A Loewner Equation for Infinitely Many Slits

It is well-known that the growth of a slit in the upper half-plane can be encoded via the chordal Loewner equation, which is a differential equation for schlicht functions with a certain normalisation. We prove that a multiple slit Loewner equation can be used to encode the growth of the union Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma $$\end{document} of multiple slits in the upper half-plane if the slits have pairwise disjoint closures. Under certain assumptions on the geometry of Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma $$\end{document}, our approach allows us to derive a Loewner equation for infinitely many slits as well.


Introduction and Main Result
In his celebrated paper [11], Loewner developed a fruitful approach to tackle extremal problems involving schlicht functions f , defined in the unit disk D, with the normalisation f (0) = 0 and f (0) = 1. This led to the so-called (radial) Loewner equation. A similar theory has been established by Kufarev et al. (cf. [9]) for schlicht functions f , defined on the upper half-plane H, and satisfying the hydrodynamic normalisation: (1.1) This led to the so-called chordal Loewner equation. However, the chordal Loewner equation has received much more attention since O. Schramm wrote his seminal paper [17]. Before going any further and stating our results, we will need some notation. A multislit is a possibly finite sequence of slits j , such that j j is a hull. Given ( j ) j , we let := j j , and also call a multislit. Moreover, if for a multislit , the set clos j can be separated from clos( \ j ) by open sets for each j, then is called admissible (see Fig. 1). If we wish to emphasise that a multislit consists of only finitely many slits, then we speak of an n slit. In what follows, every multislit is assumed to be admissible.
Recently, several authors, in particular mathematical physicists gazing towards conformal field theory, have studied a Loewner equation for multiple slits, to generate growing hulls or n slits; see, for example, [5,8,15,16]. However, the following geometric question has apparently received little attention: for what kind of parametrisations can any given multislit be encoded in a Loewner equation? In the radial case, there are some results for finitely many slits, see [2,3]. In the chordal case, it is, to the best of the authors' knowledge, only known that for n slits, there exists a certain (not effectively computable) parametrisation, such that a generalised Loewner equation is satisfied, see [14,Theorem 1.1]. To state our results, we recall the following well-known fact (cf. [10, p. 69]).  Furthermore, we need the following notation. Definition 1.3 Let be a multislit. We call γ = (γ j ) j a parametrisation of if γ j is a parametrisation of j for every j. By a slight abuse of notation, we let t := j γ j (0, t]. 1 We call a parametrisation γ of a Loewner parametrisation of if t → hcap( t ) is Lipschitz continuous for t ∈ [0, 1]. Corollary 3.3 will show that these "normalised" parametrisations can be achieved to encode a given multislit in a Loewner equation. Given a multislit , we write g t := g t , and denote by h t the inverse of g t . Consequently, we also denote by h the inverse of g . Our main result is the following: j λ j (t) = ∂ ∂t hcap t almost everywhere, such that for almost every t ∈ [0, 1], and all z ∈ H\ it holds that Informally speaking, the weight function λ j (t) corresponds to the "speed" in which γ j (t) grows at the time t ∈ [0, 1], and the driving function U j = g t • γ j keeps track of the position of the tip γ j (t) of the slit γ j (0, t]. Moreover, we would like to mention that the a.e. differentiability part of the theorem above is a consequence of a more general phenomenon occurring in Loewner theory, cf. [ [2,3] for results in the radial case). Moreover, Theorem 1.4 is best possible in the sense that t → g t (z) (z ∈ H\ ) is in general not (everywhere) differentiable, see Remark 5.3.
The paper is structured as follows. First, we collect some basic tools in Sect. 2. These are needed to study the difference quotient of t → g t (z), where g t is the map from Theorem 1.4. To this end, we use classical results from (geometric) function theory, e.g., the theory of prime ends, kernel convergence, normal families, and the Nevanlinna representation. In Sect. 3, we construct the driving functions. In Sect. 4, we construct functions that will later on turn out to be the weight functions. These are the main problems when passing from the 1-slit Loewner equation to the multislit version. For overcoming this obstacle, we use tools from Lipschitz analysis. Eventually, we put the pieces together in Sect. 5 to derive the results stated above.

Preliminaries and basic tools
Let us mention, for the sake of clarity, that we equipĈ with its natural topology; in particular, the boundary of H is understood to contain the point ∞. Moreover, we fix an arbitrary admissible multislit with Loewner parametrisation γ throughout Sects. 2-5. The next theorem deals with biholomorphic extensions of the maps g and h , and is a direct consequence of the well-known Schwarz reflection principle combined with the classical theory of prime ends (cf. [12]). Let us mention that Loewner remarked that a certain group property was essential for his approach to derive his equation. 2 Therefore, similar to Loewner, we shall study The function ϕ t,T is often easier to handle than g t , inter alia, since, as we shall see, it admits a continuous extension to R for t ≤ T . To this end, it is convenient to introduce some notation which we shall use in what follows without further mention.

Definition 2.3
Let be an admissible multislit with parametrisation γ . Take 0 ≤ t ≤ T ≤ 1, and let j ∈ N be given. We define (see Fig. 2) Fig. 2 Similarly coloured symbols are mapped onto each other. For instance, the green-coloured sets C t,T,k are first formed to a slit by h T , and then, the map g t "bites" a piece away from the slit γ j (0, T ] and manipulates the remainder slit γ j (t, T ] to form a silt J t,T,k (see also [6, Fig. 1] (colour figure online) T ] under h T (in the sense of Theorem 2.2, and observe that the normalisation in Proposition 1.2 implies that the point ∞ is not contained in C t,T, j ).
We can deduce the following properties for these quantities.  (1), it suffices to show that J t,T is locally path-connected as ∂ H\J t,T = R ∪ J t,T . Therefore, we only need to show that any given J t,T,k can be separated from clos(J t,T \J t,T,k ) by some neighbourhood U . However, this is evident, since was assumed to be an admissible multislit.
We now show (2). By (1), the map ϕ t,T extends continuously toR. Note that by the path-connectedness of H\J t,T , we can consider a simple curve J − k which connects the tip of a given J t,T,k , i.e., the point g t (γ k (T )), with its starting point, i.e., the point g t (γ k (t)), from the left. 3 The has to be the interval [α, β]. By applying the same reasoning to a curve J + k that connects the tip of J t,T,k with its starting point from the right, we get that the preimage of J t,T,k under ϕ t,T | clos ω + k is an interval of the form β, α . In view of Theorem 2.2, we find that C t,T, j = [α, β] ∪ β, α , so C t,T, j is an interval. Theorem 2.2 yields that these intervals are disjoint.
Furthermore, there is a simple but crucial integral representation for ϕ t,T : we have that Proof (1) is a simple calculation. The first equality in (2) follows from the well-known Nevanlinna representation, cf. [13,Theorem 5.3] or [7], via the Stieltjes inversion formula. The second equality is due to the decomposition expanding the left-hand side and the right-hand side of Lemma 2.5 (2) into Laurent series, and comparing coefficients yields the claim. Now, we are in the position to conclude a simple, but very useful lemma.
for some suitably chosen ξ j , ξ j ∈ C t,T, j . 4 Proof Using Lemma 2.5 (1) and (2), we can write Considering the integral the claim follows after splitting the integrand into real and imaginary parts, and applying the mean value theorem.
Using the previous lemma, we can derive a crucial fact about the differentiability of τ → g τ (z). Namely, we have: Corollary 2.7 Let 0 ≤ t ≤ T ≤ s ≤ 1, and γ be a parametrisation of an admissible multislit . Then Combining the formula above with the estimate Lemma 2.5 yields Im ϕ T,s (z) ≥ Im z, and hence The additional assertion follows from Rademacher's theorem.

Driving Functions
By the previous lemma, there exists such that we can write

Now, it is natural to proceed by showing that both factors in the product
Im ϕ t,T (ξ ) dξ converge as |t − T | tends to 0. In light of this thought, we prove, in the first part of the next section, that C t,T, j tends 5 to some point U j (T ) as t T , or to U j (t) for T t. After that, we turn our attention to the more delicate problem of deciding whether exists for t T , or T t. Moreover, we need the following simple estimate which controls how much g A , for a given hull A, can differ from the identity map z → z.

Lemma 3.1 Let A be a hull. Then, it holds that
In particular, for every 1-slit and every fixed t 0 ∈ [0, 1], the maps g t converge locally uniformly to the identity mapping on H as t → 0.
Proof If 0 ∈ clos(A), then the claim follows from [10, Corollary 3.44]. By taking c ∈ R, such that B := A − c satisfies 0 ∈ clos(B), we can deduce the general case from g B (z) = g A (z + c) − c and g A (z + c) − (z + c) = g B (z) − z.

Theorem 3.2 If T respectively t is fixed, then, for fixed k, there is a δ > 0, such that for all t ∈ [T − δ, T ], respectively, T ∈ [t, t + δ], we can separate C t,T,k from C t,T \C t,T,k by a (fixed) open set for any k.
Proof We consider the case t T .
is becoming smaller as t T , it suffices to separate C t,T,k from C t,T \C t,T,k for some t. Note that we can separate J t,T,k from J t,T \J t,T,k . Hence, by continuity of ϕ t,T , the assertion is clear in the case of t T . In the remaining case, we can separate J t,T,k = g t (γ k (t, T ]), which are getting smaller from J t,T \J t,T,k by a simple curve J . Using Carathéodory's Kernel theorem, 6 we get that the simple curvesJ t,T,k := ϕ −1 t,T • J converge to J , which separates C t,T,k from C t,T \C t,T,k .
The next corollary demonstrates how one can normalise a given parametrisation of a multislit. Proof Let L := hcap . Note that, for R > 0 sufficiently large, one has the representation By Carathéodory's kernel theorem, f is continuous. By assumption, f is strictly increasing, and hence a homoemorphism. Therefore,γ j (t) : 6 By the kernel theorem, we refer to the following statement. Let n ⊂ C denote a sequence of domains. Let X and denote domains, where X is unbounded. If n converges in the kernel sense to a domain , and the sequence f n : X → n of biholomorphisms satisfying (1.1) is locally bounded, then it converges locally uniformly to the unique biholomorphism f : X → with (1.1). This can be proved in the same manner as the ordinary kernel convergence theorem. Now, we can characterise the limit behaviour of C t,T,k and J t,T,k as t T , or T t as follows:

Lemma 3.4 The following statements hold:
(1) C t,T,k shrinks 7 to U k (T ) as t T .
(2): Let J k,t,T denote the union of all J t,T, j with j = k. We consider the function g J k,t,T . Then, we have By Theorem 2.1, the map hJ t,T,k extends to a biholomorphism from C\C t,T,k onto We claim thatJ t,T,k tends to U k (t) as T t. By Lemma 3.5 (1) with ϕ := hJ Carathéodory's kernel theorem yields that g J k,t,T converges locally uniformly on C\(J k,t,T ∪ J * k,t,T ) to the identity mapping on C\ j =k U j (t) as T t, i.e., we have 8 Therefore, (2) is proved if we show that diam(J t,T,k ) converges to zero as T t.
Due to J t,T,k ⊆ C\(J k,t,T ∪ J * k,t,T ) and Carathéodory's kernel theorem, g J k,t,T converges uniformly on J t,T ,k , for some fixed T ∈ (t, 1], to id J t,T ,k . Hence By (3), J t,T,k shrinks to U k (t) as T t, which implies diam(J t,T,k ) → 0 as T t. Therefore, we get diam(J t,T,k ) → 0 as T t. By (3.3), we infer that C t,T,k tends to U k (t).
(4): Lemma 3.5 (2) yields, similarly as above, the inclusioñ and henceJ t,T,k tends to U k (T ) as t T . To conclude that J t,T,k tends to U k (T ) as t T , it is enough to show that for any given sequence t n T , there is a subsequence for which this claim holds. After choosing an appropriate subsequence which we denote again by (t n ) n , the maps g J k,tn ,T tend to some schlicht g on a compact set K containing J t,T,k . Therefore,J t n ,T,k = g J k,tn ,T (J t n ,T,k ) implies by taking n → ∞ that J t n ,T,k has to converge to some point. By arguing similarly, for k = k, we get that for some appropriate subsequence of (t n ) n , all J t n ,T,k converge to points. Hence, g J k,tn ,T converges to id C . This implies that J t n ,T,k tends to U k (T ).
The result above immediately implies the following important corollary. Proof For 0 ≤ t ≤ T ≤ 1, we get U j (t) ∈ C t,T, j and U j (T ) ∈ J t,T, j . Therefore, Lemma 3.4 implies the right continuity and left continuity of t → U j (t), and guarantees, furthermore, that lim t τ U j (t) = U j (τ ) = lim T τ U j (T ).
derivative z →φ t,1 (z) in the "dynamical boundary points" U k (t). However, doing so requires some involved analysis. Theorem 4.1 Let γ be a Loewner parametrisation of an admissible multislit . If τ → ϕ τ,1 (u k ) is differentiable at t on a sequence (u k ) k converging to a point in the set ϕ −1 t,1 (U j (t)), then the limits exist, and are both equal to In particular, λ j is defined for almost every t ∈ [0, 1].
Proof By Corollary 2.7, the second assertion is an immediate consequence of the first one, which we prove in several steps. To simplify the notation, we assume j = 1. The case j = 1 can be treated similarly.
(iv) We let a := lim k→∞ a k ∈ C, and show that By arguing as we did in the Steps (ii) to (iv), simply replacing τ by T , and τ t by T t, we find that Im ϕ τ,t (ξ ) dξ.

The Multislit Equation
Now, we combine the results from the previous sections, and deduce a generalised Loewner equation. Proof Let (z k ) k be a sequence of pairwise distinct elements of H, such that M := {z k : k ∈ N} is dense in H, i.e., clos M = clos H. Then, t → ∂ t ϕ t,s (w) exists for every w ∈ M, and every t ∈ D for some set D ⊆ [0, 1] of full measure. Suppose that t n ∈ [0, 1] tends to t ∈ D as n → ∞, but t n = t for all n. Then n : H −→ C, z −→ ϕ t n ,s (z) − ϕ t,s (z) t n − t is holomorphic. Since ( n (z k )) n is convergent, and n is locally bounded due to Corollary 2.7, n converges locally uniformly to some holomorphic : H → C. Using the identity principle, we see that is independent of the choice of (t n ) n , which yields the claim.
We are now able to prove our main result.
Proof of Theorem 1.4. We work in several steps. We fix s ∈ [0, 1]. By the relation ϕ t,s = g t • h s , it suffices to deduce the Loewner equation for ϕ t,s instead of g t .
Proof Theorem 4.1 yields that (1) implies (2). By carefully reviewing the proof of Theorem 1.4, we see that we could have written τ t instead of τ t. Hence, we conclude that (2) implies (1).

Remark 5.3
One can show, by replacing hcap by the logarithmic mapping radius lmr, and arguing along the lines of [3], that the weight function λ k of an admissible multislit exists in t if and only if the map is differentiable at 0, and in this case, X k,t (0) = λ k (t). In view of this and since one can vary γ k suitably, it is clear that Theorem 1.4 is best possible in the sense that one cannot get a Loewner equation for all t for an arbitrary Loewner parametrisation.