quad F_T(\lambda (0)+i)=\lambda (0)+i. \end{aligned}$$

(3.7)

If (3.7) holds, we call $\beta$ the normalized global backward chordal trace generated by $\lambda$. From (3.7) we see that $\beta (0)=\lambda (0)$, and $\beta$ does not pass through $\lambda (0)+i$.

For the radial case, Lemma 3.6 still holds with $\mathbb {H}$ replaced by $\mathbb {D}$, and (3.4) replaced by $T=\infty$. For the construction, we choose $z_0=0$ and let $F_{t_2,t_1}(z)=e^{t_2} f_{t_2,t_1}(z)$. If $\lambda$ generates backward radial traces $\beta _t$, $0\le t<T$, which satisfy

$$\begin{aligned} \forall t_1\in [0,T),\quad \exists t_2\in (t_1,T),\quad \beta _{t_2}(t_1)\in \mathbb {D}, \end{aligned}$$

(3.8)

then we may define a continuous curve $\beta (t)$, $0\le t<T$, in $D_T$ using (3.6). If $T=\infty$, then $D_T=\mathbb {C}$, and such $\beta$ is determined by $\lambda$ up to a map $z\mapsto az+b$ for some $a,b\in \mathbb {C}$, $a\ne 0$, which means that we may define a normalized global backward radial trace once a normalization condition is fixed.

3.4 Forward and backward Loewner chains

In this section, we review a condition on a family of hulls that corresponds to continuously driven (forward) Loewner hulls, and discuss the corresponding condition for backward Loewner chains.

Let $D\subset \widehat{\mathbb {C}}$ be a simply connected domain such that $\widehat{\mathbb {C}}{\setminus } D$ contains more than one point. A relatively closed subset $H$ of $D$ is called a (boundary) hull in $D$ if $D{\setminus } H$ is simply connected. For example, a hull in $\mathbb {H}$ is an $\mathbb {H}$-hull iff it is bounded; a hull in $\mathbb {D}$ is a $\mathbb {D}$-hull iff it does not contain $0$. Let $T\in (0,\infty ]$. A family of hulls in $D$: $K_t$, $0\le t<T$, is called a Loewner chain in $D$ if

• 1. $K_0=\emptyset$ and $K_{t_1}\subsetneqq K_{t_2}$ whenever $0\le t_1<t_2<T$;

• 2. for any fixed $a\in [0,T)$ and a compact set $F\subset D{\setminus } K_a$, the extremal length (c.f. [1]) of the family of curves in $D{\setminus } K_t$ that separate $F$ from $K_{t+\varepsilon }{\setminus } K_t$ tends to $0$ as $\varepsilon \rightarrow 0$, uniformly in $t\in [0,a]$.

If $K_t$, $0\le t<T$, is a Loewner chain in $D$, and $a\in [0,T)$, then we also call the restriction $K_t$, $0\le t\le a$, a Loewner chain in $D$.

There are two important properties for Loewner chains. If $K_t$, $0\le t<T$, is a Loewner chain in $D$, and $u$ is a continuous increasing function defined on $[0,T)$ with $u(0)=0$, then $K_{u^{-1}(t)}$, $0\le t<u(T)$, is also a Loewner chain in $D$, which is called a time-change of $(K_t)$ via $u$. If $W$ maps $D$ conformally onto $E$, then $W(K_t)$, $0\le t<T$, is a Loewner chain in $E$.

An $\mathbb {H}$-(resp. $\mathbb {D}$-)Loewner chain is a Loewner chain in $\mathbb {H}$ (resp. $\mathbb {D}$) such that each hull is an $\mathbb {H}$-(resp. $\mathbb {D}$-)hull. An $\mathbb {H}$-(resp. $\mathbb {D}$-)Loewner chain $(K_t)$ is said to be normalized if ${{\mathrm{hcap}}}(K_t)=2t$ (resp. ${{\mathrm{dcap}}}(K_t)=t$) for each $t$.

The conditions for the conformal invariance property of $\mathbb {H}$-(resp. $\mathbb {D}$-)Loewner chains can be slightly weakened as below.

Proposition 3.7

If $K_t$, $0\le t<T$, is an $\mathbb {H}$-(resp. $\mathbb {D}$-)Loewner chain, and $W$ is an $\mathbb {R}$-(resp. $\mathbb {T}$-)symmetric conformal map, whose domain contains $K^{{{\mathrm{db}}}}_t$ for each $t$ and whose image does not contain $\infty$ (resp. $0$), then $W(K_t)$, $0\le t<T$, is also an $\mathbb {H}$-(resp. $\mathbb {D}$-)Loewner chain.

The following proposition combines some results in [8, 12].

Proposition 3.8

Let $T\in (0,\infty ]$. The following are equivalent.

• (i) $K_t$, $0\le t<T$, are chordal (resp. radial) Loewner hulls driven by some $\lambda \in C([0,T))$.

• (ii) $K_t$, $0\le t<T$, is a normalized $\mathbb {H}$-(resp. $\mathbb {D}$-)Loewner chain.

If either of the above holds, with $\mathring{\lambda }(t)=\lambda (t)$ (resp. $\mathring{\lambda }(t)=e^{i\lambda (t)}$ in the radial case) we have

$$\begin{aligned} \{\mathring{\lambda }(t)\}=\bigcap _{\varepsilon >0}\overline{K_{t+\varepsilon }/K_t}, \quad 0\le t<T. \end{aligned}$$

In addition, if $K_t$, $0\le t<T$, is any $\mathbb {H}$-(resp. $\mathbb {D}$-)Loewner chain, then the function $u(t):={{\mathrm{hcap}}}(K_t)/2$ (resp. $u(t):={{\mathrm{dcap}}}(K_t)$), $0\le t<T$, is continuous increasing with $u(0)=0$, which implies that $K_{u^{-1}(t)}$, $0\le t<u(T)$, is a normalized $\mathbb {H}$-(resp. $\mathbb {D}$-)Loewner chain.

Definition 3.9

A family of $\mathbb {H}$-(resp. $\mathbb {D}$-)hulls: $L_{t}$, $0\le t <T$, is called a backward $\mathbb {H}$-(resp. $\mathbb {D}$-)Loewner chain if they satisfy

• 1. $L_0=\emptyset$ and $L_{t_1}\prec L_{t_2}$ if $0\le t_1\le t_2 <T$;

• 2. $L_{t_0}:L_{t_0-t}$, $0\le t\le t_0$, is an $\mathbb {H}$-(resp. $\mathbb {D}$-)Loewner chain for any $t_0\in (0,T)$.

If $u$ is a continuous increasing function defined on $[0,T)$ with $u(0)=0$, then $L_{u^{-1}(t)}$, $0\le t <u(T)$, is also a backward $\mathbb {H}$-(resp. $\mathbb {D}$-)Loewner chain, and is called a time-change of $(L_{t})$ via $u$. A backward $\mathbb {H}$-(resp. $\mathbb {D}$-)Loewner chain $(L_{t})$ is said to be normalized if ${{\mathrm{hcap}}}(L_{t})=2t$ (resp. ${{\mathrm{dcap}}}(L_{t})=t$) for any $t\in [0,T)$.

Using Lemma 3.3 and Proposition 3.8, we obtain the following.

Proposition 3.10

Let $T\in (0,\infty ]$. The following are equivalent.

• (i) $L_{t}$, $0\le t<T$, are backward chordal (resp. radial) Loewner hulls driven by some $\lambda \in C([0,T))$.

• (ii) $L_{t}$, $0\le t <T$, is a normalized backward $\mathbb {H}$-(resp. $\mathbb {D}$-)Loewner chain.

If either of the above holds, with $\mathring{\lambda }(t)=\lambda (t)$ (resp. $\mathring{\lambda }(t)=e^{i\lambda (t)}$ in the radial case) we have

$$\begin{aligned} \{\mathring{\lambda }(t)\}=\bigcap _{\varepsilon >0}\overline{L_{t}:L_{t-\varepsilon }}, \quad 0< t<T, \end{aligned}$$

(3.9)

In addition, if $L_t$, $0\le t<T$, is any backward $\mathbb {H}$-(resp. $\mathbb {D}$-)Loewner chain, then the function $u(t):={{\mathrm{hcap}}}(K_t)/2$ (resp. $u(t):={{\mathrm{dcap}}}(K_t)$), $0\le t<T$, is continuous increasing with $u(0)=0$, which implies that $L_{u^{-1}(t)}$, $0\le t<u(T)$, is a normalized backward $\mathbb {H}$-(resp. $\mathbb {D}$-)Loewner chain.

We say that $f_t$ and $L_t$, $0\le t<T$, are backward chordal (resp. radial) Loewner maps and hulls, via a time change $u$, driven by $\lambda$, if $u$ is a continuous increasing function defined on $[0,T)$ with $u(0)=0$, such that $f_{u^{-1}(t)}$ and $L_{u^{-1}(t)}$, $0\le t<u(T)$, are backward chordal (resp. radial) Loewner maps and hulls driven by $\lambda \circ u^{-1}$. From the above proposition, if $(L_t)$ is any $\mathbb {H}$-(resp. $\mathbb {D}$-)Loewner chain, then $L_t$, $0\le t<T$, are backward chordal (resp. radial) Loewner hulls, via a time change $u(t):={{\mathrm{hcap}}}(L_t)/2$ (resp. ${{\mathrm{dcap}}}(L_t)$), driven by $\lambda$, which satisfies (3.9).

3.5 Simple curves and weldings

An $\mathbb {H}$-simple (resp. $\mathbb {D}$-simple) curve is a half-open simple curve in $\mathbb {H}$ (resp. $\mathbb {D}{\setminus }\{0\}$), whose open side approaches a single point on $\mathbb {R}$ (resp. $\mathbb {T}$). Every $\mathbb {H}$ (resp. $\mathbb {D}$)-simple curve $\beta$ is an $\mathbb {H}$ (resp. $\mathbb {D}$)-hull, whose base $B_\beta$ is a single point, and whose support $S_\beta$ is an $\mathbb {R}$ (resp. $\mathbb {T}$-)interval. Here an $\mathbb {T}$-interval is an arc on $\mathbb {T}$. The function $f_\beta$ extends continuously from $\mathbb {H}$ (resp. $\mathbb {D}$) to $\overline{\mathbb {H}}$ (resp. $\overline{\mathbb {D}}$), which maps $S_\beta$ onto $\overline{\beta }$, sends the two ends of $S_\beta$ to $B_\beta$, and sends a unique point, say $z_\beta \in S_\beta$ to the tip of $\beta$. The point $z_\beta$ divides $S_\beta$ into two $\mathbb {R}$(resp. $\mathbb {T}$-)intervals such that the restriction of $f_\beta$ to either interval is a homeomorphism onto $\overline{\beta }$. Thus, there is a unique involution (an auto homeomorphism whose inverse is itself) $\phi _\beta$ of $S_\beta$, which fixes only one point: $z_\beta$, swaps the two end points of $S_\beta$, and satisfies that $y=\phi _\beta (x)$ implies that $f_\beta (x)=f_\beta (y)$. We call $\phi _\beta$ the welding induced by $\beta$.

Suppose $K$ is an $\mathbb {H}$- or $\mathbb {D}$-simple curve. Let $W$ be as in Theorems 2.12, 2.20, or 2.23. Then $W^\mathcal{H}(K)$ is also an $\mathbb {H}$- or $\mathbb {D}$-simple curve. The equality $W^K\circ f_K= f_{W^\mathcal{H}(K)}\circ W$ holds after continuous extension from $\mathbb {H}$ or $\mathbb {D}$ to its closure. So the weldings induced by $K$ and $W^\mathcal{H}(K)$ satisfy $\phi _{W^\mathcal{H}(K)}=W\circ \phi _K\circ W^{-1}$.

Suppose the hulls $(L_t)$ generated by a backward chordal (resp. radial) Loewner process driven by $\lambda$ are all $\mathbb {H}$(resp. $\mathbb {D}$)-simple curves. Then the process generates backward chordal (resp. radial) traces $(\beta _t)$ such that every $\beta _t$ is a simple curve, and $L_{t}=\beta _{t}([0,t))$, $0\le t<T$.

Let $\phi _{t}$ be the welding induced by $L_t$, which is an involution of $S_t:=S_{L_t}$. Recall that $(S_t)$ is an increasing family because $L_{t_1}\prec L_{t_2}$ for $t_1<t_2$. If $0\le t_1<t_2<T$, then from $f_{t_2,t_1}\circ f_{t_1}=f_{t_2}$ we see that $\phi _{t_2}|_{S_{t_1}}=\phi _{t_1}$. Thus, there is a unique involution $\phi$ of $S_T:=\bigcup _t S_{t}$ such that $\phi |_{S_t}=\phi _t$ for each $t\in [0,T)$. In other words, $y=\phi (x)$ implies that $f_t(x)=f_t(y)$ for some $t\ge 0$, where $f_t$ is the continuous extension of the Loewner map at time $t$ from $\mathbb {H}$(resp. $\mathbb {D}$) to $\overline{\mathbb {H}}$ (resp. $\overline{\mathbb {D}}$). We say that $\phi$ is the welding induced by this process. In the case that $S_T=\mathbb {R}$ (resp. $\mathbb {T}{\setminus }\{z_0\}$ for some $z_0\in \mathbb {T}$), we will extend $\phi$ to an involution of $\widehat{\mathbb {R}}:=\mathbb {R}\cup \{\infty \}$ (resp. $\mathbb {T}$) such that $\infty$ (resp. $z_0$) is the other fixed point of $\phi$.

Here is another way to view the welding $\phi$. For every $t\in (0,T)$, $\phi$ swaps the two end points of $S_t$. Let $\mathring{\lambda }(0)=\lambda (0)$ (resp. $e^{i\lambda (0)}$). Since $f_t(\mathring{\lambda }(0))=\beta _t(0)$ is the tip of $L_t$ for each $t$, we see that $\mathring{\lambda }(0)$ is the only fixed point of $\phi$. On the other hand, it is easy to see that, $x$ and $y$ are end points of $S_t$ if and only if $\tau _x=\tau _y=t$, $0<t<T$; and every point on $S_T{\setminus } \{\mathring{\lambda }(0)\}$ is an end point of some $S_t$, $0<t<T$. Thus, for $x\ne y\in S_T{\setminus } \{\mathring{\lambda }(0)\}$, $y=\phi (x)$ if and only if $\tau _x=\tau _y$, i.e., $x$ and $y$ are swallowed at the same time.

Let $\kappa \in (0,4]$. Since the backward chordal (resp. radial) SLE$_\kappa$ traces are $\mathbb {H}$(resp. $\mathbb {D}$-)simple curves, so the process induces a random welding, which we call a backward chordal (resp. radial) SLE$_\kappa$ welding. In the chordal case, For any $x\in \mathbb {R}{\setminus }\{\lambda (0)\}=\mathbb {R}{\setminus }\{0\}$, the process $X^x_t:=\lambda (t)-f_t(x)$ is a rescaled Bessel process of dimension $1-\frac{4}{\kappa }<1$, which implies that a.s. $X^x_t\rightarrow 0$ at some finite time. Thus, $S_\infty =\mathbb {R}$. which implies that a chordal SLE$_\kappa$ welding is an involution of $\widehat{\mathbb {R}}$ with two fixed points: $\lambda (0)=0$ and $\infty$. In the radial case, since $T=\infty$, Lemma 3.5 says that $S_\infty =\mathbb {T}$ or $\mathbb {T}{\setminus }\{z_0\}$ for some $z_0\in \mathbb {T}$. The first case can not happen since $\phi$ has only one fixed point on $S_\infty$. Thus, a radial SLE$_\kappa$ welding is an involution of $\mathbb {T}$ with two fixed points, one of which is $e^{i\lambda (0)}=1$.

Suppose a backward chordal (resp. radial) Loewner process generates $\mathbb {H}$ (resp. $\mathbb {D}$)-simple backward traces $\beta _{t}$, $0\le t<T$. Then (3.5) (resp. (3.8)) is satisfied because $\beta _{t_2}(t_1)$ lies in $\mathbb {H}$ (resp. $\mathbb {D}$) if $t_2>t_1$. It is clear that the curve $\beta$ defined by (3.6) is simple, and $D_t=D_T{\setminus } \beta ([t,T))$ for $0\le t<T$. Let $\phi$ be the welding induced by the process. If $y=\phi (x)$, there is $t\in [0,T)$ such that $y,x\in S_t$ and $f_t(y)=f_t(x)$. From $F_{T,t}\circ f_t=F_T$, we get $F_T(y)=F_T(x)$. This means that $\phi$ can be realized by the conformal map $F_T$.

If a backward chordal (resp. radial) Loewner chain $(L_t)$ is composed of $\mathbb {H}$ (resp. $\mathbb {D}$)-simple curves, then $(L_t)$ induces a welding $\phi$, which is an involution of $\bigcup S_{L_t}$, and agrees with $\phi _{L_t}$ on $S_{L_t}$ for each $t$. To see this, one may first normalized the backward Loewner chain so that it is generated by a backward Loewner process.

4 Conformal transformations

In this section, we will study how a backward SLE$(\kappa ;\rho )$ process changes under a Möbius transformation, and derive the backward SLE counterpart of the results of [17]. For this purpose, we will first define the conformal transformation of a backward Loewner chain. The rest of the arguments are the same as the ones for forward SLE, up to negating $\kappa$. We will use the ideas in [17] and some results in [8].

Proposition 4.1

Suppose $L_{t}$, $0\le t<T$, is a backward $\mathbb {H}$-(resp. $\mathbb {D}$-)Loewner chain, $W$ is an $\mathbb {R}$-(resp. $\mathbb {T}$-)symmetric conformal map whose domain contains every $S_{L_{t}}$, and $\infty \not \in W(S_{L_t})$ for $0\le t<T$. Then $W^\mathcal{H}(L_t)$, $0\le t <T$, is also a backward $\mathbb {H}$-(resp. $\mathbb {D}$-)Loewner chain.

Proof

From Theorem 2.12, $W^{L_t}$ and $W^\mathcal{H}(L_t)$ are well defined. Since $L_0=\emptyset$, $W^\mathcal{H}(L_0)=\emptyset$. Let $0\le t_1\le t_2<T$. Since $L_{t_1}\prec L_{t_2}$, from Lemma 2.14, $W^\mathcal{H}(L_{t_1})\prec W^\mathcal{H}(L_{t_2})$. Fix $t_0\in (0,T)$. Since $L_{t_0}:L_{t_0-t}$, $0\le t\le t_0$, is an $\mathbb {H}$-(resp. $\mathbb {D}$-)Loewner chain, from Lemma 2.14 and Proposition 3.7 we see that

$$\begin{aligned} W^\mathcal{H}(L_{t_0}):W^\mathcal{H}(L_{t_0-t})=W^{L_{t_0}}(L_{t_0}:L_{t_0-t}),\quad 0\le t\le t_0, \end{aligned}$$

is also an $\mathbb {H}$-(resp. $\mathbb {D}$-)Loewner chain. This finishes the proof. $\square$

We call $W^\mathcal{H}(L_t)$, $0\le t <T$, the conformal transformation of $L_t$, $0\le t<T$, under $W$. Using Lemma 2.21 instead of Lemma 2.14, we can show that a similar proposition holds.

Proposition 4.2

Suppose $L_{t}$, $0\le t<T$, is a backward $\mathbb {H}$-(resp. $\mathbb {D}$-)Loewner chain, $W$ is a Mobius transform that maps $\mathbb {H}$ onto $\mathbb {D}$ (resp. maps $\mathbb {D}$ onto $\mathbb {H}$) such that $\infty \not \in W(S_{L_t})$ for $0\le t<T$. Then $W^\mathcal{H}(L_t)$, $0\le t <T$, is a backward $\mathbb {D}$-(resp. $\mathbb {H}$-)Loewner chain.

Suppose $(L_t)$ is composed of $\mathbb {H}$- or $\mathbb {D}$-simple curves. Then $(W^\mathcal{H}(L_t))$ is also composed of $\mathbb {H}$ or $\mathbb {D}$-simple curves. Let $\phi$ and $\phi _W$ be the weldings induced by these two chains, which are involutions of $\bigcup S_{L_t}$ and $\bigcup S_{W^\mathcal{H}(L_t)}$, respectively. Since for each $t\in (0,T)$, $\phi |_{S_{L_t}}=\phi _{L_t}$, $\phi _W|_{S_{W^\mathcal{H}(L_t)}}=\phi _{W^\mathcal{H}(L_t)}$, $S_{W^\mathcal{H}(L_t)}=W(S_{L_t})$, and $\phi _{W^\mathcal{H}(L_t)}=W\circ \phi _{L_t}\circ W^{-1}$, we see that $\bigcup S_{W^\mathcal{H}(L_t)}=W(\bigcup S_{L_t})$ and

$$\begin{aligned} \phi _W=W\circ \phi \circ W^{-1}. \end{aligned}$$

(4.1)

This means that the conformal transformation preserves the welding.

The following proposition is essentially Lemma 2.8 in [8].

Proposition 4.3

Let $W$ be an $\mathbb {R}$-symmetric conformal map, whose domain contains $z_0\in \mathbb {R}$, such that $W(z_0)\ne \infty$. Then

$$\begin{aligned} \lim _{H\rightarrow z_0} \frac{{{\mathrm{hcap}}}(W(H))}{{{\mathrm{hcap}}}(H)}=|W'(z_0)|^2, \end{aligned}$$

where $H\rightarrow z_0$ means that ${{\mathrm{diam}}}(H\cup \{z_0\})\rightarrow 0$ with $H$ being a nonempty $\mathbb {H}$-hull.

Using the integral formulas for capacities of $\mathbb {H}$-hulls and $\mathbb {D}$-hulls, it is not hard to derive the following similar proposition.

Proposition 4.4

• (i) Let $W$ be a conformal map on a $\mathbb {T}$-symmetric domain $\Omega$, which satisfies $I_\mathbb {R}\circ W=W\circ I_{\mathbb {T}}$ and $W(\Omega \cap \mathbb {D})\subset \mathbb {H}$. Let $z_0\in \Omega \cap \mathbb {T}$ be such that $W(z_0)\ne \infty$. Then

  $$\begin{aligned} \lim _{H\rightarrow z_0} \frac{{{\mathrm{hcap}}}(W(H))}{{{\mathrm{dcap}}}(H)}=2|W'(z_0)|^2, \end{aligned}$$

  where $H\rightarrow z_0$ means that ${{\mathrm{diam}}}(H\cup \{z_0\})\rightarrow 0$ with $H$ being a nonempty $\mathbb {D}$-hull.

• (ii) Proposition 4.3 holds with $\mathbb {R}$ replaced by $\mathbb {T}$, ${{\mathrm{hcap}}}$ replaced by ${{\mathrm{dcap}}}$, and $H\rightarrow z_0$ understood as in (i).

4.1 Transformations between backward $\mathbb {H}$-Loewner chains

Suppose $L_t$ and $f_t$, $0\le t<T$, are backward chordal Loewner hulls and maps driven by $\lambda \in C([0,T))$. From Proposition 3.10, $(L_t)$ is a backward $\mathbb {H}$-Loewner chain. Let $W$ be an $\mathbb {R}$-symmetric conformal map, whose domain $\Omega$ contains the support of every $L_t$. Write $W_t$ for $W^{L_t}$. The domain of $W_t$ is $\Omega ^{L_t}$, which contains $L^{{{\mathrm{db}}}}_t$. If $t>0$, $\lambda (t)\in L^{{{\mathrm{db}}}}_t$, so $\lambda (t)$ is contained in the domain of $W_t$. This is also true for $t=0$ because $W_0=W$ and $\{\lambda (0)\}=S_0\subset S_t=S_{L_t}\subset \Omega$ for any $t\in (0,T)$. Let $L^*_t=W^\mathcal{H}(L_t)=W_t(L_t)$, $0\le t<T$. From Proposition 4.1, $(L^*_t)$ is a backward $\mathbb {H}$-Loewner chain, and

$$\begin{aligned} W_t(L_t:L_{t-\varepsilon })=L^*_t:L^*_{t-\varepsilon },\quad 0\le t-\varepsilon <t<T. \end{aligned}$$

(4.2)

From Proposition 3.10, $L^*_{t}$, $0\le t<T$, are backward chordal Loewner hulls via a time change $u(t):={{\mathrm{hcap}}}(L^*_t)/2$, driven by some $\lambda ^*$, which satisfies

$$\begin{aligned} \{\lambda ^*(t)\}=\bigcap _{\varepsilon >0}\overline{L^*_{t}:L^*_{t-\varepsilon }}, \quad 0< t<T. \end{aligned}$$

From (3.9), (4.2), and continuity, we find that

$$\begin{aligned} \lambda ^*(t)=W_t(\lambda (t)),\quad 0\le t<T. \end{aligned}$$

(4.3)

Since $(L_t)$ and $(L^*_{u^{-1}(t)})$ are normalized, we know that ${{\mathrm{hcap}}}(L_t:L_{t-\varepsilon })=2\varepsilon$ and ${{\mathrm{hcap}}}(L^*_t:L^*_{t-\varepsilon })=2u(t)-2u(t-\varepsilon )$. From (4.2) and Proposition 4.3, we find that

$$\begin{aligned} u'(t)=W_t'(\lambda (t))^2,\quad 0\le t<T. \end{aligned}$$

(4.4)

Let $f^*_t=f_{L^*_t}$. From the definition of $W_t=W^{L_t}$, we have the equality

$$\begin{aligned} W_t\circ f_t=f^*_t\circ W, \end{aligned}$$

(4.5)

which holds in $\Omega {\setminus } S_{L_t}$. Differentiating (4.5) w.r.t. $t$, and using (4.3) and (4.4), we get

$$\begin{aligned}{}[\partial _t W_t](f_t(z))+W_t'(f_t(z))\frac{-2}{f_t(z)-\lambda (t)}&= \frac{-2u'(t)}{f^*_t(W(z))-\lambda ^*(t)}\\ {}&= \frac{-2W_t'(\lambda (t))^2}{W_t(f_t(z))-W_t(\lambda (t))}. \end{aligned}$$

Thus, for any $w=f_t(z)\in f_t(\Omega {\setminus } S_{L_t})=\Omega ^{L_t}{\setminus } L^{{{\mathrm{db}}}}_t$,

$$\begin{aligned} \partial _t W_t(w)=\frac{-2W_t'(\lambda (t))^2}{W_t(w)-W_t(\lambda (t))}-W_t'(w) \frac{-2}{w-\lambda (t)}. \end{aligned}$$

(4.6)

By analytic extension, the above equality holds for any $w\in \Omega ^{L_t}{\setminus } \{\lambda (t)\}$. Letting $w\rightarrow \lambda (t)$, we find that

$$\begin{aligned} \partial _t W_t(\lambda (t))=3W_t''(\lambda (t)),\quad 0\le t<T. \end{aligned}$$

(4.7)

Differentiating (4.6) w.r.t. $w$ and letting $w\rightarrow \lambda (t)$, we get

$$\begin{aligned} \frac{\partial _t W_t'(\lambda (t))}{W_t'(\lambda (t))}=-\frac{1}{2}\Big (\frac{W_t''(\lambda (t))}{W_t'(\lambda (t))}\Big )^2+\frac{4}{3} \frac{W_t'''(\lambda (t))}{W_t'(\lambda (t))}. \end{aligned}$$

(4.8)

4.2 Transformations involving backward $\mathbb {D}$-Loewner chains

Now suppose $L_t$, $0\le t<T$, are backward radial Loewner hulls driven by $\lambda$. Let $f_t$ and $\widetilde{f}_t$ be the corresponding radial Loewner maps and covering maps. Suppose $W$ is a $\mathbb {T}$-symmetric conformal map, whose domain $\Omega$ contains the support of every $L_t$. Let $W_t=W^{L_t}$, $L^*_t=W_t(L_t)=W^\mathcal{H}(L_t)$, and $u(t)={{\mathrm{dcap}}}(L^*_t)$, $0\le t<T$. Then $L^*_{t}$, $0\le t<T$, are backward radial Loewner hulls via a time change $u(t):={{\mathrm{dcap}}}(L^*_t)$, driven by some $\lambda ^*$, which satisfies

$$\begin{aligned} \{e^{i\lambda ^*(t)}\}=\bigcap _{\varepsilon >0} \overline{ L^*_t:L^*_{t-\varepsilon }},\quad 0<t<T. \end{aligned}$$

Let $f^*_t$ (resp. $\widetilde{f}^*_t$), $0\le t<T$, denote the backward radial (resp. covering radial) Loewner hulls via the time change $u$ driven by $\lambda ^*$. The argument in the last subsection still works with Proposition 4.4 in place of Proposition 4.3. We can conclude that $e^{i\lambda (t)}$ lies in the domain of $W_t$ for $0\le t<T$; $W_t(e^{i\lambda (t)})=e^{i\lambda ^*(t)}$; $u'(t)=|W_t'(e^{i\lambda (t)})|^2$; and (4.5) still holds. Suppose $\widetilde{W}$ is an $\mathbb {R}$-symmetric conformal map defined on $\widetilde{\Omega }=(e^i)^{-1}(\Omega )$, which satisfies $e^i\circ \widetilde{W}=W\circ e^i$. Define $\widetilde{W}_t$ to be the analytic extension of $\widetilde{f}^*_t\circ \widetilde{W}\circ \widetilde{f}_t^{-1}$ to $\widetilde{\Omega }_t:=(e^i)^{-1}(\Omega ^{L_t})$. Then we get

$$\begin{aligned} \widetilde{W}_t\circ \widetilde{f}_t=\widetilde{f}^*_t\circ \widetilde{W}; \end{aligned}$$

(4.9)

Comparing this with (4.5) we find $e^i\circ \widetilde{W}_t=W_t\circ e^i$. So $\lambda (t)$ lies in the domain of $\widetilde{W}_t$, and

$$\begin{aligned} u'(t)=\widetilde{W}_t'(\lambda (t))^2,\quad 0\le t<T. \end{aligned}$$

(4.10)

Since $W_t(e^{i\lambda (t)})=e^{i\lambda ^*(t)}$, from the continuity, there is $n\in \mathbb {N}$ such that $\widetilde{W}_t(\lambda (t))=\lambda ^*(t)+2n\pi$ for $0\le t<T$. Since $\lambda ^*$ and $\lambda ^*+2n\pi$ generate the same backward radial Loewner objects via the time change $u$, by replacing $\lambda ^*$ with $\lambda ^*+2n\pi$, we may assume that

$$\begin{aligned} \widetilde{W}_t({\lambda (t)})={\lambda ^*(t)},\quad 0\le t<T. \end{aligned}$$

(4.11)

Differentiating (4.9) w.r.t. $t$ and letting $w=\widetilde{f}_t(z)$, we get

$$\begin{aligned} \partial _t \widetilde{W}_t(w)=-\widetilde{W}_t'(\lambda (t))^2\cot _2({\widetilde{W}_t(w)-\widetilde{W}_t(\lambda (t))})+\widetilde{W}_t'(w)\cot _2({w-\lambda (t)}), \end{aligned}$$

(4.12)

which holds for $w\in (e^i)^{-1}(\Omega ^{L_t}{\setminus }\{e^{i\lambda (t)}\})$. Letting $w\rightarrow \lambda (t)$, we get

$$\begin{aligned} \partial _t \widetilde{W}_t(\lambda (t))=3\widetilde{W}_t''(\lambda (t)),\quad 0\le t<T. \end{aligned}$$

(4.13)

Differentiating (4.12) w.r.t. $w$ and letting $w\rightarrow \lambda (t)$, we get

$$\begin{aligned} \frac{\partial _t \widetilde{W}_t'(\lambda (t))}{\widetilde{W}_t'(\lambda (t))}=-\frac{1}{2}\left( \frac{W_t''(\lambda (t))}{W_t'(\lambda (t))}\right) ^2+\frac{4}{3} \frac{ W_t'''(\lambda (t))}{W_t'(\lambda (t))}+\frac{1}{6}\widetilde{W}_t'(\lambda (t))^2-\frac{1}{6}. \end{aligned}$$

(4.14)

The number $\frac{1}{6}$ comes from the Laurent series of $\cot _2(z)$: $\frac{2}{z}-\frac{z}{6}+O(z^3)$.

Let $(L_t)$, $(f_t)$, and $(\widetilde{f}_t)$ be as above. Now suppose $W$ is a Möbius transformation that maps $\mathbb {D}$ onto $\mathbb {H}$ such that $W^{-1}(\infty )\not \in S_{L_t}$ for every $t$. Let $W^{L_t}$ be as in Theorem 2.20. Let $W_t=W^{L_t}$ and $L^*_t=W_t(L_t)=W^\mathcal{H}(L_t)$, $0\le t<T$. Then $L^*_{t}$, $0\le t<T$, are backward chordal Loewner hulls via a time change $u(t):={{\mathrm{hcap}}}(L^*_t)/2$, driven by some $\lambda ^*$. Let $f^*_t=f_{L^*_t}$. Then (4.5) still holds, and we have $u'(t)=|W_t'(e^{i\lambda (t)})|^2$ and $W_t(e^{i\lambda (t)})=e^{i\lambda ^*(t)}$. Let $\widetilde{W}=W\circ e^i$ and $\widetilde{W}_t=W_t\circ e^i$. We get (4.10), (4.11), and $\widetilde{W}_t\circ \widetilde{f}_t=f^*_t\circ \widetilde{W}$. Differentiating this equality w.r.t. $t$ and letting $w=\widetilde{f}_t(z)$ tend to $\lambda (t)$, we find that (4.13) still holds.

4.3 Möbius invariance of backward SLE$(\kappa ;\rho )$ processes

We now define backward chordal and radial SLE$(\kappa ;\mathbf {\rho })$ processes, where $\mathbf {\rho }=(\rho _1,\ldots ,\rho _n)\in \mathbb {R}^n$. Let $x_0,q_1,\ldots ,q_n\in \mathbb {R}$ such that $q_k\ne x_0$ for all $k$. Let $\lambda (t), 0\le t<T$, be the maximal solution of the equation

$$\begin{aligned} d\lambda (t)=\sqrt{\kappa }dB(t)+\sum _{k=1}^n \frac{-\rho _k}{\lambda (t)-f^\lambda _t(q_k)}\,dt;\quad \lambda (0)=x_0. \end{aligned}$$

(4.15)

Here $f^\lambda _t$, $0\le t<T$, are the backward chordal Loewner maps driven by $\lambda$. Then we call the backward chordal Loewner process driven by $\lambda$ the chordal SLE$(\kappa ;\mathbf {\rho })$ process started from $x_0$ with force points $(q_1,\ldots ,q_n)$, or simply started from $(x_0;q_1,\ldots ,q_n)$. We allow some $q_k$ to be $\infty$. In that case, $f^\lambda _t(q_k)$ is always $\infty$, and the term $\frac{-\rho _k}{\lambda (t)-f^\lambda _t(q_k)}$ vanishes.

Let $x_0,q_1,\ldots ,q_n\in \mathbb {R}$ be such that $q_k\not \in x_0+2\pi \mathbb {Z}$ for all $k$. Let $\lambda (t)$, $0\le t<T$, be the maximal solution of the equation

$$\begin{aligned} d\lambda (t)=\sqrt{\kappa }dB(t)+\sum _{k=1}^n \frac{-\rho _k}{2} \cot _2({\lambda (t)-\widetilde{f}^\lambda _t(q_k)})\,dt;\quad \lambda (0)=x_0. \end{aligned}$$

(4.16)

Here $\widetilde{f}^\lambda _t$, $0\le t<T$, are the covering backward radial Loewner maps driven by $\lambda$. Then the backward radial Loewner process driven by $\lambda$ is called the radial SLE$(\kappa ;\mathbf {\rho })$ process started from $e^{ix_0}$ with marked points $(e^{iq_1},\ldots ,e^{iq_n})$, or simply started from $(e^{ix_0};e^{iq_1},\ldots ,e^{iq_n})$.

The existence of backward chordal (resp. radial) SLE$_\kappa$ traces and Girsanov’s Theorem imply the existence of a backward chordal (resp. radial) SLE$(\kappa ;\mathbf {\rho })$ traces. The traces are $\mathbb {H}$(resp. $\mathbb {D}$)-simple curves if $\kappa \in (0,4]$.

The following lemma is easy to check.

Lemma 4.5

Let $W$ be a Möbius transformation. Then the following hold.

• (i) For any $z\in \mathbb {C}\cap W^{-1}(\mathbb {C})$ and $w\in \widehat{\mathbb {C}}$,

  $$\begin{aligned} \frac{2W'(z)}{W(z)-W(w)}-\frac{2}{z-w}=\frac{W''(z)}{W'(z)}. \end{aligned}$$

• (ii) Let $\widetilde{W}=W\circ e^i$. For any $z\in \mathbb {C}\cap \widetilde{W}^{-1}(\mathbb {C})$ and $w\in \mathbb {C}$,

  $$\begin{aligned} \frac{2\widetilde{W}'(z)}{\widetilde{W}(z)-\widetilde{W}(w)}- \cot _2(z-w)= \frac{\widetilde{W}''(z)}{\widetilde{W}'(z)}. \end{aligned}$$

• (iii) Suppose an analytic function $\widetilde{W}:\Omega \rightarrow \mathbb {C}$ satisfies $e^i\circ \widetilde{W}=W\circ e^i$ in $\Omega$. Then for any $z,w\in \Omega$,

  $$\begin{aligned} \widetilde{W}'(z)\cot _2(\widetilde{W}(z)-\widetilde{W}(w))-\cot _2(z-w)= \frac{\widetilde{W}''(z)}{\widetilde{W}'(z)}. \end{aligned}$$

Theorem 4.6

Let $L_t$, $0\le t<T$, be the backward chordal SLE$(\kappa ;\mathbf {\rho })$ hulls started from $(x_0;q_1,\ldots ,q_n)$. Suppose $\sum \rho _k=-\kappa -6$. Let $W$ be a Möbius transformation from $\mathbb {H}$ onto $\mathbb {H}$ such that $\{\infty ,W^{-1}(\infty )\}\subset \{q_1,\ldots ,q_n\}$. Then, after a time change, $W^\mathcal{H}(L_t)$, $0\le t<T$, are the backward chordal SLE$(\kappa ;\mathbf {\rho })$ hulls started from $(W(x_0);W(q_1),\ldots ,W(q_n))$.

Proof

Since $W^{-1}(\infty )$ is a force point, it is not contained in the support of any $L_t$. So $\infty \not \in W(S_{L_t})$, $0\le t<T$. Let $\lambda$ be the driving function, and $f_t=f^\lambda _t$, $0\le t<T$, be the corresponding maps. We may and now adopt the notation in Sect. 4.1. Let $(\mathcal{F}_t)$ be the complete filtration generated by $B(t)$ in (4.15). Then $(\lambda _t)$ and $(L_t)$ are $(\mathcal{F}_t)$-adapted. From Corollary 2.19 (i), $(W^\mathcal{H}(L_t))$ is also $(\mathcal{F}_t)$-adapted. Since $W_t=W^{L_t}=f_{W^\mathcal{H}(L_t)}\circ W\circ g_{L_t}$ on $\Omega ^{L_t}{\setminus } L^{{{\mathrm{db}}}}_t$, $(W_t)$ is $(\mathcal{F}_t)$-adapted. So we may apply Itô’s formula (c.f. [14]). From (4.3) and (4.7), we get

$$\begin{aligned} d\lambda ^*(t)=W_t'(\lambda (t))d\lambda (t)+\left( \frac{\kappa }{2}+3\right) W_t''(\lambda (t))dt,\quad 0\le t<T. \end{aligned}$$

Applying (4.15) and Lemma 4.5 (i), and using the condition that $\sum \rho _k=-\kappa -6$, we find that

$$\begin{aligned} d\lambda ^*(t)&= W_t'(\lambda (t))\sqrt{\kappa }dB(t)+\sum _{k=1}^n \frac{-\rho _k W_t'(\lambda (t))^2}{W_t(\lambda (t))-W_t\circ f^\lambda _t(q_k)}\,dt\\&= W_t'(\lambda (t))\sqrt{\kappa }dB(t)+\sum _{k=1}^n \frac{-\rho _k W_t'(\lambda (t))^2}{\lambda ^*(t)- f^{*}_t\circ W(q_k)}\,dt, \quad 0\le t<T. \end{aligned}$$

From (4.3) we get $\lambda ^*(0)=W_0(\lambda (0))=W(x_0)$. Since $L^*_t=W^\mathcal{H}(L_t)$ and $f^*_t$ are backward chordal Loewner hulls and maps via the time change $u$ driven by $\lambda ^*$, from (4.4) and the above equation, we conclude that, after a time change, $W^\mathcal{H}(L_t)$, $0\le t<T$, are the backward chordal SLE$(\kappa ;\mathbf {\rho })$ hulls started from $(W(x_0);W(q_1),\ldots ,W(q_n))$ and stopped at some time.

It remains to show that the above process is completed. If not, the process can be extended without swallowing the force points $W(q_1),\ldots ,W(q_n)$. From the condition, $W(\infty )$ is among these force points. So $(W^{-1})^\mathcal{H}$ is well defined at the hulls of the extended process. From Propositions 4.1 and 3.10, this implies that the backward chordal Loewner hulls $L_t$, $0\le t<T$, can be extended without swallowing any of $q_1,\ldots ,q_n$, which is a contradiction. $\square$

The following theorem can be proved using the above proof with minor modifications: we now use the argument in Sect. 4.2 instead of that in Sect. 4.1, apply Lemma 4.5 (ii) and (iii) instead of (i), and use Proposition 4.2 in addition to Proposition 4.1.

Theorem 4.7

Suppose $\sum \rho _k=-\kappa -6$. Let $(L_t)$ be the backward radial SLE$(\kappa ;\mathbf {\rho })$ hulls started from $(e^{ix_0};e^{iq_1},\ldots ,e^{iq_n})$. Let $W$ map $\mathbb {D}$ conformally onto $\mathbb {H}$ (resp. $\mathbb {D}$) such that $\{W^{-1}(\infty )\}\cap \mathbb {T}\subset \{e^{iq_1},\ldots ,e^{iq_n}\}$. Then, after a time change, $(W^\mathcal{H}(L_t))$ are the backward chordal (resp. radial) SLE$(\kappa ;\mathbf {\rho })$ hulls started from $(W(e^{ix_0});W(e^{iq_1}),\ldots ,W(e^{iq_n}))$.

Corollary 4.8

Let $(L_t)$ be the backward radial SLE$(\kappa ;-\kappa -6)$ hulls started from $(e^{ix_0};e^{iq_0})$. Let $W$ map $\mathbb {D}$ conformally onto $\mathbb {H}$ such that $W(e^{ix_0})=0$ and $W(e^{iq_0})=\infty$. Then, after a time change, $(W^\mathcal{H}(L_t))$ are the backward chordal SLE$_\kappa$ hulls started from $0$.

Remarks

• 1. The above theorems are the backward SLE counterpart of the work in [17] for forward SLE$(\kappa ;\mathbf {\rho })$ processes. The condition in their paper for Möbius invariance is $\sum \rho _k=\kappa -6$. This is one reason why we may view backward SLE$_\kappa$ as SLE$_{-\kappa }$.

• 2. The definition of backward SLE$(\kappa ;\mathbf {\rho })$ process differ from Sheffield’s definition in [18] by a minus sign in (4.15) and (4.16) before the $\rho _k$’s. If Sheffield’s definition were used, the condition for conformal invariance would be $\sum \rho _k=\kappa +6$ instead of $\sum \rho _k=-\kappa -6$.

• 3. We may allow interior force points as in [17]. For the chordal (resp. radial) SLE$(\kappa ;\mathbf {\rho })$ process, if $q_k\in \mathbb {H}$ (resp. $e^{iq_k}\in \mathbb {D}$) is a force point, we use ${{\mathrm{Re}}}f^\lambda _t(q_k)$ (resp. ${{\mathrm{Re}}}\widetilde{f}^\lambda _t(q_k)$) instead of $f^{\lambda }_t(q_k)$ (resp. $\widetilde{f}^{\lambda }_t(q_k)$) in (4.15) (resp. (4.16)). In the radial case, adding $0$ to be a force point or change the force for $0$ does not affect the process. Theorems 4.6 and 4.7 still hold if some or all force points lie inside $\mathbb {H}$ or $\mathbb {D}$. For the proofs, we apply Lemma 4.5 with real parts taken on the displayed formulas. One particular example is the following corollary.

Corollary 4.9

Let $L_t$, $0\le t<\infty$, be a backward radial SLE$_\kappa$ process. Let $W$ be a Möbius transformation that maps $\mathbb {D}$ onto $\mathbb {H}$ such that $W(1)\ne \infty$. Let $T$ be the maximum number such that $W^{-1}(\infty )\not \in S_{L_t}$, $0\le t<T$. Then, after a time change, $W^\mathcal{H}(L_t)$, $0\le t<T$, are the backward chordal SLE$(\kappa ;-\kappa -6)$ hulls started from $(W(1);W(0))$.

• 4. Using the properties of Bessel process and applying Girsanov’s theorem, one may define backward chordal or radial SLE$(\kappa ;\mathbf {\rho })$ processes with exactly one degenerate force point, if the corresponding force $\rho _1$ satisfies $\rho _1<-2$ (which corresponds to a Bessel or Bessel-like process of dimension $d=1-\frac{4+2\rho _1}{\kappa }>1$). Theorems 4.6 and 4.7 still hold when a degenerate force point exists. Unlike forward SLE$(\kappa ;\mathbf {\rho })$ process, it is impossible to define a backward SLE$(\kappa ;\mathbf {\rho })$ process with two different degenerate force points.

• 5. Consider the radial case with one force point. Suppose the force $\rho _1\le -\frac{\kappa }{2}-2$. Let $X_t=\lambda (t)-\widetilde{f}^\lambda _t(q_1)$. Then $X_t$ is a Bessel-like process with dimension $d=1-\frac{4+2\rho _1}{\kappa }\ge 2$, which implies that $X_t$ never hits $2\pi \mathbb {Z}$. So $T=\infty$ and ${e^{iq_1}}\not \in S_t$ for any $t$. From Lemma 3.5, $S_\infty =\mathbb {T}{\setminus } \{e^{iq_1}\}$. If, in addition, $\kappa \in (0,4]$, then a backward radial SLE$(\kappa ;\rho _1)$ process induces a random welding $\phi$, which is a involution of $\mathbb {T}$ with exactly two fixed point, $e^{i\lambda (0)}$ and $e^{iq_1}$, which are the initial point and the force point of the process.

5 Commutation relations

Definition 5.1

Let $\kappa _1,\kappa _2>0$, $n\in \mathbb {N}$, and $\mathbf {\rho }_1,\mathbf {\rho }_2\in \mathbb {R}^n$. Let $z_1,z_2,w_k$, $2\le k\le n$, be distinct points on $\mathbb {R}$ (resp. $\mathbb {T}$). We say that a backward chordal (resp. radial) SLE$(\kappa _1;\mathbf {\rho }_1)$ started from $(z_1;z_2,w_2,\ldots ,w_n)$ commutes with a backward chordal (resp. radial) SLE$(\kappa _2;\mathbf {\rho }_2)$ started from $(z_2;z_1,w_2,\ldots ,w_n)$ if there exists a coupling of two processes $(L_1(t);0\le t<T_1)$ and $(L_2(t);0\le t<T_2)$ such that

• (i) For $j=1,2$, $(L_j(t),0\le t<T_j)$ is a complete backward chordal (resp. radial) SLE$(\kappa _j;\mathbf {\rho }_j)$ process started from $(z_j;z_{3-j},w_2,\ldots ,w_n)$.

• (ii) For $j\ne k\in \{1,2\}$, if $\bar{t}_k<T_k$ is a stopping time w.r.t. the complete filtration $(\mathcal{F}^k_t)$ generated by $(L_k(t))$, then conditioned on $\mathcal{F}^k_{\bar{t}_k}$, after a time change, $f_k({\bar{t}_k},\cdot )^\mathcal{H}(L_j({t_j}))$, $0\le t_j<T_j(\bar{t}_k)$, has the distribution of a partial backward chordal (resp. radial) SLE$(\kappa _j;\mathbf {\rho }_j)$ process started from

  $$\begin{aligned} (f_k({\bar{t}_k},(z_j));\mathring{\lambda }_k(\bar{t}_k),f_k({\bar{t}_k},w_2),\ldots ,f_k({\bar{t}_k},w_n)), \end{aligned}$$

  where $f_k({\bar{t}_k},\cdot ):=f_{L_k({\bar{t}_k})}$, $T_j(\bar{t}_k):=\sup \{t_j<T_j:S_{L_j({t_j})}\cap S_{L_{k}({\bar{t}_k})}=\emptyset \}$, $\mathring{\lambda }_k(\bar{t}_k)=\lambda _k(\bar{t}_k)$ in the chordal case (resp. $e^{i\lambda _k(\bar{t}_k)}$ in the radial case), and $\lambda _k$ is the driving function for $(L_k(t))$.

Here a partial backward SLE$(\kappa ;\mathbf {\rho }_j)$ process is a complete SLE$(\kappa ;\mathbf {\rho }_j)$ process stopped at a positive stopping time. If the commutation holds for any distinct points $z_1,z_2,w_k$, $2\le k\le n$ on $\mathbb {R}$ (resp. $\mathbb {T}$), then we simply say that backward chordal (resp. radial) SLE$(\kappa _1;\mathbf {\rho }_1)$ commutes with backward chordal (resp. radial) SLE$(\kappa _2;\mathbf {\rho }_2)$.

Remark The definition is similar to the definition of the commutation relation between forward SLE$(\kappa ;\rho )$ processes that first appeared in [5], where implicitly $g_k({\bar{t}_k},L_j({t_j}))=g_{L_k(\bar{t}_k)}(L_j(t_j))$ was used instead of the $f_k({\bar{t}_k},\cdot )^\mathcal{H}(L_j({t_j}))$ here, and $T_j(\bar{t}_k)$ was defined to be the first $t_j$ such that $L_j(t_j)^{{{\mathrm{db}}}}$ intersects $L_k(\bar{t}_k)^{{{\mathrm{db}}}}$.

Theorem 5.2

For any $\kappa >0$, backward chordal (resp. radial) SLE$(\kappa ;-\kappa -6)$ commutes with backward chordal (resp. radial) SLE$(\kappa ;-\kappa -6)$.

We will prove this theorem in the next two subsections.

5.1 Ensemble

In this subsection we will study how two backward SLE processes interact with each other. The argument relies extensively on ideas and techniques from [9]. We first consider the radial case. Fix $\kappa >0$ and $z_1\ne z_2\in \mathbb {T}$. Write $z_j=e^{i\widetilde{z}_j}$, $j=1,2$. For $j=1,2$, let $L_j(t)$, $0\le t<T_j$, be a backward radial SLE$(\kappa ;-\kappa -6)$ process started from $(z_j;z_{3-j})$; let $\lambda _j$ be the driving function, and let $f_j(t,\cdot )$ and $\widetilde{f}_j(t,\cdot )$, $0\le t<T_j$, be the corresponding maps and covering maps. At first, we suppose that the two processes are independent. Then for $j=1,2$, $\lambda _j$ satisfies $\lambda _j(0)=\widetilde{z}_j$ and the SDE:

$$\begin{aligned} d\lambda _j(t)=\sqrt{\kappa }dB_j(t)-\frac{-\kappa -6}{2} \cot _2(\lambda _j(t)-\widetilde{f}_j(t,\widetilde{z}_{3-j}))dt,\quad 0\le t<T_j, \end{aligned}$$

(5.1)

where $B_1(t)$ and $B_2(t)$ are independent standard Brownian motions. For $j=1,2$, let $(\mathcal{F}^j_t)$ denote the complete filtration generated by $B_j(t)$.

Define $\mathcal{D}=\{(t_1,t_2)\in [0,T_1)\times [0,T_2): S_{L_1(t_1)}\cap S_{L_2(t_2)}=\emptyset \}$. Then for $(t_1,t_2)\in \mathcal D$, we have $(L_1(t_1),L_2(t_2))\in \mathcal{P}_*$. So we may define

$$\begin{aligned} (L_{1,t_2}(t_1),L_{2,t_1}(t_2))=f^\mathcal{P}(L_1(t_1),L_2(t_2)). \end{aligned}$$

Let $f_{1,t_2}(t_1,\cdot )=f_{L_{1,t_2}(t_1)}$ and $f_{2,t_1}(t_2,\cdot )=f_{L_{2,t_1}(t_2)}$. From a radial version of Theorem 2.16, we see that

$$\begin{aligned} f_{1,t_2}(t_1,\cdot )\circ f_2(t_2,\cdot )=f_{L_1(t_1)\vee L_2(t_2)}=f_{2,t_1}(t_2,\cdot )\circ f_1(t_1,\cdot ). \end{aligned}$$

(5.2)

Recall that $L_1(t_1)\vee L_2(t_2)$ is the quotient union of $L_1(t_1)$ and $L_2(t_2)$, i.e., the unique hull which is the disjoint union of two hulls such that the corresponding two quotient hulls are $L_1(t_1)$ and $L_2(t_2)$. Fix $j\ne k\in \{1,2\}$. From a radial version of Corollary 2.19 (ii), the random map $f_{j,t_k}(t_j,\cdot )$ is $\mathcal{F}^j_{t_j}\times \mathcal{F}^k_{t_k}$-measurable. Let $u_{j,t_k}(t_j)={{\mathrm{dcap}}}(L_{j,t_k}(t_j))$. From Propositions 3.10 and 4.1, for any fixed $t_k\in [0,T_k)$, $f_{j,t_k}(t_j,\cdot )$ are backward radial Loewner maps via the time change $u_{j,t_k}$. Let $\widetilde{f}_{j,t_k}(t_j,\cdot )$ be the corresponding covering maps. So $e^i\circ \widetilde{f}_{j,t_k}(t_j,\cdot )= f_{j,t_k}(t_j,\cdot )\circ e^i$. From continuity, we see that $\widetilde{f}_{j,t_k}(t_j,\cdot )$ is also $\mathcal{F}^j_{t_j}\times \mathcal{F}^k_{t_k}$-measurable, and from (5.2) we have

$$\begin{aligned} \widetilde{f}_{1,t_2}(t_1,\cdot )\circ \widetilde{f}_2(t_2,\cdot )=\widetilde{f}_{2,t_1}(t_2,\cdot )\circ \widetilde{f}_1(t_1,\cdot ). \end{aligned}$$

(5.3)

Define ${{\mathrm{m}}}$ on $\mathcal D$ by ${{\mathrm{m}}}(t_1,t_2)={{\mathrm{dcap}}}(L_{1}(t_1)\vee L_{2}(t_2))$. From (5.2) we get

$$\begin{aligned} {{\mathrm{m}}}(t_1,t_2)=u_{1,t_2}(t_1)+t_2=u_{2,t_1}(t_2)+t_1. \end{aligned}$$

(5.4)

Apply the argument in the first paragraph of Sect. 4.2 with $\lambda =\lambda _j$, $L_{t_j}=L_j(t_j)$, $W=f_k(t_k,\cdot )$, and $\widetilde{W}=\widetilde{f}_k(t_k,\cdot )$, where $t_k\in [0,T_k)$ is fixed. Then we have correspondence: $L^*_{t_j}=L_{j,t_k}(t_j)$, $u=u_{j,t_k}$, and $\widetilde{f}^*_{t_j}=\widetilde{f}_{j,t_k}(t_j,\cdot )$. Since $\widetilde{W}_{t_j}$ is an analytic extension of $\widetilde{f}^*_{t_j}\circ \widetilde{W}\circ \widetilde{f}_{t_j}^{-1}$, from (5.3), we find that $\widetilde{W}_{t_j}=\widetilde{f}_{k,t_j}(t_k,\cdot )$. Thus, $e^{i\lambda _j(t_j)}$ (resp. $\lambda _j(t_j)$) lies in the domain of $f_{k,t_j}(t_k,\cdot )$ (resp. $\widetilde{f}_{k,t_j}(t_k,\cdot )$) as long as $(t_1,t_2)\in \mathcal D$.

Write $\widetilde{F}_{k,t_k}(t_j,\cdot )=\widetilde{f}_{k,t_j}(t_k,\cdot )$. We will use $\partial _t$ to denote the partial derivative w.r.t. the first variable inside the parentheses, and use $'$ and the superscript $(h)$ to denote the partial derivatives w.r.t. the second variable inside the parentheses. For $h=0,1,2,3$, define $A_{j,h}$ on $\mathcal D$ by

$$\begin{aligned} A_{j,h}(t_1,t_2)=\widetilde{f}_{k,t_j}^{(h)}(t_k,\lambda _j(t_j))=\widetilde{F}_{k,t_k}^{(h)}(t_j,\lambda _j(t_j)). \end{aligned}$$

(5.5)

Use $Sf$ to denote the (partial) Schwarzian derivative of $f$. Define $A_{j,S}$ on $\mathcal D$ by

$$\begin{aligned} A_{j,S}(t_1,t_2)=S\widetilde{f}_{k,t_j}(t_k,\lambda _j(t_j))=S\widetilde{F}_{k,t_k}(t_j,\lambda _j(t_j)) \end{aligned}$$

(5.6)

From Sect. 4.2, we know that $L_{j,t_k}(t_j)$ are backward radial Loewner hulls via the time change $u_{j,t_k}$ driven by $\lambda _{j,t_k}$, which can be chosen such that

$$\begin{aligned} \lambda _{j,t_k}(t_j)=A_{j,0}(t_1,t_2). \end{aligned}$$

(5.7)

Moreover, from (4.10), (4.13), and (4.14), we have

$$\begin{aligned} u_{j,t_k}'(t_j)&= A_{j,1} ^2,\end{aligned}$$

(5.8)

$$\begin{aligned} \partial _{t} \widetilde{F}_{k,t_k}(t_j,\lambda _j(t_j))&= 3A_{j,2},\end{aligned}$$

(5.9)

$$\begin{aligned} \frac{ \partial _{t} \widetilde{F}_{k,t_k}'(t_j,\lambda _j(t_j))}{\widetilde{F}_{k,t_k}'(t_j,\lambda _j(t_j))}&= -\frac{1}{2}\Big (\frac{A_{j,2}}{A_{j,1}}\Big )^2 +\frac{4}{3} \frac{A_{j,3}}{A_{j,1}} +\frac{1}{6}A_{j,1}^2-\frac{1}{6}, \end{aligned}$$

(5.10)

where all $A_{j,h}$ are valued at $(t_1,t_2)$.

From now on, we fix an $(\mathcal{F}^k_t)$-stopping time $t_k$ with $t_k<T_k$. Then the process of conformal maps $(\widetilde{F}_{k,t_k}(t_j,\cdot ))$ is $(\mathcal{F}^j_{t_j}\times \mathcal{F}^k_{t_k})_{t_j\ge 0}$-adapted. Let $T_j(t_k)$ be the maximal number such that for any $t_j<T_j(t_k)$, we have $(t_1,t_2)\in \mathcal D$. Then $T_j(t_k)$ is an $(\mathcal{F}^j_{t_j}\times \mathcal{F}^k_{t_k})_{t_j\ge 0}$-stopping time. Recall that $(\lambda _j(t))$ is an $(\mathcal{F}^j_{t_j})$-adapted local martingale with $\langle \lambda _j\rangle _t =\kappa t$. From now on, we will apply Itô’s formula repeatedly. All SDEs below are $(\mathcal{F}^j_{t_j}\times \mathcal{F}^k_{t_k})_{t_j\ge 0}$-adapted, and $t_j$ runs in the interval $[0,T_j(t_k))$.

From (5.7), (5.5), and (5.9), we get

$$\begin{aligned} d\lambda _{j,t_k}(t_j)=A_{j,1} d\lambda _j(t_j)+\Big (\frac{\kappa }{2}+3\Big ) A_{j,2}dt,\quad 0\le t<T_j(t_k). \end{aligned}$$

(5.11)

From (5.5) and (5.10) we get

$$\begin{aligned} \frac{\partial _{t_j} A_{j,h}}{A_{j,h}}=\frac{A_{j,2}}{A_{j,1}}d\lambda _j +\left[ -\frac{1}{2}\left( \frac{A_{j,2}}{A_{j,1}}\right) ^2 +\left( \frac{\kappa }{2}+\frac{4}{3}\right) \frac{A_{j,3}}{A_{j,1}}+\frac{1}{6}A_{j,1}^2-\frac{1}{6}\right] dt_j. \end{aligned}$$

(5.12)

Let

$$\begin{aligned} \alpha =\frac{6-(-\kappa )}{2(-\kappa )},\quad {{\mathrm{c}}}=\frac{(8-3(-\kappa ))(-\kappa -6)}{2(-\kappa )}. \end{aligned}$$

Note that if $-\kappa$ is replaced by $\kappa$, then ${{\mathrm{c}}}$ becomes the central charge for forward SLE$_\kappa$, which runs in the interval $(-\infty , 1]$. The ${{\mathrm{c}}}$ here falls in the interval $[25,\infty )$. Since $A_{j,S}=\frac{A_{j,3}}{A_{j,1}}-\frac{3}{2}(\frac{A_{j,2}}{A_{j,1}})^2$, from (5.12) we get

$$\begin{aligned} \frac{\partial _{t_j} A_{j,1}^{\alpha }}{ A_{j,1}^{\alpha }}=\alpha \frac{A_{j,2}}{A_{j,1}}d \lambda _j+ \left[ -\frac{{{\mathrm{c}}}}{6} A_{j,S} +\frac{\alpha }{6} A_{j,1}^2 - \frac{\alpha }{6}\right] dt_j. \end{aligned}$$

(5.13)

Now we study $\partial _{t_j} A_{k,h}$ and $\partial _{t_j} A_{k,S}$. From (5.5) we have $A_{k,h}(t_1,t_2)=\widetilde{F}_{j,t_j}^{(h)}(t_k,\lambda _k(t_k))$. Recall that $\widetilde{F}_{j,t_j}(t_k,\cdot )=\widetilde{f}_{j,t_k}(t_j,\cdot )$, and $\widetilde{f}_{j,t_k}(t_j,\cdot )$ are backward covering radial Loewner maps via the time change $u_{j,t_k}$ driven by $\lambda _{j,t_k}$. From (5.7) and (5.8), we get

$$\begin{aligned} \partial _t \widetilde{f}_{j,t_k}(t_j,z)=-A_{j,1}^2\cot _2(\widetilde{f}_{j,t_k}(t_j,z)-A_{j,0}). \end{aligned}$$

(5.14)

Differentiate the above formula w.r.t. $z$, we get

$$\begin{aligned} \frac{\partial _t \widetilde{f}_{j,t_k}'(t_j,z)}{\widetilde{f}_{j,t_k}'(t_j,z)}=-A_{j,1}^2\cot _2'(\widetilde{f}_{j,t_k}(t_j,z)-A_{j,0}) . \end{aligned}$$

(5.15)

Differentiating the above formula w.r.t. $z$, we get

$$\begin{aligned} \partial _t \frac{ \widetilde{f}_{j,t_k}''(t_j,z)}{\widetilde{f}_{j,t_k}'(t_j,z)}=-A_{j,1}^2\cot _2''(\widetilde{f}_{j,t_k}(t_j,z)-A_{j,0})\widetilde{f}_{j,t_k}'(t_j,z). \end{aligned}$$

Since $Sf=(\frac{f''}{f'})'-\frac{1}{2}(\frac{f''}{f'})^2$, from the above formula, we get

$$\begin{aligned} \partial _t S{ \widetilde{f}_{j,t_k}(t_j,z)} =-A_{j,1}^2\cot _2'''(\widetilde{f}_{j,t_k}(t_j,z)-A_{j,0})\widetilde{f}_{j,t_k}'(t_j,z)^2. \end{aligned}$$

(5.16)

Letting $z=\lambda _k(t_k)$ in (5.14), (5.15), and (5.16), we get

$$\begin{aligned} \partial _{t_j} A_{k,0}&= -A_{j,1}^2\cot _2(A_{k,0}-A_{j,0})dt_j;\end{aligned}$$

(5.17)

$$\begin{aligned} \frac{\partial _{t_j} A_{k,1}}{A_{k,1}}&= -A_{j,1}^2\cot _2'(A_{k,0}-A_{j,0})dt_j;\end{aligned}$$

(5.18)

$$\begin{aligned} \partial _{t_j}A_{k,S}&= -A_{j,1}^2A_{k,1}^2\cot _2'''(A_{k,0}-A_{j,0})dt_j. \end{aligned}$$

(5.19)

Define $X_j$ on $\mathcal D$ such that $X_j=A_{j,0}-A_{k,0}$. Then $X_1+X_2=0$. Since $e^{i\lambda _j(t_j)}$ lies in the domain of $f_{k,t_j}(t_k,\cdot )$, $e^{iA_{j,0}}=f_{k,t_j}(t_k,e^{i\lambda _j(t_j)})$ lies in the range of $f_{k,t_j}(t_k,\cdot )$, i.e., $\widehat{\mathbb {C}}{\setminus } L_{k,t_j}(t_k)$. On the other hand, since via a time change, $L_{k,t_j}(t_k)$ are backward radial Loewner hulls driven by $\lambda _{k,t_j}(t_k)=A_{k,0}$, from Lemma 3.4 we have $e^{iA_{k,0}}\in L_{k,t_j}(t_k)$ when $t_k>0$. Thus, $e^{iA_{j,0}}\ne e^{iA_{k,0}}$ if $t_k>0$. Switching $j$ and $k$, the inequality also holds if $t_j>0$. If $t_j=t_k=0$, then $e^{iA_{j,0}}=e^{i\widetilde{z}_j}\ne e^{i\widetilde{z}_k}=e^{iA_{k,0}}$. Thus, $X_j,X_k\not \in 2\pi \mathbb {Z}$. So we may define

$$\begin{aligned} Y=|\sin _2(X_1)|^{-2\alpha }=|\sin _2(X_2)|^{-2\alpha }. \end{aligned}$$

From (5.7), (5.11), and (5.17), we get

$$\begin{aligned} \partial _{t_j} X_j=A_{j,1} d\lambda _j+\left( \frac{\kappa }{2}+3\right) A_{j,2}dt-A_{j,1}^2\cot _2(X_j)dt. \end{aligned}$$

From Itô’s formula, we get

$$\begin{aligned} \frac{\partial _{t_j} Y}{Y}&= -\alpha \cot _2(X_j)A_{j,1}d\lambda _j-\alpha \left( \frac{\kappa }{2}+3\right) A_{j,2}\cot _2(X_j)dt_j\nonumber \\&-\frac{\alpha }{2}A_{j,1}^2\cot _2^2(X_j)dt_j+\frac{\alpha \kappa }{4}A_{j,1}^2dt_j. \end{aligned}$$

(5.20)

Define $Q$ and $F$ on $\mathcal D$ such that $Q=\cot _2'''(X_1)=\cot _2'''(X_2)$ and

$$\begin{aligned} F(t_1,t_2)=\exp \left( \int _{0}^{t_2}\!\int _0^{t_1} A_{1,1}(s_1,s_2)^2A_{2,1}(s_1,s_2)^2Q(s_1,s_2)ds_1ds_2\right) . \end{aligned}$$

(5.21)

Since $S\widetilde{F}_{k,t_k}(0,\cdot )={{\mathrm{id}}}$, from (5.6) we have $A_{j,S}=0$ when $t_j=0$. From (5.19) we get

$$\begin{aligned} \frac{\partial _{t_j} F}{F}=-A_{j,S}dt_j. \end{aligned}$$

(5.22)

Define a positive function $\widehat{M}$ on $\mathcal D$ by

$$\begin{aligned} \widehat{M}=A_{1,1}^\alpha A_{2,1}^\alpha Y F^{-\frac{{{\mathrm{c}}}}{6}}e^{\frac{{{\mathrm{c}}}}{12} {{\mathrm{m}}}}. \end{aligned}$$

(5.23)

From (5.4), (5.8), (5.13), (5.18), (5.20), and (5.22), we have

$$\begin{aligned} \frac{\partial _{t_j}\widehat{M}}{\widehat{M}}=\alpha \frac{A_{j,2}}{A_{j,1}}d \lambda _j-\alpha \cot _2(X_j)A_{j,1}d\lambda _j- \frac{\alpha }{6} dt_j. \end{aligned}$$

(5.24)

When $t_k=0$, we have $A_{j,1}=1$, $A_{j,2}=0$, ${{\mathrm{m}}}=t_j$, and $X_j=\lambda _j(t_j)-\widetilde{f}_{j}(t_j,\widetilde{z}_k)$, so the RHS of (5.24) becomes

$$\begin{aligned} \frac{1}{\kappa }\left( \frac{\kappa }{2}+3\right) \cot _2(\lambda _j(t_j)-\widetilde{f}_{j}(t_j,\widetilde{z}_k))d\lambda _j- \frac{\alpha }{6}dt_j. \end{aligned}$$

(5.25)

Define another positive function $M$ on $\mathcal D$ by

$$\begin{aligned} M(t_1,t_2)=\frac{\widehat{M}(t_1,t_2) \widehat{M}(0,0)}{\widehat{M}(t_1,0)\widehat{M}(0,t_2)}. \end{aligned}$$

(5.26)

Then $M(\cdot ,0)\equiv M(0,\cdot )\equiv 1$. From (5.1), (5.24), and (5.25), we have

$$\begin{aligned} \frac{\partial _{t_j} M}{ M}&= \Bigg [ -\Bigg (3+\frac{\kappa }{2}\Bigg ) \frac{A_{j,2}}{A_{j,1}}-\frac{-\kappa -6}{2}\cot _2(X_j)A_{j,1}\nonumber \\&+\frac{-\kappa -6}{2}\cot _2(\lambda _j(t_j)-\widetilde{f}_{j}(t_j,\widetilde{z}_k)) \Bigg ] \frac{d B_j(t_j)}{\sqrt{\kappa }}. \end{aligned}$$

(5.27)

So when $t_k\in [0,p)$ is a fixed $(\mathcal{F}^k_t)$-stopping time, $M$ as a function in $t_j$ is an $(\mathcal{F}^j_{t_j}\times \mathcal{F}^k_{t_k})_{t_j\ge 0}$-local martingale.

5.2 Coupling measures

Let ${{\mathrm{JP}}}$ denote the set of disjoint pairs of closed arcs $(J_1,J_2)$ on $\mathbb {T}$ such that $z_j=e^{i\widetilde{z}_j}$ is contained in the interior of $J_j$, $j=1,2$. Let $T_j(J_j)$ denote the first time that $S_{L_j(t)}$ intersects $\overline{\mathbb {T}{\setminus } J_j}$. Then for every $(J_1,J_2)\in {{\mathrm{JP}}}$, if $t_j\le T_{j}(J_j)$, then $S_{L_j(t_j)}\subset J_j$, which implies that $L_j(t_j)\in \mathcal{H}_{J_j}$. So $[0,T_{1}(J_1)]\times [0,T_{2}(J_2)]\subset \mathcal D$.

Proposition 5.3

(Boundedness) For any $(J_1,J_2)\in {{\mathrm{JP}}}$, $|\ln (M)|$ is uniformly bounded on $[0,T_1(J_1)]\times [0,T_2(J_2)]$ by a constant depending only on $J_1$ and $J_2$.

Proof

Fix $(J_1,J_2)\in {{\mathrm{JP}}}$. In this proof, all constants depend only on $(J_1,J_2)$, and we say a function is uniformly bounded if its values on $[0,T_1(J_1)]\times [0,T_2(J_2)]$ are bounded in absolute value by a constant. From (5.23) and (5.26), it suffices to show that $\ln (A_{1,1})$, $\ln (A_{2,1})$, $\ln (Y)$, $\ln (F)$, and ${{\mathrm{m}}}$ are all uniformly bounded.

Note that if $t_j\le T_j(J_j)$, then $L_j(t_j)\in \mathcal{H}_{J_j}$. From a radial version of Theorem 2.16 (iii), we have

$$\begin{aligned} \{L_1(t_1)\vee L_2(t_2):t_j\in [0, T_j(J_j)],\,j=1,2\}\subset \mathcal{H}_{J_1\cup J_2}. \end{aligned}$$

(5.28)

Since $J_1\cup J_2\subsetneqq \mathbb {T}$, from Lemma 10.2, the righthand side is a compact set. So the lefthand side is relatively compact. Since $H\mapsto {{\mathrm{dcap}}}(H)$ is continuous, and ${{\mathrm{m}}}(t_1,t_2)={{\mathrm{dcap}}}(L_1(t_1)\vee L_2(t_2))$, we see that ${{\mathrm{m}}}$ is uniformly bounded. For $j=1,2$, since $t_j\le {{\mathrm{m}}}$, $T_j(J_j)$ is also uniformly bounded.

Let $S_1$ and $S_2$ be the two components of $\mathbb {T}{\setminus }(J_1\cup J_2)$. For $s=1,2$, let $E_s\subset S_s$ be a compact arc. From Lemma 10.3, $L_n\rightarrow L$ in $\mathcal{H}_{J_1\cup J_2}$ implies that $f_{L_n}\mathop {\longrightarrow }\limits ^\mathrm{l.u.}f_L$ in $\mathbb {C}{\setminus } (J_1\cup J_2)$, which then implies that $f_{L_n}'\mathop {\longrightarrow }\limits ^\mathrm{l.u.}f_L'$ in $\mathbb {C}{\setminus } (J_1\cup J_2)$. From (5.28), the compactness of $\mathcal{H}_{J_1\cup J_2}$, and that $E_1\cup E_2$ are compact subsets of $\mathbb {C}{\setminus }(J_1\cup J_2)$, we conclude that there is a constant $c_1>0$ such that $|f_{L_1(t_1)\vee L_2(t_2)}'(z)|\ge c$ for any $t_j\le T_j(J_j)$, $j=1,2$, and $z\in E_1\cup E_2$. Thus, for $t_j\in [0, T_j(J_j)]$, $j=1,2$, the length of $f_{L_1(t_1)\vee L_2(t_2)}(E_s)$, $s=1,2$, is bounded below by a constant $c_2>0$. Suppose $t_j\in (0,T_j(t_j)]$, $j=1,2$. From Lemma 3.4, $e^{iA_{j,0}}\in B_{L_{j,t_{3-j}}(t_j)}$, $j=1,2$. Note that $f_{L_1(t_1)\vee L_2(t_2)}(E_1\cup E_2)$ disconnects $B_{L_{1,t_2}(t_1)}$ from $B_{L_{2,t_1}(t_2)}$ on $\mathbb {T}$. Thus, there is a constant $c_3>0$ such that $|e^{iA_{1,0}(t_1,t_2)}-e^{iA_{2,0}(t_1,t_2)}|\ge c_3$ for $t_j\in (0,T_j(t_j)]$, $j=1,2$. From continuity, this still holds if $t_j\in [0, T_j(J_j)]$, $j=1,2$. Thus, $\ln (Y)=-2\alpha \ln |\sin _2(X_j)|$, $|\cot _2'(X_j)|$, and $|\cot _2'''(X_j)|$, $j=1,2$, are all uniformly bounded.

We may find a Jordan curve $\sigma$, which is disjoint from $J_1\cup J_2$, such that its interior contains $J_1$ and its exterior contains $J_2$. From compactness, $\sup _{z\in \sigma }\ln |f_j'(t_j,z)|$ and $\sup _{z\in \sigma } \ln |f_{L_1(t_1)\vee L_2(t_2)}'(z)|$ are both uniformly bounded. From (5.2) we see that the value $\sup _{w\in f_j(t_j,\sigma )} \ln |f_{3-j,t_j}'(t_k,w)|$ is also uniformly bounded. Note that the interior of $f_j(t_j,\sigma )$ contains ${L_j(t_j)}^{{{\mathrm{db}}}}$, which contains $e^{i\lambda _j(t_j)}$ if $t_j>0$. From maximum principle, there is $c_4\in (0,\infty )$ such that $A_{j,1}(t_1,t_2)=|f_{3-j,t_j}'(t_{3-j},e^{i\lambda _j(t_j)})|\le c_4$ if $t_j\in (0,T_j(J_j)]$ and $t_{3-j}\in [0,T_{3-j}(J_{3-j})]$. From continuity, $A_{j,1}$ is uniformly bounded, $j=1,2$. From (5.18) and the uniformly boundedness of $|\cot _2'(X_j)|$ we see that $\ln (A_{j,1})$ is uniformly bounded, $j=1,2$. From (5.21) and the uniformly boundedness of $|\cot _2'''(X_j)|$ we see that $\ln (F)$ is also uniformly bounded, which completes the proof. $\square$

Let $\mu _j$ denote the distribution of $(\lambda _j)$, $j=1,2$. Let $\mu =\mu _1\times \mu _2$. Then $\mu$ is the joint distribution of $(\lambda _1)$ and $(\lambda _2)$, since $\lambda _1$ and $\lambda _2$ are independent. Fix $(J_1,J_2)\in {{\mathrm{JP}}}$. From the local martingale property of $M$ and Proposition 5.3, we have $\mathbf{E}\,_\mu [M(T_1(J_1),T_2(J_2))]=M(0,0)=1$. Define $\nu _{J_1,J_2}$ by $d\nu _{J_1,J_2}/d\mu =M(T_1(J_1),T_2(J_2))$. Then $\nu _{J_1,J_2}$ is a probability measure. Let $\nu _1$ and $\nu _2$ be the two marginal measures of $\nu _{J_1,J_2}$. Then $d\nu _1/d\mu _1=M(T_1(J_1),0)=1$ and $d\nu _2/d\mu _2=M(0,T_2(J_2))=1$, so $\nu _j=\mu _j$, $j=1,2$. Suppose temporarily that the joint distribution of $(\lambda _1)$ and $(\lambda _2)$ is $\nu _{J_1,J_2}$ instead of $\mu$. Then the distribution of each $(\lambda _j)$ is still $\mu _j$.

Fix an $(\mathcal{F}^2_t)$-stopping time $t_2\le T_2(J_2)$. From (5.1), (5.27), and Girsanov theorem (c.f. [14]), under the probability measure $\nu _{J_1,J_2}$, there is an $(\mathcal{F}^1_{t_1}\times \mathcal{F}^2_{t_2})_{t_1\ge 0}$-Brownian motion $\widetilde{B}_{1,t_2}(t_1)$ such that $\lambda _1(t_1)$, $0\le t_1\le T_1(J_1)$, satisfies the $(\mathcal{F}^1_{t_1}\times \mathcal{F}^2_{ t_2})_{t_1\ge 0}$-adapted SDE:

$$\begin{aligned} d\lambda _1(t_1)=\sqrt{\kappa }d \widetilde{B}_{1,t_2}(t_1)-\left( 3+\frac{\kappa }{2}\right) \frac{A_{1,2} }{A_{1,1} }dt_1-\frac{-\kappa -6}{2}\cot _2(X_1) A_{1,1} dt_1, \end{aligned}$$

which together with (5.5), (5.7), (5.9), and Itô’s formula, implies that

$$\begin{aligned} d\lambda _{1,t_2}(t_1)=A_{1,1} \sqrt{\kappa }d\widetilde{B}_{1,t_2}(t_1) -\frac{-\kappa -6}{2}\cot _2(X_1)A_{1,1} ^2dt_1. \end{aligned}$$

From (5.5) and (5.7) we get $X_1=A_{1,1}-A_{2,1}=\lambda _{1,t_2}(t_1)-\widetilde{f}_{1,t_2}(t_1,\lambda _2(t_2))$. Note that $\lambda _{1,t_2}(0)=\widetilde{f}_{2,0}(t_2,\widetilde{z}_1)=\widetilde{f}_2(t_2,\widetilde{z}_1)$. Since $L_{1,t_2}(t_1)$ and $\widetilde{f}_{1,t_2}(t_1,\cdot )$ are backward radial Loewner hulls and covering maps via the time change $u_{1,t_2}$, from (5.8) and the above equation, we find that, under the measure $\nu _{J_1,J_2}$, conditioned on $\mathcal{F}^1_{t_2}$ for any $(\mathcal{F}^2_t)$-stopping time $t_2\le T_2(J_2)$, via the time change $u_{1,t_2}$, $L_{1,t_2}(t_1)=f_2(t_2,\cdot )^\mathcal{H}(L_1(t_1))$, $0\le t_1\le T_1(J_1)$, is a partial backward radial SLE$(\kappa ;\frac{-\kappa -6}{2})$ process started from $e^i\circ \widetilde{f}_2(t_2,\widetilde{z}_1)=f_2(t_2,z_1)$ with marked point $e^i(\lambda _2(t_2))$. Similarly, the above statement holds true if the subscripts “$1$” and “$2$” are exchanged.

The joint distribution $\nu _{J_1,J_2}$ is a local coupling such that the desired properties in the statement of Theorem 5.2 holds true up to the stopping times $T_1(J_1)$ and $T_2(J_2)$. Then we can apply the maximum coupling technique developed in [20] to construct a global coupling using the local couplings within different pairs $(J_1,J_2)$. To be more specific, the construction is composed of two steps:

• 1. Prove that for any finite sequence $(J_1^{(k)},J_2^{(k)})$, $1\le k\le n$, in ${{\mathrm{JP}}}$, there is a global coupling, say $\nu ^{(n)}$, of $(L_1(t_1))$ and $(L_2(t_2))$, such that for every $1\le k\le n$, if the two processes in the coupling are stopped at $T_1(J_1^{(k)})$ and $T_2(J_2^{(k)})$, respectively, then we get the joint distribution $\nu _{J_1,J_2}$. Such coupling is obtained by doing operations on the process $M$. From Theorem 4.5 in [21], we see that there is a bounded positive process $M^{(n)}$ defined on $[0,T_1]\times [0,T_2]$ such that

  • (a) When $t_1$ is fixed, $M^{(n)}$ is a martingale in $t_2$, and vice versa.

  • (b) When $t_1$ or $t_2$ equals $0$, $M^{(n)}$ is constant $1$.

  • (c) For every $1\le k\le n$, $M^{(n)}$ agrees with $M$ on $[0,T_1(J_1^{(k)})]\times [0,T_2(J_2^{(k)})]$.

  Then the $\nu ^{(n)}$ is defined by weighting the independent coupling of $(L_1(t_1))$ and $(L_2(t_2))$ by $M^{(n)}(T_1,T_2)$.

• 2. Choose a dense sequence $(J_1^{(k)},J_2^{(k)})$, $k\in \mathbb {N}$, in ${{\mathrm{JP}}}$. For each $n\in \mathbb {N}$, we get a global coupling $\nu ^{(n)}$ using the previous step for the pairs $(J_1^{(k)},J_2^{(k)})$ up to $n$. Then we choose a suitable topology such that the space of coupling measures is tight. Then the desired commutation coupling is any subsequential limit of the sequence $(\nu ^{(n)})$.

The reader is referred to Section 4.3 in [21] for more details of the technique. This finishes the proof of Theorem 5.2 in the radial case.

Now we briefly describe the proof for the chordal case. The proof in this case is simpler because there are no covering maps. Suppose the two backward chordal SLE$(\kappa ;-\kappa -6)$ processes start from $(z_j;z_k)$, where $z_1\ne z_2\in \mathbb {R}$. Formula (5.1) holds with all tildes removed and the function $\cot _2$ replaced by $z\mapsto \frac{2}{z}$. The domain $\mathcal D$ and the $\mathbb {H}$-hulls $L_{1,t_2}(t_1)$ and $L_{2,t_1}(t_2)$ are defined in the same way. Then (5.2) still holds. From Corollary 2.19 (ii), $f_{1,t_2}(t_1,\cdot )$ and $f_{2,t_1}(t_2,\cdot )$ are $\mathcal{F}^1_{t_1}\times \mathcal{F}^2_{t_2}$-measurable. Define ${{\mathrm{m}}}(t_1,t_2)={{\mathrm{hcap}}}(L_1(t_1)\vee L_2(t_2))/2$. Then (5.4) holds with $u_{j,t_k}(t_j):={{\mathrm{hcap}}}(L_{j,t_k}(t_j))/2$.

Now we apply the argument in Sect. 4.1 with $W=f_k(t_k,\cdot )$. Then $W_t=f_{k,t_j}(t_k,\cdot )$. Let $F_{k,t_k}(t_j,\cdot )=f_{k,t_j}(t_k,\cdot )$, and define $A_{j,h}$ and $A_{j,S}$ using (5.5) and (5.6) with all tildes removed. Using (4.3), (4.4), (4.7), and (4.8), we see that (5.7) still holds here; (5.8) and (5.9) hold with all tildes removed; and (5.10) holds without the tildes and the terms $+\frac{1}{6}A_{j,1}^2-\frac{1}{6}$. Then we get the SDEs (5.11) and (5.13) without the terms $+\frac{\alpha }{6} A_{j,1}^2 - \frac{\alpha }{6}$. Formulas (5.17), (5.18), and (5.19) hold with $\cot _2$ replaced by $z\mapsto \frac{2}{z}$. We still define $X_j=A_{j,1}-A_{k,1}$. Then $X_j\ne 0$ in $\mathcal D$. Define $Y$ on $\mathcal D$ by $Y=|X_1|^{-2\alpha }=|X_2|^{-2\alpha }$. Then (5.20) holds with $\cot _2$ replaced by $z\mapsto \frac{2}{z}$ and the term $+\frac{\alpha \kappa }{4}A_{j,1}^2dt_j$ removed. Define $F$ using (5.21) with $Q=-\frac{12}{X_1^4}=-\frac{12}{X_2^4}$. Then (5.22) still holds. Define $\widehat{M}$ using (5.23) without the factor $e^{\frac{{{\mathrm{c}}}}{12} {{\mathrm{m}}}}$. Then (5.24) holds with $\cot _2$ replaced by $z\mapsto \frac{2}{z}$ and the term $- \frac{\alpha }{6} dt_j$ removed. Define $M$ using (5.26). Then (5.27) holds with all tildes removed and $\cot _2$ replaced by $z\mapsto \frac{2}{z}$.

We define ${{\mathrm{JP}}}$ to be the set of disjoint pairs of closed real intervals $(J_1,J_2)$ such that $z_j$ is contained in the interior of $J_j$. Then Proposition 5.3 holds with a similar proof, where Lemma 9.2 is applied here, and we can show that $|X_1|$ is uniformly bounded away from $0$. The argument on the local couplings hold with all tildes and $e^i$ removed and $\cot _2$ replaced by $z\mapsto \frac{2}{z}$. Finally, we may apply the maximum coupling technique to construct a global coupling with the desired properties. This finishes the proof in the chordal case.

5.3 Other results

Besides Theorem 5.2, one may also prove the following two theorems, which are similar to the couplings for forward SLE that appear in [5, 21].

Theorem 5.4

Let $\kappa _1,\kappa _2>0$ satisfy $\kappa _1\kappa _2=16$, and $c_1,\ldots ,c_n\in \mathbb {R}$ satisfy $\sum _{k=1}^n c_k=\frac{3}{2}$. Let $\mathbf {\rho }_j=(\frac{\kappa _j}{2},c_1(-\kappa _j-4),\ldots ,c_n(-\kappa _j-4))$, $j=1,2$. Then backward chordal (resp. radial) SLE$(\kappa _1;\mathbf {\rho }_1)$ commutes with backward chordal (resp. radial) SLE$(\kappa _2;\mathbf {\rho }_2)$.

Theorem 5.5

Let $\kappa >0$ and $\mathbf {\rho }\in \mathbb {R}^n$, whose first coordinate is $2$. Then backward chordal (resp. radial) SLE$(\kappa ;\mathbf {\rho })$ commutes with backward chordal (resp. radial) SLE$(\kappa ;\mathbf {\rho })$.

6 Reversibility of backward chordal SLE

Theorem 6.1

Let $\kappa \in (0,4]$ and $z_1\ne z_2\in \mathbb {T}$. Suppose a backward radial SLE$(\kappa ;-\kappa -6)$ process $(L_1(t))$ started from $(z_1;z_2)$ commutes with a backward radial SLE$(\kappa ;-\kappa -6)$ process $(L_2(t))$ started from $(z_2;z_1)$. Then a.s. they induce the same welding.

Proof

For $j=1,2$, let $S^j_t=S_{L_j(t)}$ and $f^j_t=f_{L_j(t)}$. Let $T_j(\cdot )$, $j=1,2$, be as in Definition 5.1. Let $\phi _j$ be the welding induced by $(L_j(t))$. Since $-\kappa -6\le -\kappa /2-2$, from the last remark in Sect. 4.3, we see that, for $j=1,2$, a.s. $T_j=\infty$, $S^j_\infty =\mathbb {T}{\setminus }\{z_{3-j}\}$, and $\phi _j$ is an involution of $\mathbb {T}$ with exactly two fixed points: $z_1$ and $z_2$.

Fix $t_2>0$. Since $(L_1(t))$ and $(L_2(t))$ commute, the following is true. Conditioned on $(L_2(t))_{t\le t_2}$, $(f^2_{t_2})^\mathcal{H}(L_1(t_1))$, $0\le t_1<T_1(t_2)$, is a partial backward radial SLE$(\kappa ;-\kappa -6)$ process, after a time change, started from $(f^2_{t_2}(z_1);e^{i\lambda _2(t_2)})$, where $\lambda _2$ is a driving function for $(L_2(t_2))$. We have

$$\begin{aligned} S:= \bigcup _{0\le t_1<T_1(t_2)} S_{(f^2_{t_2})^\mathcal{H}(L_1(t_1))}=f^2_{t_2}\left( \bigcup _{0\le t_1<T_1(t_2)} S^1_{t_1})=f^2_{t_2}(S^1_{T_1(t_2)^-}\right) . \end{aligned}$$

(6.1)

Recall that $f^2_{t_2}$ is a homeomorphism from $\mathbb {T}{\setminus } S^2_{t_2}$ onto $\mathbb {T}{\setminus } B_{L_2(t_2)}=\mathbb {T}{\setminus }\{e^{i\lambda _2(t_2)}\}$. From the definition of $T_1(t_2)$, we see that $S^1_{T_1(t_2)}$ intersects $S^2_{t_2}\ne \emptyset$ at one or two end points of both arcs. If they intersect at only one point, then $S^1_{T_1(t_2)^-}$ is a proper subset of $\mathbb {T}{\setminus } S^2_{t_2}$, and these two arcs share an end point. From (6.1), this then implies that the arc $S$ is a proper subset of $\mathbb {T}{\setminus } B_{L_2(t_2)}$, and $B_{L_2(t_2)}$ is an end point of $S$. Recall that, after a time change, $(f^2_{t_2})^\mathcal{H}(L_1(t_1))$, $0\le t_1<T_1(t_2)$, is a partial backward radial SLE$(\kappa ;-\kappa -6)$ process. Since $S\ne \mathbb {T}{\setminus } B_{L_2(t_2)}$, the process is not complete. Then we conclude that $S$ is contained in a closed arc on $\mathbb {T}$ that does not contain $B_{L_2(t_2)}$ because the force point is not swallowed by the process at any finite time, which contradicts that $B_{L_2(t_2)}$ is an end point of $S$. Thus, a.s. $S^1_{T_1(t_2)}$ and $S^2_{t_2}$ share two end points. Since $\phi _j$ swaps the two end points of any $S^j_t$, $j=1,2$, we see that a.s. $\phi _2=\phi _1$ on $\partial _{\mathbb {T}} S^2_{t_2}$. Let $t_2>0$ vary in the set of rational numbers, we see that a.s. $\phi _2=\phi _1$ on $\bigcup _{t\in \mathbb {Q}_{>0}} \partial _{\mathbb {T}} S^2_{t_2}$, which is a dense subset of $\mathbb {T}$. The conclusion follows since $\phi _1$ and $\phi _2$ are continuous. $\square$

We now state the reversibility of backward chordal SLE$_\kappa$ for $\kappa \in (0,4]$ in terms of its welding. Recall that a backward chordal SLE$_\kappa$ welding is an involution of $\widehat{\mathbb {R}}$ with two fixed points: $0$ and $\infty$.

Theorem 6.2

Let $\kappa \in (0,4]$, and $\phi$ be a backward chordal SLE$_\kappa$ welding. Let $h(z)=-1/z$. Then $h\circ \phi \circ h$ has the same distribution as $\phi$.

Proof

Let $(L_1(t))$ and $(L_2(t))$ be commuting backward radial SLE$(\kappa ;-\kappa -6)$ processes as in Theorem 5.2, which induce the weldings $\psi _1$ and $\psi _2$, respectively. The above theorem implies that a.s. $\psi _1=\psi _2$. For $j=1,2$, let $W_j$ be a Möbius transformation that maps $\mathbb {D}$ onto $\mathbb {H}$ such that $W_j(z_j)=0$ and $W_j(z_{3-j})=\infty$, and $W_2=h\circ W_1$. From Corollary 4.8, $K_j(t):=W_j^\mathcal{H}(L_j(t))$, $0\le t<\infty$, is a backward chordal SLE$_\kappa$, after a time change, which then induces backward chordal SLE$_\kappa$ welding $\phi _j$, $j=1,2$. Then $\phi _1$ and $\phi _2$ have the same law as $\phi$. From (4.1), we get $\phi _j=W_j\circ \psi _j\circ W_j^{-1}$, $j=1,2$, which implies that a.s. $\phi _2=h\circ \phi _1\circ h$. The conclusion follows since $\phi _1$ and $\phi _2$ has the same distribution as $\phi$. $\square$

Lemma 6.3

Let $\kappa >0$. Let $f_t$, $0\le t<\infty$, be backward chordal SLE$_\kappa$ maps. Then for every $z_0\in \mathbb {H}$, a.s. (3.4) holds.

Proof

Let $Z_t=f_t(z_0)-\lambda (t)$, $X_t={{\mathrm{Re}}}Z_t$, and $Y_t={{\mathrm{Im}}}Z_t$. Then

$$\begin{aligned} dX_t=-\sqrt{\kappa }dB(t)-\frac{2X_t}{X_t^2+Y_t^2}\,dt,\quad dY_t=\frac{2Y_t}{X_t^2+Y_t^2}\,dt \end{aligned}$$

Let $R_t=|f_t'(z_0)|$. Then $\frac{dR_t}{R_t}=\frac{2(X_t^2-Y_t^2)}{(X_t^2+Y_t^2)^2}dt$. Let $N_t=Y_t/R_t$ and $A_t=X_t/Y_t$. Then

$$\begin{aligned} \frac{dN_t}{N_t}=\frac{4Y_t^2}{(X_t^2+Y_t^2)^2}\,dt,\quad dA_t=-\frac{\sqrt{\kappa }dB(t)}{Y_t}-\frac{4A_t}{X_t^2+Y_t^2}\,dt. \end{aligned}$$

Let $u(t)=\ln (Y_t)$. Then $u'(t)=\frac{2}{X_t^2+Y_t^2}$. Let $T=\sup u([0,\infty ))$ and define $\widehat{N}_s=N_{u^{-1}(s)}$ and $\widehat{A}_s=A_{u^{-1}(s)}$ for $0\le s<T$. Then

$$\begin{aligned} \frac{d\widehat{N}_s}{\widehat{N}_s}=\frac{2}{\widehat{A}_s^2+1}\,ds,\quad d\widehat{A}_s=-\sqrt{1+\widehat{A}_s^2}\sqrt{\kappa /2} d\widehat{B}(s)-2\widehat{A}_sds, \end{aligned}$$

where $\widehat{B}(s)$ is another Brownian motion. We claim that $T=\infty$. Suppose $T<\infty$. Then $\lim _{t\rightarrow \infty } Y(t)=e^T\in \mathbb {R}$. From the SDE for $A_s$, we see that a.s. $\lim _{s\rightarrow T} A_s\in \mathbb {R}$, which implies that $\lim _{t\rightarrow \infty } A_t\in \mathbb {R}$ and $\lim _{t\rightarrow \infty } X_t\in \mathbb {R}$ as $X_t=Y_tA_t$. Then we have a.s. $s'(t)=\frac{2}{X_t^2+Y_t^2}$ tends to a finite positive number as $t\rightarrow \infty$, which contradicts that $T=\sup \{s(t), 0\le t < \infty \}<\infty$. So the claim is proved. Using Itô’s formula, we see that $\widehat{A}_s$, $0\le s < \infty$, is recurrent. Since $(\ln (\widehat{N}_s))'=\frac{2}{\widehat{A}_s^2+1}$, we see that a.s. $\widehat{N}_s\rightarrow \infty$ as $s\rightarrow \infty$. So a.s. $N_t=\frac{{{\mathrm{Im}}}f_t(z_0)}{|f_t'(z_0)|}\rightarrow \infty$ as $t\rightarrow \infty$, i.e., (3.4) holds. $\square$

If $\kappa \in (0,4]$, then since the backward chordal traces are simple, (3.5) holds. From the above lemma and Sect. 3.3, we see that, for $\kappa \in (0,4]$, the backward chordal SLE$_\kappa$ a.s. generates a normalized global backward chordal trace $\beta$, which we call a normalized global backward chordal SLE$_\kappa$ trace. Recall that $\beta (t)$, $0\le t<\infty$, is simple with $\beta (0)=0$, and $i\not \in \beta$; and there is $F_\infty :\mathbb {H}\mathop {\twoheadrightarrow }\limits ^\mathrm{Conf}\mathbb {C}{\setminus }\beta$, whose continuation maps $\mathbb {R}$ onto $\beta$ such that (3.7) holds, and for any $x\in \mathbb {R}$, $F_\infty (x)=F_\infty (\phi (x))\in \beta$. Now we state the reversibility of the backward chordal SLE$_\kappa$ for $\kappa \in (0,4)$ in terms of $\beta$.

Theorem 6.4

Let $\kappa \in (0,4)$, and $\beta$ be a normalized global backward chordal SLE$_\kappa$ trace. Let $h(z)=-1/z$. Then $h(\beta {\setminus }\{0\})$ has the same distribution as $\beta {\setminus }\{0\}$ as random sets.

Proof

For $j=1,2$, let $\phi _j$ be a backward chordal SLE$_\kappa$ welding and $\beta _j$ be the corresponding normalized global trace. Then $\beta _j$ is a simple curve with one end point $0$, and there exists $F_j:\mathbb {H}\mathop {\twoheadrightarrow }\limits ^\mathrm{Conf}\mathbb {C}{\setminus }\beta _j$ such that $F_j(i)=i$, $F_j(0)=0$, and $F_j(x)=F_j(\phi _j(x))$ for $x\in \mathbb {R}$. From Theorem 6.2 we may assume that $\phi _2=h\circ \phi _1\circ h^{-1}$. Now it suffices to show that $h(\beta _2{\setminus }\{0\})=\beta _1{\setminus }\{0\}$.

Define $G=h\circ F_2\circ h\circ F_1^{-1}$. Then $G$ is a conformal map defined on $\mathbb {C}{\setminus }\beta _1$. It has continuation to $\beta _1{\setminus }\{0\}$. In fact, if $z\in \mathbb {C}{\setminus }\beta _1$ and $z\rightarrow z_0\in \beta _1{\setminus }\{0\}$, then $F_1^{-1}(z)\rightarrow \{x,\phi _1(x)\}$ for some $x\in \mathbb {R}{\setminus }\{0\}$, which then implies that $h\circ F_1^{-1}(z)\rightarrow \{h(x),h\circ \phi _1(x)\}$; since $\phi _2\circ h=h\circ \phi _1$, we find that $F_2\circ h\circ F_1^{-1}(z)$ tends to some point on $\beta _2{\setminus } \{0\}$, so $G(z)$ tends to some point on $h(\beta _2{\setminus }\{0\})$. It was proved in [15] that a forward SLE$_\kappa$ trace is the boundary of a Hölder domain. Then the same is true for backward chordal SLE$_\kappa$ traces and the normalized global trace. From the results in [6], we see that $\beta _1{\setminus }\{0\}$ is conformally removable, which means that $G$ extends to a conformal map from $(\mathbb {C}{\setminus }\beta _1)\cup (\beta _1{\setminus }\{0\})=\mathbb {C}{\setminus }\{0\}$ onto $\mathbb {C}{\setminus }\{0\}$, and maps $\beta _1{\setminus }\{0\}$ to $h(\beta _2{\setminus }\{0\})$. Since $G(i)=i$, either $G={{\mathrm{id}}}$ or $G=h$. Suppose $G=h$. Then $F_1=F_2\circ h$. Since $F_1(0)=F_2(0)=0$, for $j=1,2$, $F_j$ maps a neighborhood of $0$ in $\mathbb {H}$ onto a neighborhood of $0$ in $\mathbb {C}$ without a simple curve. Since $F_1=F_2\circ h$, $F_1$ also maps a neighborhood of $\infty$ in $\mathbb {H}$ onto a neighborhood of $0$ without a simple curve, which contradicts the univalent property of $F_1$. Thus, $G={{\mathrm{id}}}$, and we get $h(\beta _2{\setminus }\{0\})=G(\beta _1{\setminus }\{0\})=\beta _1{\setminus }\{0\}$, as desired. $\square$

Now we propose a couple of questions. First, let’s consider backward chordal SLE$_\kappa$ for $\kappa >4$. Since the process does not generate simple backward chordal traces, the random welding $\phi$ can not be defined. However, the lemma below and the discussion in Sect. 3.3 show that we can still define a global backward chordal SLE$_\kappa$ trace.

Lemma 6.5

Let $\kappa \in (0,\infty )$. Suppose $\beta _{t}$, $0\le t<\infty$, are backward chordal traces driven by $\lambda (t)=\sqrt{\kappa }B(t)$. Then a.s. (3.5) holds.

Proof

If $\kappa \in (0,4]$, a.s. the traces are simple, so (3.5) holds. Now suppose $\kappa >4$. Let $f_t$ and $L_t$ be the corresponding maps and hulls. It suffices to show that, for any $t_0>0$, a.s. there exists $t_1>t_0$ such that $\beta _{t_1}([0,t_0])\subset \mathbb {H}$.

Let $g_t$ and $K_t$, $0\le t<\infty$, be the forward chordal Loewner maps and hulls driven by $\sqrt{\kappa }B(t)$. From Theorem 6.1 in [22], for any deterministic time $t_1\in (0,\infty )$, the continuation of $g_{t_1}^{-1}$ a.s. maps the interior of $S_{K_{t_1}}$ into $\mathbb {H}$. From Lemma 3.1 and the property of Brownian motion, we see that, for any $t_1\in (0,\infty )$, $f_{t_1}$ has the same distribution as $\lambda (t_1)+g_{t_1}^{-1}(\cdot -\lambda (t_1))$, which implies that the continuation of $f_{t_1}$ a.s. maps the interior of $S_{L_{t_1}}$ into $\mathbb {H}$.

Since a.s. $\bigcup _{n=1}^\infty S_n=S_\infty =\mathbb {R}\supset \lambda ([0,t_0])$, and $(S_t)$ is an increasing family of intervals, we see that a.s. there is $N\in \mathbb {N}$ such that the interior of $S_N$ contains $\lambda ([0,t_0])$. Let $t_1=N$. Then $f_{t_1}$ maps $\lambda ([0,t_0])$ into $\mathbb {H}$, which implies that $\beta _{t_1}(t)=f_{t_1}(\lambda (t))\in \mathbb {H}$ for $0\le t\le t_0$. $\square$

Question 6.6

Do we have the reversibility of the global backward chordal SLE$_\kappa$ trace for $\kappa >4$?

Second, let’s consider backward radial SLE$_\kappa$ processes. One can show that (3.8) a.s. holds. Since $T=\infty$, we may define a global backward radial SLE$_\kappa$ trace.

Question 6.7

Does a global backward radial SLE$_\kappa$ trace satisfy some reversibility property of any kind?

Recall that the forward radial SLE$_\kappa$ trace does not satisfy the reversibility property in the usual sense. However, it’s proved in [24] that, for $\kappa \in (0,4]$, the whole-plane SLE$_\kappa$, as a close relative of radial SLE$_\kappa$, satisfies reversibility.

Finally, it is worth mentioning the following simple fact. Recall that, if $\kappa \in (0,4]$, a backward radial SLE$_\kappa$ welding is an involution of $\mathbb {T}$ with two fixed points, one of which is $1$. The following theorem gives the distribution of the other fixed point $\zeta$, and says that a backward radial SLE$_\kappa$ process conditioned on $\zeta$ is a backward radial SLE$(\kappa ;-4)$ process with force point $\zeta$. It is similar to Theorem 3.1 in [22], and we omit its proof.

Theorem 6.8

Let $\kappa \in (0,4]$. Let $\mu$ denote the distribution of a backward radial SLE$_\kappa$ process. For $\theta \in (0,2\pi )$, let $\nu _\theta$ denote the distribution of a backward radial SLE$(\kappa ;-4)$ process started from $(1;e^{i\theta })$. Let $f(\theta )=C\sin _2(\theta )^{4/\kappa }$, where $C>0$ is such that $\int _0^{2\pi } f(\theta )d\theta =1$. Then

$$\begin{aligned} \mu =\int _0^{2\pi } \nu _\theta f(\theta )d\theta . \end{aligned}$$

Appendices

Appendices

1.1 Appendix A: Carathéodory topology

Definition 7.1

Let $(D_n)_{n=1}^\infty$ and $D$ be domains in $\mathbb {C}$. We say that $(D_n)$ converges to $D$, and write $D_n\mathop {\longrightarrow }\limits ^\mathrm{Cara}D$, if for every $z\in D$, ${{\mathrm{dist}}}(z,\mathbb {C}{\setminus } D_n)\rightarrow {{\mathrm{dist}}}(z,\mathbb {C}{\setminus } D)$. This is equivalent to the following:

• (i) every compact subset of $D$ is contained in all but finitely many $D_n$’s;

• (ii) for every point $z_0\in \partial D$, there exists $z_n\in \partial D_n$ for each $n$ such that $z_n\rightarrow z_0$.

Remark A sequence of domains may converge to two different domains. For example, let $D_n=\mathbb {C}{\setminus }((-\infty ,n])$. Then $D_n\mathop {\longrightarrow }\limits ^\mathrm{Cara}\mathbb {H}$, and $D_n\mathop {\longrightarrow }\limits ^\mathrm{Cara}-\mathbb {H}$ as well. But two different limit domains of the same domain sequence must be disjoint from each other, because if they have nonempty intersection, then one contains some boundary point of the other, which implies a contradiction.

Lemma 7.2

Suppose $D_n\mathop {\longrightarrow }\limits ^\mathrm{Cara}D$, $f_n:D_n\mathop {\twoheadrightarrow }\limits ^\mathrm{Conf}E_n$, $n\in \mathbb {N}$, and $f_n\mathop {\longrightarrow }\limits ^\mathrm{l.u.}f$ in $D$. Then either $f$ is constant on $D$, or $f$ is a conformal map on $D$. In the latter case, let $E=f(D)$. Then $E_n\mathop {\longrightarrow }\limits ^\mathrm{Cara}E$ and $f_n^{-1}\mathop {\longrightarrow }\limits ^\mathrm{l.u.}f^{-1}$ in $E$.

Remark The above lemma resembles the Carathéodory kernel theorem [13, Theorem 1.8], but the domains here don’t have to be simply connected. The main ingredients in the proof are Rouché’s theorem and Koebe’s $1/4$ theorem. The lemma also holds in the case that $D_n$ and $D$ are domains of any Riemann surface, if the metric in the underlying space is used in place of the Euclidean metric for Definition 7.1 and locally uniformly convergence. In particular, if we use the spherical metric, then Lemma 7.2 holds for domains of $\widehat{\mathbb {C}}$.

1.2 Appendix B: Topology on interior hulls

Let $\mathcal{H}$ denote the set of all interior hulls in $\mathbb {C}$. Recall that for any $H\in \mathcal H$, $\phi _H^{-1}$ is defined on $\{|z|>{{\mathrm{rad}}}(H)\}$, and for a nondegenerate interior hull, $\psi _H(z)=\varphi _H^{-1}(z)=\phi _H^{-1}({{\mathrm{rad}}}(H)z)$ is defined on $\{|z|>1\}$. It’s shown in Section 2.5 of [23] that there is a metric $d_\mathcal{H}$ on $\mathcal H$ such that for any $H_n,H\in \mathcal H$, the followings are equivalent:

• 1. $d_\mathcal{H}(H_n,H)\rightarrow 0$

• 2. ${{\mathrm{rad}}}(H_n)\rightarrow {{\mathrm{rad}}}(H)$ and $\phi _{H_n}^{-1}\mathop {\longrightarrow }\limits ^\mathrm{l.u.}\phi _H^{-1}$ in $\{|z|>{{\mathrm{rad}}}(H)\}$.

• 3. $\mathbb {C}{\setminus } H_n\mathop {\longrightarrow }\limits ^\mathrm{Cara}\mathbb {C}{\setminus } H$.

In particular, we see that ${{\mathrm{rad}}}$ is a continuous function on $(\mathcal{H},d_\mathcal{H})$. Thus, for nondegenerate interior hulls, $d_\mathcal{H}(H_n,H)\rightarrow 0$ iff $\psi _{H_n}\mathop {\longrightarrow }\limits ^\mathrm{l.u.}\psi _H$ in $\{|z|>1\}$. The following lemma is Lemma 2.2 in [23].

Lemma 8.1

For any $F\in \mathcal H$, the set $\{H\in \mathcal{H}:H\subset F\}$ is compact.

Corollary 8.2

For any $F\in \mathcal H$ and $r>0$, the set $\{H\in \mathcal{H}:H\subset F,{{\mathrm{rad}}}(H)\ge r\}$ is compact.

1.3 Appendix C: Topology on $\mathbb {H}$-hulls

From Section 5.2 in [19], there is a metric $d_\mathcal{H}$ on the space of $\mathbb {H}$-hulls such that $d_\mathcal{H}(H_n\rightarrow H_\infty )$ iff $f_{H_n}\mathop {\longrightarrow }\limits ^\mathrm{l.u.}f_{H_\infty }$ in $\mathbb {H}$. From Lemma 7.2, this implies that $\mathbb {H}{\setminus } H_n\mathop {\longrightarrow }\limits ^\mathrm{Cara}\mathbb {H}{\setminus } H_\infty$. But $\mathbb {H}{\setminus } H_n\mathop {\longrightarrow }\limits ^\mathrm{Cara}\mathbb {H}{\setminus } H_\infty$ does not imply $d_\mathcal{H}(H_n\rightarrow H_\infty )$. A counterexample is $H_n=\{z\in \mathbb {H}:|z-2n|\le n\}$ and $H_\infty =\emptyset$. Since $H_1\cdot H_2=H_3$ iff $f_{H_1}\circ f_{H_2}=f_{H_3}$, the dot product is continuous.

Formula (5.1) in [19] states that for any $\mathbb {H}$-hull $H$, there is a positive measure $\mu _H$ supported by $S_H^{{{\mathrm{cv}}}}$, the convex hull of $S_H$, such that for any $z\in \mathbb {C}{\setminus } S_H^{{{\mathrm{cv}}}}$,

$$\begin{aligned} f_H(z)=z+\int \frac{-1}{z-x} d\mu _H(x). \end{aligned}$$

(9.1)

In particular, if $H$ is bounded by a crosscut, then $\mu _H$ is absolutely continuous w.r.t. the Lebesgue measure, and $d\mu _H/dx=\frac{1}{\pi }{{\mathrm{Im}}}f_H(x)$, where the value of $f_H$ on $S_H^{{{\mathrm{cv}}}}$ is the continuation of $f_H$ from $\mathbb {H}$. If $H$ is approximated by a sequence of $\mathbb {H}$-hulls $(H_n)$, then $\mu _H$ is the weak limit of $(\mu _{H_n})$. We may choose each $H_n$ to be bounded by a crosscut, whose height is not bigger than $h+1/n$, where $h$ is the height of $H$. Then each $\mu _{H_n}$ has a density function, whose $L^\infty$ norm is not bigger than $(h+1/n)/\pi$. Thus, $\mu _H$ also has a density function, whose $L^\infty$ norm is not bigger than $h/\pi$. We use $\rho _H$ to denote the density function of $\mu _H$. Since $f_H:\mathbb {C}{\setminus } S_H^{{{\mathrm{cv}}}}\mathop {\twoheadrightarrow }\limits ^\mathrm{Conf}\mathbb {C}{\setminus } H^{{{\mathrm{db}}},{{\mathrm{cv}}}}$ and $f_H'(\infty )=1$, we see that ${{\mathrm{rad}}}(H^{{{\mathrm{db}}},{{\mathrm{cv}}}})={{\mathrm{rad}}}(S_H^{{{\mathrm{cv}}}})=|S_H^{{{\mathrm{cv}}}}|/4$. Thus, ${{\mathrm{diam}}}(H^{{{\mathrm{db}}},{{\mathrm{cv}}}})\le 4{{\mathrm{rad}}}(H^{{{\mathrm{db}}},{{\mathrm{cv}}}})=|S_H^{{{\mathrm{cv}}}}|$. On the other hand, the diameter of $H^{{{\mathrm{db}}},{{\mathrm{cv}}}}$ is at least twice the height of $H$. So $\Vert \rho _H\Vert _{\infty }\le \frac{|S_H^{{{\mathrm{cv}}}}|}{2\pi }$.

By approximating any $\mathbb {H}$-hull $H$ using a sequence of $\mathbb {H}$-hulls $(H_n)$, each of which is the union of finitely many mutually disjoint $\mathbb {H}$-hulls bounded by crosscuts in $\mathbb {H}$, we see that $\mu _H$ is in fact supported by $S_H$. By continuation, (9.1) holds for any $z\in \mathbb {C}{\setminus } S_H$. Furthermore, the support of $\mu _H$ is exactly $S_H$ because from (9.1) $f_H$ extends analytically to the complement of the support of $\mu _H$, while from Lemma 2.6 $f_H$ can not be extended analytically beyond $\mathbb {C}{\setminus } S_H$. So we obtain the following lemma.

Lemma 9.1

For any $\mathbb {H}$-hull $H$, $\mu _H$ has a density function $\rho _H$, whose support is $S_H$, and whose $L^\infty$ norm is no more than $\frac{|S_H^{{{\mathrm{cv}}}}|}{2\pi }$. Moreover, (9.1) holds for any $z\in \mathbb {C}{\setminus } S_H$.

The following lemma extends Lemma 5.4 in [19], and we now give a proof.

Lemma 9.2

For any compact $F\subset \mathbb {R}$, $\mathcal{H}_F:=\{H:S_H\subset F\}$ is compact, and $H_n\rightarrow H$ in $\mathcal{H}_F$ implies that $f_{H_n}\mathop {\longrightarrow }\limits ^\mathrm{l.u.}f_H$ in $\mathbb {C}{\setminus } F$.

Proof

Suppose $(H_n)$ is a sequence in $\mathcal{H}_F$. Let $|F^{{{\mathrm{cv}}}}|$ denote the length of the convex hull of $F$. Then for each $n$, $\rho _{H_n}$ is supported by $S_{H_n}\subset F$, and the $L^\infty$ norm of $\rho _{H_n}$ is no more than $\frac{|S_{H_n}^{{{\mathrm{cv}}}}|}{2\pi }\le \frac{|F^{{{\mathrm{cv}}}}|}{2\pi }$. Thus, $(\rho _{H_n})$ contains a subsequence $(\rho _{H_{n_k}})$, which converges in the weak-$*$ topology to a function $\rho$ supported by $F$. From (9.1) we see that $f_{H_{n_k}}$ converges uniformly on each compact subset of $\mathbb {C}{\setminus } F$, and if $f$ is the limit function, then $f(z)-z=\int _F \frac{-1}{z-x} \rho (x)dx$, $z\in \mathbb {C}{\setminus } F$. So $f(z)-z\rightarrow 0$ as $z\rightarrow \infty$. This means that $f$ can not be constant. From Lemma 7.2, $f$ is a conformal map on $\mathbb {C}{\setminus } F$. Since $f(z)-z\rightarrow 0$ as $z\rightarrow \infty$, $\infty$ is a simple pole of $f$. Thus, $f(\mathbb {C}{\setminus } F)$ contains a neighborhood of $\infty$. Let $G=\mathbb {C}{\setminus } f(\mathbb {C}{\setminus } F)$. Then $G$ is compact. Since every $f_{H_{n_k}}$ is $\mathbb {R}$-symmetric, so is $f$. Let $H=G\cap \mathbb {H}$. Then $f$ maps $\mathbb {H}$ conformally onto $\mathbb {H}{\setminus } H$. This implies that $H$ is an $\mathbb {H}$-hull and $f=f_H$ on $\mathbb {H}$ because $f(z)-z\rightarrow 0$ as $z\rightarrow \infty$. Since $f$ extends $f_H|_\mathbb {H}$, from Lemma 2.6, we see that $S_H\subset F$ and $f=f_H$ in $\mathbb {C}{\setminus } F$. Since $f_{H_{n_k}}\mathop {\longrightarrow }\limits ^\mathrm{l.u.}f_H$ in $\mathbb {H}$, we get $H_{n_k}\rightarrow H\in \mathcal{H}_F$. This shows that $\mathcal{H}_F$ is compact. The above argument also gives $f_{H_{n_k}}\mathop {\longrightarrow }\limits ^\mathrm{l.u.}f_H$ in $\mathbb {C}{\setminus } F$. If $H_n\rightarrow H$, then any subsequence $(H_{n_k})$ of $(H_n)$ contains a subsequence $(H_{n_{k_l}})$ such that $f_{H_{n_{k_l}}}\mathop {\longrightarrow }\limits ^\mathrm{l.u.}f_H$ in $\mathbb {C}{\setminus } F$, which implies that $f_{H_n}\mathop {\longrightarrow }\limits ^\mathrm{l.u.}f_H$ in $\mathbb {C}{\setminus } F$. $\square$

1.4 Appendix D: Topology on $\mathbb {D}$-hulls

Define a metric $d_\mathcal{H}$ on the space of $\mathbb {D}$-hulls such that

$$\begin{aligned} d_\mathcal{H}(H_1,H_2)=\sum _{n=1}^\infty \frac{1}{2^n}\sup _{|z|\le 1-1/n}\{|f_{H_1}(z)-f_{H_2}(z)|\}. \end{aligned}$$

(10.1)

It is clear that $d_\mathcal{H}(H_n,H)\rightarrow 0$ iff $f_{H_n}\mathop {\longrightarrow }\limits ^\mathrm{l.u.}f_H$ in $\mathbb {D}$. From Lemma 7.2, this implies that $\mathbb {D}{\setminus } H_n\mathop {\longrightarrow }\limits ^\mathrm{Cara}\mathbb {D}{\setminus } H$. On the other hand, from Lemma 10.1 below, one see that $\mathbb {D}{\setminus } H_n\mathop {\longrightarrow }\limits ^\mathrm{Cara}\mathbb {D}{\setminus } H$ also implies that $H_n\rightarrow H$. Since $f_{H_n}\mathop {\longrightarrow }\limits ^\mathrm{l.u.}f_H$ in $\mathbb {D}$ implies that $f_{H_n}'(0)\rightarrow f_H'(0)$, we see that ${{\mathrm{dcap}}}$ is a continuous function. Moreover, the dot product is also continuous.

Lemma 10.1

For any $M<\infty$, $\{H:{{\mathrm{dcap}}}(H)\le M\}$ is compact.

Proof

Suppose $(H_n)$ is a sequence of $\mathbb {D}$-hulls with ${{\mathrm{dcap}}}(H_n)\le M$ for each $n$. Then $f_{H_n}'(0)=e^{-{{\mathrm{dcap}}}(H_n)}\ge e^{-M}$. Since $(f_{H_n})$ is uniformly bounded in $\mathbb {D}$, it contains a subsequence $(f_{H_{n_k}})$, which converges locally uniformly in $\mathbb {D}$. Let $f$ be the limit. Then $f'(0)=\lim _{k\rightarrow \infty } f_{H_{n_k}}'(0)\ge e^{-M}$. Thus, $f$ is not constant. From Lemma 7.2, $f$ is conformal in $\mathbb {D}$. Since $f(0)=\lim _{k\rightarrow \infty } f_{H_{n_k}}(0)$ and $f'(0)>0$, we see that $f=f_H|_\mathbb {D}$ for some $\mathbb {D}$-hull $H$. Since $f'(0)\ge e^{-M}$, ${{\mathrm{dcap}}}(H)\le M$. From $f_{H_{n_k}}\mathop {\longrightarrow }\limits ^\mathrm{l.u.}f_H$ in $\mathbb {D}$ we get $H_{n_k}\rightarrow H$. $\square$

Remark We may compactify the space of $\mathbb {D}$-hulls by adding one element $H_\infty$ with the associated function $f_{H_\infty }\equiv 0$ in $\mathbb {D}$, and defining the metric $d_\mathcal{H}$ in the extended space using (10.1).

Lemma 10.2

For any compact $F\subsetneqq \mathbb {T}$, $\mathcal{H}_F:=\{H:S_H\subset F\}$ is compact.

Proof

Let $H\in \mathcal{H}_F$. From conformal invariance, the harmonic measure of $\mathbb {T}{\setminus } H^{{{\mathrm{db}}}}$ in $\mathbb {D}{\setminus } H$ seen from $0$ equals to the harmonic measure of $\mathbb {T}{\setminus } S_H$ in $\mathbb {D}$ seen from $0$, which is bounded below by $|\mathbb {T}{\setminus } F|/|\mathbb {T}|>0$. This implies that the distance between $0$ and $H$ is bounded below by a positive constant $r$ depending on $F$, which then implies that ${{\mathrm{dcap}}}(H)$ is bounded above by $-\ln (r)<\infty$. From Lemma 10.1, we see that $\mathcal{H}_F$ is relatively compact.

It remains to show that $\mathcal{H}_F$ is bounded. Let $(H_n)$ be a sequence in $\mathcal{H}_F$, which converges to $H$. We need to show that $H\in \mathcal{H}_F$. Since $H_{n}\in \mathcal{H}_F$, each $f_{H_{n}}$ is analytic in $\widehat{\mathbb {C}}{\setminus } F$. We have $f_{H_{n}}\mathop {\longrightarrow }\limits ^\mathrm{l.u.}f_H$ in $\mathbb {D}$. From $\mathbb {T}$-symmetry, $f_{H_{n}}\mathop {\longrightarrow }\limits ^\mathrm{l.u.}f_H$ in $\mathbb {D}^*$. Let $J=\{|z|=2\}\subset \mathbb {D}^*$. Then $f_{H_{n_k}}\rightarrow f_H$ uniformly on $J$. Sine $f_{H_{n}}$ maps $\{|z|< 2\}{\setminus } F$ into the Jordan domain bounded by $f_{H_{n}}(J)$, we see that the family $(f_{H_{n}})$ is uniformly bounded in $\{|z|< 2\}{\setminus } F$. So it contains a subsequence $(f_{H_{n_{k}}})$, which converges locally uniformly in $\{|z|< 2\}{\setminus } F$. The limit function is analytic in $\{|z|< 2\}{\setminus } F$ and agrees with $f_H$ on $\mathbb {D}$, which implies that $f_H$ extends analytically across $\mathbb {T}{\setminus } F$. So $S_H\subset F$, i.e., $H\in \mathcal{H}_F$. $\square$

There is an integral formula for $\mathbb {D}$-hulls which is similar to (9.1). For any $\mathbb {D}$-hull $H$, there is a positive measure $\mu _H$ with support $S_H$ such that

$$\begin{aligned} f(z)=z\cdot \exp \left( \int _\mathbb {T}-\frac{x+z}{x-z}d\mu _H(x)\right) ,\quad z\in \mathbb {C}{\setminus } S_H, \end{aligned}$$

(10.2)

and $H_n\rightarrow H$ iff $\mu _{H_n}\rightarrow \mu _H$ weakly. Moreover, $\mu _H$ is absolutely continuous w.r.t. the Lebesgue measure on $\mathbb {T}$, and the density function is bounded. From this integral formula, it is easy to get the following lemma.

Lemma 10.3

For any compact $F\subset \mathbb {T}$, $H_n\rightarrow H$ in $\mathcal{H}_F$ implies that $f_{H_n}\mathop {\longrightarrow }\limits ^\mathrm{l.u.}f_H$ in $\mathbb {C}{\setminus } F$.