1 Introduction

Given a surface M, there are many interesting questions with regard to representing a given conformal structure by a Riemannian metric.

A classical question in this context, for \(M=S^2\) known as Nirenberg’s problem, asks what functions can occur as Gauss curvatures of such metrics on closed surfaces, and over the past decades this problem has been studied by many different authors, we refer in particular to [1, 6, 13, 24, 25] as well as the more recent work of [2, 7] and the references therein for an overview of existing results. We also note that the corresponding problem on surfaces with boundary was investigated in [8] .

Another classical problem in this context, but of a quite different flavour, is to ask how to ‘best’ represent a given conformal structure by a Riemannian metric. For closed surfaces this problem is addressed by the classical uniformisation theorem that allows us to represent every conformal structure by a (unique for genus at least 2) metric of constant Gauss curvature \(K_g\equiv 1,0,-1\), while for complete surfaces this problem was addressed by Mazzeo and Taylor in [15]. On surfaces with boundary, Osgood, Philips and Sarnak introduced in [17] two different notions of uniformisation, with uniform metrics of type I characterised by having constant Gauss curvature and geodesic boundary curves, while uniform metrics of type II are flat and have boundary curves of constant geodesic curvature. The corresponding heat flows were analysed by Brendle in [3], who proved that these flows admit global solutions which converge to the corresponding uniform metric in the given conformal class. As observed by Brendle in [4], for the two different types of uniform metrics introduced in [17], only one of the terms on the left-hand side of the Gauss–Bonnet formula

$$\begin{aligned} \int _M K dv_g+\int _{{\partial M}} k_g dS_g=2\pi \chi (M) \end{aligned}$$

gives a contribution and so the two types of uniform metrics can be seen the opposite ends of a whole family of metrics for which all terms in the above formula have the same sign. Brendle [4] proved also in this more general setting that solutions of the corresponding heat flows exist for all times and converge, now to metrics with \(K_g\equiv \bar{K}\) and \(k_g\equiv \bar{k}\), where the signs of \(\bar{K}\) and \(\bar{k}\) both agree with the sign of \(\chi (M)\). We note that the same restriction on the signs of the curvatures is also present in the work of Cherrier [8].

Here we propose an alternative way of representing conformal structures on surfaces of general type with boundary which is motivated by applications to geometric flows, such as Teichmüller harmonic map flow [19, 23] or Ricci-harmonic map flow [16], in which the surface (M, g) plays the role of a time dependent domain on which a further PDE is solved. For this purpose the described ways of uniformisation on surfaces with boundary suffer the serious drawback that a degeneration of the conformal structure, which can occur even for curves of metrics with finite length, can lead to a degeneration of the metric near the boundary curves, with boundary curves turning into punctures in the limit, so that the very set on which the boundary condition is imposed can be lost.

To resolve this problem, we propose to represent conformal classes on surfaces of general type instead by hyperbolic metrics for which each boundary curve is a curve of positive constant geodesic curvature, chosen so that each of the boundary curves gives a fixed positive contribution to the Gauss–Bonnet formula. As we shall see below, this alternative approach has the advantage that the resulting metrics will remain well controlled near the boundary even if the conformal class degenerates in a way that would cause the boundary curves of the corresponding uniform metrics of type I or II to collapse. The existence of a unique representative of each conformal class with these desired properties is ensured by our first main result.

Theorem 1.1

Let M be a compact oriented surface of genus \(\gamma \) with boundary curves \(\Gamma _1,..,\Gamma _k\) and negative Euler characteristic and let \(d>0\) be any fixed number. Then for any conformal structure \(\texttt {c} \) on M, there exists a unique hyperbolic metric g compatible with \(\texttt {c} \) for which

$$\begin{aligned} k_g\vert _{\Gamma _i}\cdot L_g(\Gamma _i)\equiv d \text { on } \Gamma _i \text { for every } i=1\ldots k. \end{aligned}$$
(1.1)

Denoting by \({\mathcal M}_{-1}^d\) the set of all hyperbolic metrics on M satisfying (1.1), we furthermore have that for every \(g\in {\mathcal M}_{-1}^d\) and every \(i\in \{1,\ldots ,k\}\), there exists a unique simple closed geodesic \(\gamma _i\) in the interior of M which is homotopic to \(\Gamma _i\) and this geodesic is surrounded by a collar neighbourhood \(\mathcal {C}(\gamma _i)\) that is described in Lemma 3.2 and its length is related to the length of the corresponding boundary curve by

$$\begin{aligned} L_g(\Gamma _i)^2-L_g(\gamma _i)^2=d^2. \end{aligned}$$
(1.2)

For hyperbolic surfaces with boundary curves of constant geodesic curvature \(k_g\), the relation (1.2) between the lengths of a boundary curve and the corresponding geodesic is equivalent to (1.1) and we note that (1.1) implies that the area of the region enclosed by \(\Gamma _i\) and \(\gamma _i\) is always equal to d. We remark that the quantity (1.2) appears also naturally if one studies horizontal curves of metrics on closed surfaces, that is curves of hyperbolic metrics which move \(L^2\) orthogonally to the action of diffeomorphisms, as for such curves, the analogue of (1.2) is valid at the infinitesimal level, see Remark 3.4 for details.

While the uniform metrics of type I could be viewed as the extremal case \(d=0\) of \({\mathcal M}_{-1}^d\), the resulting compactifications of the moduli space are very different. For our class of metrics, the analogue of the Deligne–Mumford compactness theorem takes the following form.

Theorem 1.2

Let M be a compact oriented surface of genus \(\gamma \) with boundary curves \(\Gamma _1,..,\Gamma _k\) and negative Euler characteristic, let \({\mathcal M}_{-1}^d\), \(d>0\), be the set of all hyperbolic metrics on M for which (1.1) is satisfied and suppose that \(g^{(j)}\) is a sequence in \({\mathcal M}_{-1}^d\) for which the lengths of the boundary curves are bounded above uniformly.

Then, after passing to a subsequence, \((M,g^{(j)})\) converges to a complete hyperbolic surface \((\Sigma ,g_\infty )\) with the same number of boundary curves, all satisfying (1.1), where \(\Sigma \) is obtained from M by removing a collection \(\mathscr {E}=\{\sigma ^{j}, j=1,\ldots ,\kappa \}\) of \(\kappa \in \{0,\ldots ,3(\gamma -1)+2k\}\) pairwise disjoint homotopically non-trivial simple closed curves in the interior of M and the convergence is to be understood as follows:

For each j, there exists a collection \(\mathscr {E}^{(j)}=\{\sigma _{i}^{(j)}, i=1,\ldots ,\kappa \}\) of pairwise disjoint simple closed geodesics in \((M,g^{(j)})\) of length \(L_{g^{(j)}}(\sigma ^{(j)}_{i}) \rightarrow 0\text { as }j \rightarrow \infty \) and a diffeomorphism \(f_j:\Sigma \rightarrow M\setminus \cup _{i=1}^\kappa \sigma ^{(j)}_{i} \) such that

$$\begin{aligned} f_j^*g^{(j)} \rightarrow g^\infty \text { smoothly locally on }\Sigma . \end{aligned}$$

The above results assure that the metrics are well controlled near the boundary even if the conformal structure degenerates, see also Remark 3.3, and that in particular no boundary curve can be ‘lost’. Both of these properties are crucial in applications where (M, g) plays the role of the domain of a PDE with prescribed boundary conditions, e.g. if one wants to extend ideas of Teichmüller harmonic map flow, introduced for maps from tori by Ding, Li and Liu in [9] and in the joint work [19] of Topping and the author for maps from general closed surfaces, to the setting of maps from general surfaces with boundary in order to flow to solutions of the Douglas–Plateau problem.

In particular, if one hopes to prove global existence results, as obtained for closed surfaces in [20] and [21] for Teichmüller harmonic map flow and in [5] for Ricci-harmonic map flow, for geometric flows on surfaces with boundary, it is important that the most delicate region for the PDE, i.e. the boundary region, and the most delicate region for the evolution of the domain metric, which for hyperbolic metrics are sets of small injectivity radius, do not overlap but are instead far apart as is the case for our class of metrics \({\mathcal M}_{-1}^d\).

This paper is organised as follows. In Sect. 2 we consider the problem of finding hyperbolic metrics in a given conformal class with prescribed positive geodesic curvatures \(k_g\vert _{\Gamma _i}=c_i\) and analyse the properties of such metrics. The main difficulty here lies in the fact that for \(c_i>0\) , the boundary condition has the wrong sign to apply known existence results as found, e.g. in [8] and the corresponding variational problems contain negative boundary terms that have to be analysed carefully. Based on the results and estimates proven in Sect. 2, we will then give the proofs of the main results in Sect. 3.

2 Hyperbolic Surfaces with Boundary Curves of Prescribed Positive Curvature

In this section we prove the existence and uniqueness of hyperbolic metrics for which the boundary curves have prescribed positive constant geodesic curvature and establish several key properties of these metrics, which will be the basis of the proofs of our main results given in Sect. 3. We show in particular.

Lemma 2.1

Let M be an oriented surface with boundary \(\partial M=\bigcup _{i=1}^k \Gamma _i\) with \(\chi (M)=2(1-\gamma )-k<0\) and let \(\texttt {c} \) be any conformal structure on M. Then for any \(c=(c_1,..,c_k)\in [0,1)^k\), there exists a unique metric \(g_c\) on M compatible with \(\texttt {c} \) so that

$$\begin{aligned} \left\{ \begin{array}{rll} K_{g_c}&{}=-1 &{}\text { in } M\\ k_{g_c}&{}=c_i &{} \text { on } \Gamma _i, \quad i=1,..,k. \end{array} \right. \end{aligned}$$
(2.1)

This result is of course true also for \(c_i<0\), and in that case is indeed easier to prove as the boundary term in the corresponding variational integral has the right sign. We are, however, not interested in the properties of representatives with \(c_i\le 0\) as their boundary curves can collapse if the conformal structure degenerates, the very feature of the existing approaches of uniformisation that we want to avoid with our construction.

We recall that under a conformal change \(g=e^{2u}g_0\) the Gauss curvature transforms by

$$\begin{aligned} K_g=e^{-2u}(K_{g_0}-\Delta _{g_0}u) \end{aligned}$$
(2.2)

while, denoting by \(n_{g_0}\) the outer unit normal of \((M,g_0)\), the geodesic curvature \(k_g\) is characterised by

$$\begin{aligned} \frac{\partial u}{\partial n_{g_0}}+k_{g_0}=k_g\cdot e^{u}. \end{aligned}$$
(2.3)

In the following we let \(g_0\) be the unique metric so that \((M,g _0)\) is hyperbolic with geodesic boundary curves (which can, e.g. be obtained by doubling the surface and applying the classical uniformisation theorem), and write for short \(n=n_{g_0}\).

Thus \(g=e^{2u}g_0\) satisfies (2.1) if and only if

$$\begin{aligned} \left\{ \begin{array}{lll} -\Delta _{g_0}u&{}=1-e^{2u} &{}\text { in } M\\ \frac{\partial u}{\partial n}&{}=c_i e^u &{} \text { on } \Gamma _i, \quad i=1,..,k. \end{array} \right. \end{aligned}$$
(2.4)

Lemma 2.1 is hence an immediate consequence of the following more refined result on solutions of the above PDE that we will prove in the present section.

Proposition 2.2

Let \((M,g_0)\) be an oriented hyperbolic surface with geodesic boundary curves \(\Gamma _1,\ldots , \Gamma _k\). Then for any \(c=(c_1,\ldots ,c_k)\in [0,1)^k\) the equation (2.4) has a unique weak solution \(u_c\in H^1(M,g_0)\) and this solution is smooth up to the boundary of M. In addition, the map

$$\begin{aligned}{}[0,1)^k\ni c \mapsto u_c\in H^1(M,g_0) \end{aligned}$$

is of class \(C^1\).

We begin by establishing the existence of solutions to (2.4) based on the direct method of calculus of variations. Solutions of (2.4) correspond to critical points of

$$\begin{aligned} I_c(u)=\int _M\vert d u\vert _{g_0}^2 +e^{2u}-2u dv_{g_0}-\sum _{i} 2c_i\int _{\Gamma _i} e^u dS_{g_0}, \end{aligned}$$
(2.5)

which is well defined on \(H^1(M,g_0)\) as the Moser–Trudinger inequality [26] and its trace versions, see, e.g. [14], ensure in particular that for any \(q<\infty \)

$$\begin{aligned} \sup _{u\in H^1(M,g_0), \Vert u\Vert _{H^1(M,g_0)}\le 1}\int _M e^{q \vert u\vert }dv_{g_0}+\int _{\partial M} e^{q \vert u\vert } dS_{g_0}<\infty . \end{aligned}$$
(2.6)

A well-known consequence of this estimate is that for every \(1<p<\infty \), the maps

$$\begin{aligned} H^1(M,g_0)\ni u{\mapsto } e^u\in L^p(M,g_0) \text { and } H^1(M,g_0)\ni u{\mapsto } \text {tr}_{{\partial M}}(e^u)\in L^p({\partial M},g_0) \end{aligned}$$
(2.7)

are compact operators: Any bounded sequence in \(H^1\) has a subsequence which converges weakly in \(H^1\), strongly in \(L^2\) and whose traces converge strongly in \(L^2\). The corresponding sequences \(e^{u_n}\) and \(\text {tr}_{{\partial M}}(e^{u_n})\) hence converge in measure and, thanks to (2.6) (applied, e.g. for \(q=2p\)), are p-equiintegrable so converge strongly in \(L^p\) by Vitali’s convergence theorem.

An immediate consequence of the compactness of the operators in (2.7) is that \(I_c\) is weakly lower semicontinuous on \(H^1(M,g_0)\). Hence, to establish the existence of a minimiser of \(I_c\) in \(H^1(M,g_0)\), and thus a solution of (2.4), it suffices to prove that \(I_c\) is also coercive on \(H^1(M,g_0)\). To deal with the negative boundary terms, we will use that on hyperbolic surfaces with geodesic boundary curves, the trace theorem is valid in the following form, in particular with leading order term on the right-hand side appearing with a factor of 1.

Lemma 2.3

For any \(\bar{L}<\infty \), there exists a constant \(C_1=C_1(\bar{L})<\infty \) so that the estimate

$$\begin{aligned} \int _{{\partial M}} \vert w\vert dS_{g_0}\le \int _M\vert d w\vert _{g_0} dv_{g_0}+C_1 \int _M \vert w\vert dv_{g_0}\end{aligned}$$
(2.8)

holds true for any oriented hyperbolic surface \((M,g_0)\) with geodesic boundary curves of length \(L_{g_0}(\Gamma _i)\le \bar{L}\), \(i=1,\ldots , k\), and every \(w\in W^{1,1}(M,g_0)\).

Proof of Lemma 2.3

We derive this estimate from the corresponding trace estimate

$$\begin{aligned} \int _{\{0\}\times S^1} \vert w\vert d\theta \le \int _{0}^{X}\int _{S^1} \vert \partial _s w\vert d\theta ds+X^{-1} \int _{0}^{X}\int _{S^1} \vert w\vert d\theta ds \end{aligned}$$
(2.9)

on Euclidean cylinders \([0,X]\times S^1\) and the properties of hyperbolic collars as follows. We first recall that the classical Collar lemma of Keen–Randol [18] yields the existence of pairwise disjoint neighbourhoods \(\mathcal {C}(\Gamma _i)\) of the boundary curves which are isometric to the cylinders \((-X(\ell _i),0]\times S^1\) equipped with \(\rho _{\ell _i}(s)^2 (ds^2+d\theta ^2)\), where \(\ell _i=L_{g_0}(\Gamma _i) \) and where

$$\begin{aligned} \rho _{\ell }(s)=\tfrac{\ell }{2\pi } (\cos (\tfrac{\ell }{2\pi }s))^{-1} \text { and } X(\ell )=\tfrac{2\pi }{\ell }\big (\tfrac{\pi }{2}-\arctan (\sinh (\tfrac{\ell }{2})) \big ), \end{aligned}$$
(2.10)

with the boundary curve \(\Gamma _i\) corresponding to \(\{0\}\times S^1\). We hence obtain from (2.9) that

$$\begin{aligned} \begin{aligned} \int _{\Gamma _i}\vert w\vert dS_{g_0}&= \rho _{\ell _i}(0)\int _{\{0\}\times S^1} \vert w\vert d\theta \le \int _{0}^{X(\ell _i)}\int _{S^1} \rho _{\ell _i}^{-1}\vert \partial _s w\vert \rho _{\ell _i}^2 d\theta ds\\&\quad +\frac{1}{X(\ell _i)\rho _{\ell _i}(0)} \int _{0}^{X(\ell _i)}\int _{S^1} \vert w\vert \rho _{\ell _i}^2d\theta ds\\&\le \int _{\mathcal {C}(\Gamma _i)} \vert d w\vert _{g_0} dv_{g_0}+c_{\bar{L}}^{-1} \int _{\mathcal {C}(\Gamma _i)} \vert w\vert dv_{g_0}, \end{aligned} \end{aligned}$$

for every i, where we use that \(\rho _\ell (s)\ge \rho _\ell (0) \ge \frac{c_{\bar{L}}}{X(\ell )}\) for some \(c_{\bar{L}}>0\) and \(\ell \in (0,\bar{L}]\). As the collars are disjoint, this implies the claim of the lemma. \(\square \)

Returning to the proof of the first part of Proposition 2.2, and hence of the coercivity of \(I_c\) defined in (2.5), we now set \(\bar{c}:=\max \{c_i\}<1\) and apply Lemma 2.3 to bound

$$\begin{aligned} \begin{aligned} \sum _i c_i\int _{\Gamma _i}e^udS_{g_0}&\le \bar{c}\int _{{\partial M}}e^u dS_{g_0}\le \frac{1}{2} \bar{c}\int \vert d u\vert _{g_0}^2+e^{2u} dv_{g_0}+C_1\int e^u dv_{g_0}, \end{aligned} \end{aligned}$$

where all integrals are computed over M unless specified otherwise. Writing \(-2u=2\vert u\vert -4u^+\) for \(u^+=\max \{u,0\}\), we can thus estimate

$$\begin{aligned} \begin{aligned} I_c(u)&\ge (1-\bar{c})\int \vert d u\vert _{g_0}^2+e^{2u} dv_{g_0}+ 2\int \vert u\vert dv_{g_0}-4 \int u^+ dv_{g_0}-2C_1\int e^u dv_{g_0}\\&\ge \tfrac{1}{2}(1-\bar{c})\int \vert d u\vert _{g_0}^2+e^{2u} dv_{g_0}+2\int \vert u\vert dv_{g_0}\\&\quad +\int \tfrac{1}{2}(1-\bar{c})e^{2u^+}-4u^+-2C_1e^{u^+} dv_{g_0}\\&\ge \tfrac{1}{2}(1-\bar{c})\int \vert d u\vert _{g_0}^2+e^{2u} dv_{g_0}+2\int \vert u\vert dv_{g_0}-C \end{aligned} \end{aligned}$$

for a constant C that is allowed to depend on \(\bar{c}\in [0,1)\), \(\chi (M)\), and hence \(\text {Area}(M,g)=-2\pi \chi (M)\), and an upper bound \(\bar{L}\) on the length of the boundary curves of \((M,g_0)\).

Coercivity of \(I_c\) now easily follows: If \(\Vert d u_n\Vert _{L^2(M,g_0)}\rightarrow \infty \) then clearly \(I_c(u_n)\rightarrow \infty \) while for sequences with \(\Vert u_n\Vert _{H^1(M,g_0)}\rightarrow \infty \) and \(\Vert d u_n\Vert _{L^2(M,g_0)}\le C\), the Poincaré inequality implies that also \(\vert \fint u_n dv_{g_0}\vert \rightarrow \infty \), so \(I_c(u_n)\ge 2\int \vert u_n\vert dv_{g_0}-C \ge 2\text {Area}(M,g_0)\cdot \vert \fint u_n dv_{g_0}\vert -C\rightarrow \infty .\)

This establishes the existence of a weak solution \(u_c\in H^1(M,g_0)\) to (2.4) for any \(c\in [0,1)^k\). Since the non-linearity in the Neumann problem (2.4) is subcritical, the regularity theorem [8, Théorème 1] of Cherrier applies and yields that every weak solution of (2.4) is indeed smooth up to the boundary. At the same time, we remark that we could not have used the results of [8] to establish existence of solutions, as our boundary data have the wrong sign.

Remark 2.4

As we only use that the geodesic curvature \(k_g\) is strictly less than 1, the above proof indeed shows that for any given functions \(k_i\in L^p(\Gamma _i)\), \(p>1\), for which \(k_i\le \bar{c}\) for some \(\bar{c}<1\), there exists a hyperbolic metric g compatible to \(\texttt {c} \) with \(k_g=k_i\) on \(\Gamma _i\), \(i=1,\ldots , k\).

We will prove the other claims of Proposition 2.2 at the end of the section based on the properties of the surfaces \((M,g_c)\) that we discuss now, including the following version of the collar lemma.

Lemma 2.5

Let (M, g) be an oriented hyperbolic surface with boundary curves of constant geodesic curvature \(k_g\vert _{\Gamma _i}\equiv c_i\in [0,1)\). Then for each \(i\in \{1,\ldots , k\}\), there exists a unique simple closed geodesic \(\gamma _i\) in (M, g) that is homotopic to \(\Gamma _i\) and there exist pairwise disjoint neighbourhoods \(\mathcal {C}(\Gamma _i)\) of the boundary curves \(\Gamma _i\) in (M, g) which are isometric to cylinders

$$\begin{aligned} (-X(\ell _i),Y(\ell _i,c_i)]\times S^1 \text { with metric } \rho _{\ell _i}(s)^2(ds^2+d\theta ^2) \end{aligned}$$

where \(\rho _{\ell _i}\) and \(X(\ell _i)\) are given by (2.10), \(\ell _i=L_g(\gamma _i)\), and where

$$\begin{aligned} Y(\ell _i,c_i)=\tfrac{2\pi }{\ell _i} \arcsin (c_i). \end{aligned}$$
(2.11)

In these coordinates \(\Gamma _i\) corresponds to \(\{Y(\ell _i,c_i)\}\times S^1\) while \(\gamma _i\) corresponds to \(\{0\}\times S^1\).

Proof

We note that since our surface is hyperbolic, the Dirichlet energy of maps \(u:S^1\rightarrow (M,g)\) has a unique minimiser in the homotopy class of \(\Gamma _i\), c.f. [10], which coincides with \(\Gamma _i\) if \(c_i=0\). Otherwise, \(\Gamma _i\) has positive geodesic curvature so the image of this minimiser must lie in the interior of M and hence be the desired simple closed geodesic.

We let \(\mathcal {C}^+(\Gamma _i)\) be the connected component of \({\mathcal M}_{-1}\setminus \bigcup _i \gamma _i\) that is bounded by \(\gamma _i\) and \(\Gamma _i\) and set \(M_0:= M\setminus \bigcup _i \mathcal {C}^+(\Gamma _i)\). As \((M_0,g)\) is hyperbolic with geodesic boundary, the Collar lemma [18] gives disjoint neighbourhoods \(\mathcal {C}^-(\gamma _i)\) of \(\gamma _i\) in \(M_0\) that are isometric to \(\big ((-X(\ell _i),0]\times S^1,\rho _{\ell _i}(s) (ds^2+d\theta ^2)\big )\), with \(\rho _\ell \) and \(X(\ell )\) given by (2.10). The resulting disjoint neighbourhoods \(\mathcal {C}(\Gamma _i):= \mathcal {C}^-(\gamma _i)\cup \mathcal {C}^+(\Gamma _i)\) of \(\Gamma _i\) in our original surface M are bounded by curves of constant geodesic curvature and are isometric to a subset of the complete hyperbolic cylinder \( \big ((-\tfrac{\pi ^2}{\ell _i}, \tfrac{\pi ^2}{\ell _i})\times S^1, \rho _{\ell _i}^2(ds^2+d\theta ^2)\big )\) around a geodesic of length \(\ell _i\), where such an isometry can, e.g. be obtained by using the fibration of \(\mathcal {C}(\Gamma _i)\) by the geodesics that cross \(\gamma _i\) orthogonally. We note that the only closed curves of constant geodesic curvature in such a cylinder are circles \(\{s\}\times S^1\), whose curvature is

$$\begin{aligned} k_g=\rho ^{-1}\frac{\partial }{\partial s} \log (\rho (s))+k_{g_{eucl}}=\rho ^{-2}\partial _s \rho =\sin \left( \tfrac{\ell }{2\pi }s\right) , \end{aligned}$$
(2.12)

compare (2.3); indeed, comparing the curvature of any other closed curve \(\sigma \) with the one of the circles \(\{s_{\pm }\}\times S^1\) through points \(P_\pm =(s_\pm ,\theta _\pm )\) of \(\sigma \) with extremal s coordinate, we get

$$\begin{aligned} k_g(\sigma )(P_+)\ge k_g(\{s_+\}\times S^1) =\sin \left( \tfrac{\ell }{2\pi }s_+\right) >\sin \left( \tfrac{\ell }{2\pi }s_-\right) \ge k_g(\sigma )(P_-). \end{aligned}$$

The collar neighbourhood \(\mathcal {C}(\Gamma _i)\) obtained above must hence be isometric to a cylinder \(((-X,Y]\times S^1,\rho _{\ell _i}^2(ds^2+d\theta ^2))\) where, by (2.12), X and Y are as described in the lemma. \(\square \)

We also use the following standard property of Riemann surfaces.

Remark 2.6

For any given oriented Riemann surface \((M,\texttt {c} )\) with boundary curves \(\Gamma _1,\ldots \Gamma _k\), there exists a number \(\bar{Z}\) so that the following holds true. Let U be any neighbourhood of one of the boundary curves \(\Gamma _i\) which is conformal to a cylinder \((0,Z]\times S^1\). Then \(Z\le \bar{Z}.\)

For the sake of completeness, we include a short proof of this well-known fact in the appendix. Combining it with Lemma 2.5, we get

Corollary 2.7

For any conformal structure \(\texttt {c} \) on M there exists \(\delta >0\) so that the following holds true. Let g be any hyperbolic metric on M for which \(k_g\vert _{\Gamma _i}\equiv c_i\in [0,1)\), \(i=1\ldots k\), and let \(\gamma _i\) be the geodesics in (M, g) that are homotopic to the boundary curves \(\Gamma _i\). Then

$$\begin{aligned} \ell _i:=L_{g_c}(\gamma _i)\ge \delta \text { and } L_{g_c}(\Gamma _i)=\frac{\ell _i}{\sqrt{1-c_i^2}} \ge \frac{\delta }{\sqrt{1-c_i^2}}, \end{aligned}$$
(2.13)

in particular \(L_{g_c}(\Gamma _i)\rightarrow \infty \) as \(c_i\uparrow 1\).

The bound on \(\ell _i\) follows directly from Lemma 2.5 and Remark 2.6, applied for \(Z= X(\ell _i)\rightarrow \infty \) as \(\ell _i\rightarrow 0\), while the expression for \(L_{g_c}(\Gamma _i)\) follows from (2.10) and (2.11).

For these surfaces, we can now prove the following version of the trace theorem.

Lemma 2.8

Let (M, g) be an oriented hyperbolic surface with boundary curves of constant geodesic curvature \(k_g\vert _{\Gamma _i}\equiv c_i\in [0,1)\) and let \(\mathcal {C}^+(\Gamma _i)\) be the subset of the collar \(\mathcal {C}(\Gamma _i)\) described in Lemma 2.5 that is bounded by \(\Gamma _i\) and the corresponding geodesic \(\gamma _i\). Then

$$\begin{aligned} c_i\int _{\Gamma _i} \vert w\vert dS_g\le \int _{\mathcal {C}^+(\Gamma _i)}\vert w\vert dv_g+c_i\int _{\mathcal {C}^+(\Gamma _i)}\vert dw\vert _g dv_g\end{aligned}$$
(2.14)

holds true for any \(w\in W^{1,1}(M,g)\). Furthermore, there exists \(\varepsilon >0\), allowed to depend on both the lengths \(\ell _i\) of the geodesics \(\gamma _i\) and the curvatures \(c_i\), so that for every \(w\in W^{1,1}(M,g)\)

$$\begin{aligned} \int _{{\partial M}} (k_g+\varepsilon ) \vert w\vert dS_g\le (1-\varepsilon ) \int _M \vert w\vert dv_g+(1-\varepsilon )\int _M \vert dw\vert _g dv_g. \end{aligned}$$
(2.15)

We note that the above lemma assures in particular that if \(w\in H^1(M,g)\), then

$$\begin{aligned} \int _{{\partial M}} (k_g+\varepsilon ) w^2 dS_g\le (1-\varepsilon ) \int _M \vert dw\vert _g^2+2w^2dv_g. \end{aligned}$$
(2.16)

Proof of Lemma 2.8

From Gauss–Bonnet and (2.12) we obtain that for \(s\in [0,Y_i]\), \(Y_i=Y(\ell _i,c_i)\)

$$\begin{aligned} \int _0^s\rho _{\ell _i}^2(x) dx= & {} \tfrac{1}{2\pi }\text {Area}_{g}([0,s]\times S^1)= \tfrac{1}{2\pi }\int _{\{s\}\times S^1} k_gdS_g\\= & {} \rho _{\ell _i}(s)k_g\vert _{{\{s\}\times S^1}}=\rho _{\ell _i}(s) \sin (\tfrac{\ell _i}{2\pi }s), \end{aligned}$$

\((s,\theta )\) collar coordinates on \(\mathcal {C}(\Gamma _i)\), in particular \(\int _0^{Y_i}\rho _{\ell _i}^2=\rho _{\ell _i }(Y_i)\cdot c_i\). Multiplying

$$\begin{aligned} \int _{\{Y_i\}\times S^1}\vert w\vert d\theta =\int _{\{s_0\}\times S^1} \vert w\vert d\theta +\int _{s_0}^{Y_i}\int _{S^1}\partial _s\vert w\vert d\theta ds, \end{aligned}$$
(2.17)

with \(\rho _{\ell _i}(s_0)^2\) and integrating over \(s_0\in [0,Y_i]\) using Fubini hence gives the desired bound of

$$\begin{aligned} \begin{aligned} c_i\int _{\Gamma _i}\vert w\vert dS_g=&\int _{\mathcal {C}^+(\Gamma _i)}\vert w\vert dv_g+\int _0^{Y_i}\int _{S^1} \partial _s \vert w\vert \cdot \rho _{\ell _i}(s)\sin (\tfrac{\ell _i}{2\pi }s) d\theta ds\\ \le&\int _{\mathcal {C}^+(\Gamma _i)}\vert w\vert dv_g+c_i\cdot \int _{\mathcal {C}^+(\Gamma _i)} \vert dw\vert _g dv_g. \end{aligned} \end{aligned}$$
(2.18)

Multiplying (2.17) with \(\rho _{\ell _i}(Y_i)\) and averaging over \(s_0\in (-X_i,0]\), \(X_i=X(\ell _i)\), also yields

$$\begin{aligned} \begin{aligned} \int _{\Gamma _i}\vert w\vert dS_g&\le \rho _{\ell _i}(Y_i)\cdot X_i^{-1} \int _{-X_i}^0\int _{S^1} \vert w\vert d\theta ds+\rho _{\ell _i}(Y_i)\int _{-X_i}^{Y_i}\int _{S^1}\vert \partial _s w\vert d\theta ds\\&\le C_2 \int _{\mathcal {C}(\Gamma _i)\setminus \mathcal {C}^+(\Gamma _i)}\vert w\vert dv_g+C_3 \int _{\mathcal {C}(\Gamma _i)}\vert dw\vert _g dv_g\end{aligned} \end{aligned}$$
(2.19)

now for constants \(C_{2,3}\) that depend both on \(\ell _i=2\pi \min \rho _{\ell _i}(\cdot )\) and \(L_{g}(\Gamma _i)=2\pi \rho _{\ell _i}(Y_i)\). To obtain the second claim of the lemma, we now combine (2.18), multiplied by \((1-\varepsilon )\) for some \(\varepsilon \in (0,1)\) chosen below, and (2.19), multiplied by \((1+c_i)\cdot \varepsilon \), to conclude that

$$\begin{aligned} \begin{aligned} (c_i+\varepsilon )\int _{\Gamma _i}\vert w\vert dS_g&\le (1-\varepsilon ) \int _{\mathcal {C}^+(\Gamma _i)}\vert w\vert dv_g+C_2(1+c_i)\cdot \varepsilon \int _{\mathcal {C}(\Gamma _i)\setminus \mathcal {C}^+(\Gamma _i)}\vert w\vert dv_g\\&\quad +[c_i(1-\varepsilon ) +C_3(1+c_i)\cdot \varepsilon ] \int _{\mathcal {C}(\Gamma _i)}\vert dw\vert _g dv_g. \end{aligned} \end{aligned}$$

For \(\varepsilon >0\) chosen small enough to ensure that \(2C_2\varepsilon \le 1-\varepsilon \) and \(2C_3\varepsilon \le (1-c_i)(1-\varepsilon )\), this yields the second claim (2.15) of the lemma as the collar neighbourhoods are disjoint. \(\square \)

We are now in a position to prove the following a priori bounds for PDEs related to (2.4)

Lemma 2.9

Let M be an oriented surface with boundary curves \(\Gamma _1,\ldots ,\Gamma _k\) and let g be a metric on M which satisfies (2.1) for some \(c\in [0,1)^k\).

Then there exist constants \(C_{4,5}\), allowed to depend both on c and the underlying conformal structure, so that the following holds true for any \(f\in L^2(M,g)\) and \(h\in L^2({\partial M},g)\).

  1. (i)

    Suppose that \(w\in H^1(M,g)\) is a weak solution of

    $$\begin{aligned} \begin{aligned} -\Delta _g w&=1-e^{2w}+f \quad \text { in } M \quad \text { with } \quad \frac{\partial {w}}{\partial n_g}&=k_g (e^w-1)+h \text { on } {\partial M}\quad \end{aligned} \end{aligned}$$
    (2.20)

    for which furthermore \(e^{w}\in H^1(M,g)\). Then

    $$\begin{aligned}&\int _M \vert d u\vert _{g}^2 e^w+(e^w-1)^2 (e^w+1) dv_g+\int _{{\partial M}} (e^w-1)^2 dS_g\nonumber \\&\quad \le C_4\left( \Vert f\Vert _{L^2(M,g)}^2+\Vert h\Vert _{L^2({\partial M}, g)}^2\right) . \end{aligned}$$
    (2.21)
  2. (ii)

    There exists a unique solution \(v\in H^1(M,g)\) of the linearised problem

    $$\begin{aligned} \begin{aligned} -\Delta _g v+2v&=f \quad \text { in } M\quad \text { with } \quad \frac{\partial v}{\partial n_{g}}&=k_g v+h \text { on } {\partial M}\end{aligned} \end{aligned}$$
    (2.22)

    and we have that

    $$\begin{aligned} \Vert v\Vert _{H^1(M,g)}^2 \le C_5(\Vert f\Vert _{L^2(M,g)}^2+\Vert h\Vert _{L^2({\partial M},g)}^2). \end{aligned}$$
    (2.23)

Proof of Lemma 2.9

Let w be as in the first part of the lemma and let \(\varepsilon >0\) be as in Lemma 2.8. Testing (2.20) with \(e^{w}-1\in H^1(M,g)\), we may estimate

$$\begin{aligned} \begin{aligned} I&:= \int \vert d w\vert _{g}^2e^w+(e^w-1)^2(e^w+1) dv_g=\int _{{\partial M}} \frac{\partial w}{\partial n_g}(e^w-1) dS_g\\&\qquad +\int f\cdot (e^w-1) dv_g\\&=\int _{{\partial M}} k_g (e^w-1)^2 +h(e^w-1)dS_g+\int f \cdot (e^w-1) dv_g\\&\le \int _{{\partial M}} (k_g+\varepsilon )(e^w-1)^2 dS_g-\tfrac{\varepsilon }{2}\int _{{\partial M}}(e^w-1)^2 dS_g+\tfrac{\varepsilon }{2}I\\&\qquad + \tfrac{1}{2\varepsilon }[\Vert h\Vert _{L^2({\partial M},g)}^2+\Vert f\Vert _{L^2(M,g)}^2]. \end{aligned} \end{aligned}$$

By (2.15), the first term on the right is bounded by \((1-\varepsilon )\int (1-e^w)^2+2 e^w \vert dw\vert _g \vert 1-e^w\vert dv_g\le (1-\varepsilon )I\), so the first claim (2.21) of the lemma immediately follows.

To prove the second part of the lemma, we use that the variational integral

$$\begin{aligned} F_{f,h}(v):= \int \vert d v\vert _g^2+2v^2+2fv\, dv_g-\int _{{\partial M}}k_gv^2+2vh dS_g \end{aligned}$$

associated with (2.22) is coercive, as (2.16) implies

$$\begin{aligned} \begin{aligned} F_{f,h}(v)&\ge \varepsilon \int \vert d v\vert _g^2+2v^2 dv_g -2\Vert f\Vert _{L^2(M,g)}\cdot \Vert v\Vert _{L^2(M,g)}\\&\quad - \Vert h\Vert _{L^2({\partial M},g)}\Vert v\Vert _{L^2({\partial M},g)}\\&\ge \varepsilon \Vert v\Vert _{H^1(M,g)}^2- (2\Vert f\Vert _{L^2(M,g)}+C\Vert h\Vert _{L^2({\partial M},g)})\Vert v\Vert _{H^1(M,g)}. \end{aligned} \end{aligned}$$
(2.24)

Hence \(F_{f,h}\) has a minimiser v which is of course a solution of (2.22), and satisfies \(F_{f,h}(v)\le F_{f,h}(0)=0\) which, combined with (2.24), furthermore yields the claimed a priori estimate (2.23). Finally, this solution of (2.22) is unique as the difference v of two solutions of (2.22) satisfies

$$\begin{aligned} 0=\int \vert d v\vert _g^2+2v^2 dv_g-\int _{\partial M}k_gv^2 dS_g\ge \varepsilon \Vert v\Vert _{H^1(M,g)}^2, \end{aligned}$$

again by (2.16), and must thus vanish.

As a next step towards completing the proof of Proposition 2.2, we show

Lemma 2.10

Let M be as in Lemma 2.9 and let g be any metric for which (2.1) holds true for some \(c\in [0,1)^k\). Then there exist numbers \(0<\varepsilon _0<1-\max {c_i}\) and \(C_{6}<\infty \) so that for any \(b\in [-\varepsilon _0,\varepsilon _0]^k\) and any hyperbolic metric \(\tilde{g}=e^{2w} g\) with \(k_{\tilde{g}}\vert _{\Gamma _i}=c_i+b_i\), \(i=1,\ldots , k\), we have

$$\begin{aligned} \int _M (e^w+1)(e^w-1)^2+e^w\vert d w\vert _g^2 dv_g + \Vert w\Vert _{H^1(M,g)}^2\le C_6\max \vert b_i\vert ^2. \end{aligned}$$
(2.25)

In particular, the solution of (2.1) is unique.

Proof

Let \(b\in [-\varepsilon _0,\varepsilon _0]^k\), where \(\varepsilon _0>0\) is determined later, set \(\bar{b}=\max \vert b_i\vert \) and suppose that \(\tilde{g}=e^{2w}g\) is as in the lemma. From (2.2) and (2.3), we obtain that w solves

$$\begin{aligned} \begin{aligned} -\Delta _g w&=1-e^{2w} \quad \text { in } M \quad \text { with } \quad \frac{\partial {w}}{\partial n_g}&=k_g (e^w-1)+b_i e^w \text { on } \Gamma _i, \end{aligned} \end{aligned}$$
(2.26)

i.e. satisfies (2.20) for \(f\equiv 0\) and \(h\vert _{\Gamma _i}=b_i e^w\). We note that \(e^{w}\in H^1(M,g)\), as we may characterise \(w=u_{c+b}-u_c\) as difference of smooth solutions of (2.1), so we may bound \(I:= \int _M \vert d w\vert _g^2 e^w+(e^w+1)(e^w-1)^2 dv_g+\int _{{\partial M}}(e^w-1)^2dS_g\) using the first part of Lemma 2.9 by

$$\begin{aligned} \begin{aligned} I&\le C_4 \Vert b e^w\Vert _{L^2({\partial M},g)}^2 \le 2C_4 \bar{b}^2 L_g({\partial M}){+}2C_4 \bar{b}^2\int _{{\partial M}}(e^w-1)^2 dS_g\le C\bar{b}^2+C\varepsilon _0^2 I. \end{aligned} \end{aligned}$$

For \(\varepsilon _0>0\) sufficiently small, this gives the bound \(I\le C\bar{b}^2\) on I claimed in the lemma and it remains to establish the analogue bound on the \(H^1\) norm of w. We first note that

$$\begin{aligned} \Vert e^{w/2}-1\Vert _{H^1(M)}^2= \int \tfrac{1}{4} \vert d w\vert _g^2 e^w+(e^{w/2}-1)^2 dv_g\le I \le C\bar{b}^2, \end{aligned}$$
(2.27)

where norms are computed with respect to g and integrals over M unless indicated otherwise. In particular \(\Vert e^{w/2}\Vert _{H^1(M)}\le C\), and so of course \(\Vert e^{3w/2}\Vert _{L^4(M)}+\Vert e^{w}\Vert _{L^2(\partial M)}\le C\), where all constants are allowed to depend on (M, g) but not on b. Writing (2.26) in the form

$$\begin{aligned} -\Delta _g w= & {} (1-e^{w/2})\cdot (1+e^{w/2}+e^w+e^{3w/2}) \text { on } M,\\ \frac{\partial w}{\partial n_g}= & {} c_i(e^{w/2}-1)(e^{w/2}+1)+b_ie^w \text { on } \Gamma _i, \end{aligned}$$

and testing this equation with \(w-\bar{w}_M\), \(\bar{w}_M:= \fint _M wdv_g\), thus allows us to bound

$$\begin{aligned} \begin{aligned} \Vert d w\Vert _{L^2(M)}^2&\le C\Vert w-\bar{w}_M\Vert _{L^2(M)} \Vert 1-e^{w/2}\Vert _{L^4(M)}\cdot (1+\Vert e^{3u/2}\Vert _{L^{4}(M)})\\&\qquad +\Vert w-\bar{w}_M\Vert _{L^2({\partial M})}\cdot \big [\Vert e^{w/2}-1\Vert _{L^4({\partial M})}\cdot \Vert e^{w/2}+1\Vert _{L^4({\partial M})}\\&\qquad + \bar{b} \Vert e^{w}\Vert _{L^2({\partial M})}\big ]\\&\le C\Vert w-\bar{w}_M\Vert _{H^1(M)}\cdot \big [\Vert e^{w/2}-1\Vert _{H^1(M)}+\bar{b}\big ]\le C \bar{b}\cdot \Vert d w\Vert _{L^2(M)}. \end{aligned} \end{aligned}$$

Having thus shown that \(\Vert dw\Vert _{L^2(M)}\le C\bar{b}\), it now remains to show that also \(\vert \bar{w}_M \vert \le C\bar{b}\), which of course follows if we prove that \(\int \vert w\vert dv_g\le C\bar{b}\). As \(\vert x\vert \le 2e^{-\min (x/2,0)}\vert e^{x/2}-1\vert \), \(x\in {\mathbb R}\), we already obtain from (2.27) that

$$\begin{aligned} \int _{\{w>-4\}} \vert w\vert dv_g\le 2e^2 \int _M\vert e^{w/2}-1\vert dv_g\le C\bar{b}, \end{aligned}$$

so it remains to bound the corresponding integral over \(\{w<-4\}\). To this end we note that as \(\Vert e^{w/2}\Vert _{L^4(M)}\le C\), we obtain from (2.27) that, after reducing \(\varepsilon _0\) if necessary,

$$\begin{aligned} \begin{aligned} \int (e^w-1)^2 dv_g&\le C \Vert e^{w/2}-1\Vert _{H^1(M)}^2 \le C\bar{b}^2 \le C\varepsilon _0^2 \le \frac{1}{2}\text {Area}(M), \end{aligned} \end{aligned}$$

so \(\text {Area}_g(\{w<-3\})\le \text {Area}_g(\{ (e^{w}-1)^2>\tfrac{3}{4}\})\le \tfrac{4}{3}\int (e^w-1)^2dv_g \le \tfrac{2}{3}\text {Area}_g(M).\) Hence \(v=(w+3)_{-}=\max (-(w+3),0)\) vanishes on a set of measure at least \(\alpha =\tfrac{1}{3}\text {Area}_g(M)\), so the variant of the Poincaré inequality

$$\begin{aligned} \Vert v\Vert _{L^2(M)}\le C\Vert d v\Vert _{L^2(M)}, \qquad C=C(\alpha , (M,g)) \end{aligned}$$

valid for such functions v implies that also

$$\begin{aligned} \begin{aligned} \int _{\{w<-4\}} \vert w\vert dv_g\le 4\int _{\{w<-4\}} v dv_g\le C \Vert d v\Vert _{L^2} \le C \Vert d w\Vert _{L^2}\le C\bar{b}, \end{aligned} \end{aligned}$$

which completes the proof of the lemma. \(\square \)

Lemmas 2.9 and 2.10 represent the main steps in the proof of the remaining claims of Proposition  2.2, which now follow by the following standard argument.

Let \(S:c\mapsto u_c\in H^1(M,g_0)\) be the map that assigns to each \(c\in [0,1)^k\) the unique solution \(u_c\) of (2.4). We claim that S is \(C^1\) with \(dS(c)(b)= v_{c,b}\), for \(v_{c,b}\) the unique solutions of

$$\begin{aligned} -\Delta _{g_c} v+2v=0 \text { in } M, \quad \text { with }\quad \frac{\partial v}{\partial n_{g_c}}=c_iv+b_i \text { on } \,\Gamma _i, \,i\in \{1,\ldots ,k\}. \end{aligned}$$
(2.28)

Here and in the following \(g_c=e^{2u_c}g_0\) is the unique metric satisfying (2.1).

Given \(c\in [0,1)^k\), \(b\in {\mathbb R}^k\), say with \(\vert b\vert =1\), and \(\vert \varepsilon \vert \le 1-\max c_i\), we let \(c_\varepsilon =c+\varepsilon b\) and set \(w_{\varepsilon }:=S(c_\varepsilon )-S(c)\). As \(g_{c_\varepsilon }=e^{2S(c_{\varepsilon })}g_0=e^{2w_{\varepsilon }}g_c\) is hyperbolic with \(k_{g_{c_\varepsilon }}=c_\varepsilon \), we have

$$\begin{aligned} -\Delta _{g_c} w_\varepsilon +e^{2w_\varepsilon }-1=0 \text { in } M \quad \text { with } \quad \frac{\partial w_\varepsilon }{\partial n_{g_c}}=(c_i+\varepsilon b_i)e^{w_\varepsilon }-c_i \text { on } {\Gamma _i}, \end{aligned}$$
(2.29)

compare (2.3). Hence \(\beta _\varepsilon :=S(c_\varepsilon )-S(c)-\varepsilon v_{c,b}= w_\varepsilon -\varepsilon v_{c,b}\) solves (2.22) for \(g=g_c\), \(f=1+2w_\varepsilon -e^{2w_\varepsilon }\) and \(h\vert _{\Gamma _i}=c_i(e^{w_\varepsilon }-(1+w_\varepsilon ))+\varepsilon b_i(e^{w_\varepsilon }-1)\), so for functions with

$$\begin{aligned} \vert f\vert \le 2e^{2(w_\varepsilon )_+} \cdot (w_\varepsilon )^2\le & {} 2(1+e^{2w_\varepsilon })\cdot w_\varepsilon ^2 \text { and } \vert h\vert \le \tfrac{1}{2} e^{(w_\varepsilon )_+}w_\varepsilon ^2+\varepsilon e^{(w_\varepsilon )_+}\vert w_\varepsilon \vert \\\le & {} (1+e^{w_\varepsilon })\cdot \big [w_\varepsilon ^2+\tfrac{1}{2}\varepsilon ^2\big ]. \end{aligned}$$

We recall from Lemma 2.10 that the \(H^1\) norms of \(e^{w_\varepsilon }\), \(\vert \varepsilon \vert \le \varepsilon _0\), are uniformly bounded, and hence so are \(\Vert e^{2w_\varepsilon }\Vert _{L^4(M)}\) and \(\Vert e^{w_\varepsilon }\Vert _{L^4({\partial M})}\). Using (2.23) as well as that \(w_\varepsilon =\beta _\varepsilon +\varepsilon v_{c,b}\) we thus get

$$\begin{aligned} \begin{aligned} \Vert \beta _\varepsilon \Vert _{H^1(M,g_c)}&\le C(\Vert f\Vert _{L^2(M,g_c)}+\Vert h\Vert _{L^2({\partial M},g_c)}) \le C \Vert w_\varepsilon ^2\Vert _{L^4(M,g_c)}\\&\quad +C \Vert w_\varepsilon ^2\Vert _{L^4({\partial M},g_c)}+C\varepsilon ^2\le C\Vert \beta _\varepsilon \Vert _{H^1(M,g_c)}^2\\&\quad +C\varepsilon ^2(1{+}\Vert v_{c,b}\Vert _{H^1(M,g_c)}^2) \le C\Vert \beta _\varepsilon \Vert _{H^1(M,g_c)}^2{+}C\varepsilon ^2, \end{aligned} \end{aligned}$$

where we use in the last step that (2.23) yields a bound on the norm of \(v_{c,b}\) that is independent of b. As we know a priori that \(\Vert \beta _\varepsilon \Vert _{H^1(M,g_c)}\le \Vert w_\varepsilon \Vert _{H^1(M,g_c)}+\varepsilon \Vert v_{c,b}\Vert _{H^1(M,g_c)} \le C\varepsilon \), compare Lemma 2.10, we thus conclude that \(\Vert \beta _\varepsilon \Vert _{H^1(M,g_c)}\le C\varepsilon ^2\) and thus that S is indeed Fréchet differentiable in c with \(df(c)(b)=v_{c,b}\) as claimed.

We finally remark that \(v_{c,b}\) depends continuously on c as can be readily seen by using that \(g_{\tilde{c}}=e^{2(S(\tilde{c})-S(c))}\) to view \(v_{\tilde{c},b}\) as solution of (2.22) for \(g=g_c\), \(f= 2v(1-e^{2(S(\tilde{c})-S(c))})\) and \(h\vert _{\Gamma _i}= b_i+ (e^{S(\tilde{c})-S(c)}-1)(c_i v+b_i)+e^{S(\tilde{c})-S(c)} (\tilde{c}_i-c_i)v\) and applying Lemmas 2.9 and 2.10.

3 Proof of the Main Results

Based on the results of Sect. 2, we can now show the first part of Theorem 1.1 by proving

Lemma 3.1

Let \((M,\texttt {c} )\) be a compact oriented Riemann surface with boundary curves \(\Gamma _1,\ldots ,\Gamma _k\) and denote by \(g_c\), \(c\in [0,1)^k\), the unique metric compatible to \(\texttt {c} \) for which (2.1) holds. Then the map \(f:c\mapsto (c_i\cdot L_{g_c}(\Gamma _i))_i\) is a diffeomorphism from \((0,1)^k\) to \( ({\mathbb R}^+)^k\). In particular, for every \(d>0\) there exists a unique hyperbolic metric g that is compatible with \(\texttt {c} \) and that satisfies (1.1).

Proof

We first remark that \(f:c\mapsto (c_i\cdot L_{g_c}(\Gamma _i))_i\) is a \(C^1\) map from \([0,1)^k\) to \(({\mathbb R}^+_0)^k\) as Proposition 2.2 establishes that \(c\mapsto u_c\) is \(C^1\) into \(H^1\), while the trace version of the Moser–Trudinger inequality implies that \(H^1(M,g_0) \ni u\mapsto \int _{\Gamma _i} e^u dS_{g_0}=L_{e^{2u}g_0}(\Gamma _i)\) is \(C^1\).

We now claim that \(f:(0,1)^k\rightarrow ({\mathbb R}^+)^k\) is proper: To see this we first recall that Corollary 2.7 assures that \(L_{g_c}(\Gamma _i)\rightarrow \infty \) as \(c_i\rightarrow 1\) and hence that the preimage \(f^{-1}(K)\) of any compact set \(K\subset ({\mathbb R}^+_0)^k\) is a compact set in \([0,1)^k\). As \(c\mapsto L_{g_c}(\Gamma _i)\) is continuous on \([0,1)^k\) we furthermore have a uniform upper bound on each \(L_{g_c}(\Gamma _i)\) for \(c\in f^{-1}(K)\). For compact subsets K of \(({\mathbb R}^+)^k\) we hence obtain that the components \(c_i\) of \(c\in f^{-1}(K)\) are bounded away from zero uniformly and hence that \(f^{-1}(K)\) is a compact subset of \((0,1)^k\) as required.

By Hadamard’s global inverse function theorem, see, e.g. [12, Chap. 6], the lemma thus follows provided we show that

$$\begin{aligned} \det (df(c))\ne 0 \text { for every } c\in (0,1)^k. \end{aligned}$$

So suppose that there exists \(c\in (0,1)^k\) so that \(\det (df(c))=0\). Hence there must be some non-trivial element b of the kernel of df(c), i.e. \(b\in {\mathbb R}^k\setminus \{0\}\) so that for every \(i=1,\ldots , k\)

$$\begin{aligned} 0=df(c)(b)_i=b_iL_{g_c}(\Gamma _i)+c_i\int _{\Gamma _i} \tfrac{d}{d\varepsilon }\vert _{\varepsilon =0} e^{u_c+\varepsilon v_{c,b}} dS_{g_{0}} =b_iL_{g_c}(\Gamma _i)+c_i\int _{\Gamma _i} v_{c,b} dS_{g_{c}} ,\end{aligned}$$
(3.1)

where \(v_{c,b}=dS(c)(b)\) is characterised by (2.28). Testing (2.28) with \(v_{c,b}\) and applying the trace estimate (2.16) of Lemma 2.8 , however, yields that

$$\begin{aligned} \begin{aligned} \int _M\vert d v_{c,b}\vert _{g_c}^2+2v_{c,b}^2 dv_{g_{c}} =&\int _{{\partial M}} k_{g_c}v_{c,b}^2dS_{g_c} +\sum _i b_i\int _{\Gamma _i} v_{c,b} dS_{g_{c}}\\ \le&(1-\varepsilon ) \int _M\vert d v_{c,b}\vert _{g_c}^2+2v^2 dv_{g_{c}}+ \sum _i b_i\int _{\Gamma _i} v_{c,b} dS_{g_{c}}. \end{aligned} \end{aligned}$$

Since \(v_{c,b}\) cannot vanish identically as \(b\ne 0\), there hence must be at least one \(i\in \{1,\ldots k\}\) with

$$\begin{aligned} \text {sign}(b_i) \int _{\Gamma _i}v_{c,b} dS_{g_{c}}>0 \end{aligned}$$

which contradicts (3.1) as \(c_i>0\). \(\square \)

Having thus proven that each conformal class is represented by a unique metric \(g\in {\mathcal M}_{-1}^d\), we now obtain the remaining claims of Theorem 1.1 from the following lemma which is based on Lemma 2.5 and Corollary 2.7.

Lemma 3.2

Let M be as in Theorem 1.1 and let \(g\in {\mathcal M}_{-1}^d\), \(d>0\). Then there is a unique geodesic \(\gamma _i\) in (M, g) homotopic to the boundary curve \(\Gamma _i\), its length \(\ell _i\) is related to the length of \(\Gamma _i\) by (1.2) and \(\Gamma _i\) is surrounded by a collar neighbourhood that is isometric to

$$\begin{aligned} \big ((-X(\ell _i),\overline{X}_d(\ell _i)]\times S^1, \rho _{\ell _i}(ds^2+d\theta ^2)\big ) \text { where }\overline{X}_d(\ell )=\frac{2\pi }{\ell }\left( \frac{\pi }{2}- \arctan \left( \frac{\ell }{d}\right) \right) \end{aligned}$$
(3.2)

while \(X(\ell )\) and \(\rho _\ell \) are as in (2.10).

Proof

The existence of such a geodesic was proven in Lemma 2.5 and the relation between \(\ell _i=L_g(\gamma _i)\) and \(L_i=L_g(\Gamma _i)\) follows from Corollary 2.7 which implies that \(\ell _i^2=(1-(k_g\vert _{\Gamma _i})^2)L_i^2=L_i^2-d^2.\) From Lemma 2.5 we then obtain that the boundary curve is surrounded by a collar as described in the above lemma where we know that \(\overline{X}_d\) must be so that \(k_g\vert _{\Gamma _i}=\sin (\frac{\ell }{2\pi }\overline{X}_d(\ell _i))\). Combined with (2.13) this yields the condition \(\frac{d}{\ell _i}=\frac{k_g\vert _{\Gamma _i} L_i}{\sqrt{1-(k_g\vert _{\Gamma _i})^2}L_i}=\tan (\frac{\ell }{2\pi } \overline{X}_d(\ell _i))\), so \(\overline{X}_d(\ell _i)\) must be given by (3.2).

\(\square \)

Remark 3.3

We remark that while \(X(\ell )\) and \(\overline{X}_d(\ell )\) have a similar asymptotic behaviour as \(\ell \rightarrow 0\) the behaviour of \(X(\ell )\) and \(\overline{X}_d(\ell )\) as \(\ell \rightarrow \infty \) is very different, with \(X(\ell )\) decaying exponentially, \(X(\ell )\le C\ell ^{-1} e^{-\ell /2}\), while \(\overline{X}_d(\ell )\) is of order \(\ell ^{-2}\) for large \(\ell \). This difference is significant due to its effect on the Teichmüller space and its completion with respect to the corresponding Weil–Petersson metric, which will be discussed in more detail in future work. To illustrate this, we note that for the corresponding metrics \(G_\ell \) on cylinders (which were considered in [23] and are isometric to \(([-\overline{X}_d(\ell ),\overline{X}_d(\ell )]\times S^1,\rho ^2(ds^2+d\theta ^2)\)) the change of the length of the central geodesic is controlled by \(\frac{d\ell }{dt}\le C\ell \Vert \partial _t G_{\ell (t)}\Vert _{L^2}\) if \(\ell \) is large, which excludes the possibility that \(\ell \rightarrow \infty \) along a curve of metrics of finite \(L^2\)-length. In this case we thus know that while the completion of the Teichmüller space includes the punctured limit obtained as \(\ell \rightarrow 0\), it does not include any limiting object corresponding to \(\ell \) becoming unbounded.

Remark 3.4

We note that the quantity \(L_g(\Gamma _i)^2-L_g(\gamma _i)^2\) appears naturally also for horizontal curves of hyperbolic metrics on closed surfaces, as considered in [22]. Such curves move orthogonally to the action of diffeomorphisms and hence satisfy \(\partial _t\hat{g} =\text {Re}(\Omega )\) for holomorphic quadratic differentials \(\Omega \) on \((M,\hat{g})\). Given a simple closed geodesic \(\gamma \subset (M,\hat{g})\) and a closed curve \(\Gamma \subset (\mathcal {C}(\gamma ),\hat{g})\) with constant geodesic curvature, which is hence described by some \(\{s\}\times S^1\) in collar coordinates, we can use the Fourier expansion of \(\Omega =\sum _{j\in {\mathbb Z}} b_j e^{j(s+\text {i}\theta )}dz^2\) to obtain that

$$\begin{aligned} \begin{aligned} \frac{d}{dt} \big [ L_{\hat{g}(t)}(\Gamma )^2-L_{\hat{g}(t)} (\gamma )^2\big ]&= L_{\hat{g}}(\Gamma )\int _{\{s\}\times S^1} \frac{(\partial _t\hat{g})_{\theta \theta }}{\sqrt{\hat{g}_{\theta \theta }}} d\theta -L_{\hat{g}}(\gamma )\int _{\{s\}\times S^1} \frac{(\partial _t\hat{g})_{\theta \theta }}{\sqrt{\hat{g}_{\theta \theta }}} d\theta \\&=- 2\pi \text {Re}\big (\sum _{j} b_j (e^{js}-1) \int _{S^1} e^{\text {i}j \theta } d\theta \big )=0. \end{aligned} \end{aligned}$$

We finally give the proof of the analogue of the Deligne–Mumford compactness result for our class of metrics that we stated in Theorem 1.2. This proof is based on the proof of the corresponding result for surfaces with geodesic boundary curves as carried out in [11, Sect. IV.5]. For part of this proof, it will be more convenient to work with so-called Fermi coordinates \((x,\theta )\) instead of collar coordinates \((s,\theta )\) on a collar \(\mathcal {C}(\sigma )\) around a simple closed geodesic \(\sigma \). As indicated, the angular components of these two different sets of coordinates agree, while the x coordinate of a point \(p\in \mathcal {C}(\sigma )\) is given as the signed distance \(x=x(s)=\text {dist}_g(p,\sigma )\) to \(\sigma \).

Proof of Theorem 1.2

For \((M,g^{(j)})\) as in the theorem we denote by \(\gamma ^{(j)}_i\), \(i=1,\ldots ,k\), the (unique) geodesics in \((M,g^{(j)})\) that are homotopic to \(\Gamma _i\), and note that their lengths are bounded from above by \(\ell _i^{(j)}:=L_{g^{(j)}}(\gamma _i^{(j)})\le L_{g^{(j)}}(\Gamma _i)\le C\). We can thus pass to subsequence so that \(\ell _i^{(j)}\rightarrow \ell _i^\infty \) as \(j\rightarrow \infty \) for each \(i=1,\ldots , k\), where, after relabelling, we may assume that \(\ell _i^\infty =0\) for \(1\le i\le \kappa _1\) while \(\ell _i>0\) for \(\kappa _1+1\le i\le k \) for some \(\kappa _1\in \{0,\ldots , k\}\). As above we let \(\mathcal {C}^+(\Gamma _i,g^{(j)})\), \(i=1,\ldots ,k\), be the half-collars that are bounded by \(\Gamma _i\) and the corresponding geodesic \(\gamma ^{(j)}_i\subset (M,g^{(j)})\) and set \( M^{(j)}:=M\setminus \bigcup _i \mathcal {C}^+(\Gamma ^{(j)}_i)\).

As \((M^{(j)},g^{(j)})\) is a sequence of hyperbolic surfaces with geodesic boundary, we can apply the version of the Deligne–Mumford compactness theorem as found in [11, Prop. 5.1], or alternatively first double the surface and then apply the version of Deligne–Mumford for closed surfaces that is recalled, e.g. in [20, Prop. A.3]. After passing to a subsequence we thus obtain a collection \(\mathscr {E}^{(j)}=\{\sigma _i^{(j)}, i=1,\ldots ,\kappa _2\}\), \(\kappa _2\in \{0,\ldots ,3(\gamma -1)+k\}\), of simple closed geodesics in the interior of \((M^{(j)},g^{(j)})\) whose lengths tend to zero, so that for the surfaces \(\Sigma ^{(j)}= M^{(j)}\setminus \big (\bigcup _{i=1}^{\kappa _2} \sigma _i^{(j)}\cup \bigcup _{i=1}^{\kappa _1} \gamma _i^{(j)}\big )\), which have with \(\kappa _1+2\kappa _2\) punctures, the following holds true: There exists a complete hyperbolic metric \(\hat{g}_\infty \) on \(\hat{\Sigma }:=\Sigma ^{(1)}\) and diffeomorphisms \(\hat{f}_j: \hat{\Sigma }\rightarrow \Sigma ^{(j)}\) so that

$$\begin{aligned} \hat{f}_j^{*}g^{(j)}\rightarrow \hat{g}_\infty \text { smoothly locally on }\hat{\Sigma }= \Sigma ^{(1)} \end{aligned}$$

and so that the diffeomorphisms \(\hat{f}_j\) map \(\gamma _i^{(1)}\) to \(\gamma _i^{(j)}\), \(i\ge \kappa _1+1\), while a neighbourhood of each puncture (respectively, of each pair of punctures) of \((\Sigma ^{(1)},g_\infty )\) obtained by collapsing one the \(\gamma _i^{(1)}\), \( 1\le i \le \kappa _1\) (respectively, one of the \(\sigma _i^{(1)}\)) is mapped to a neighbourhood of the corresponding puncture (respectively, pair of punctures) of \(\Sigma ^{(j)}\).

In addition, for j sufficiently large, we can modify these diffeomorphisms as described in the proof of Claim 3 on p. 75 of [11] to ensure that \(\hat{f}_j: (\hat{\Sigma },g_\infty ) \rightarrow (\Sigma ^{(j)},g^{(j)})\) is given in a neighbourhood of \(\bigcup _{i=\kappa _1+1}^k\gamma ^{(1)}_i\) by the identity in the respective Fermi coordinates.

In slight abuse of notation we now denote by \(\mathcal {C}^+(\ell )\), \(\ell \ge 0\), the unique hyperbolic half-collar which has one boundary curve of constant geodesic curvature and length L, where \(L^2-\ell ^2=d^2\), while the other boundary curve is a geodesic of length \(\ell \) if \(\ell >0\), respectively, degenerated to a hyperbolic cusp if \(\ell =0\). We then construct the limit surface \((\Sigma ,g_\infty )\) out of the limiting surface \((\hat{\Sigma },\hat{g}_\infty )\) with geodesic boundary obtained above and the half-collars \(\mathcal {C}^+(\ell _i^\infty )\), \(i=1,\ldots ,k\), by gluing the non-degenerate half-collars \(\mathcal {C}^+(\ell _i^\infty )\), \(i\ge \kappa _1+1\), to \(\hat{\Sigma }\) along the corresponding non-collapsed boundary curves of \((\hat{\Sigma },\hat{g}_\infty )\), and adding the degenerate collars \(\mathcal {C}^+(\ell _i^\infty )\), \(i\le \kappa _1\), as additional connected components of \((\Sigma ,g_\infty )\). As the connected components of \(M\setminus \bigcup _{i=1}^{\kappa _1}\gamma ^{(j)}_{i}\) are given by \(M\setminus \bigcup _{i=1}^{\kappa _1} \mathcal {C}^+(\ell _i^{(j)})\) and a collection \(\{\mathcal {C}^+(\ell _i^{(j)})\}_{i=1}^{\kappa _1}\) of degenerating half-collars, we can now extend the diffeomorphisms \(\hat{f}_j\) obtained above to the required diffeomorphisms

$$\begin{aligned} f_j:\Sigma \rightarrow M\setminus \bigg (\bigcup _{i=1}^{\kappa _1}\gamma ^{(j)}_{i}\cup \bigcup _{i=1}^{\kappa _2}\sigma ^{(j)}_{i}\bigg ) \end{aligned}$$

as follows: The degenerated connected components \(\mathcal {C}^+(\ell _i^{\infty }=0)\), \(1\le i\le \kappa _1\), which are isometric to \(([0,\infty )\times S^1, \rho _{\ell =0}^2(ds^2+d\theta ^2))\), \(\rho _0(s)=\frac{1}{d+s}\), are mapped to the degenerating half-collars \(\mathcal {C}^+(\ell _i^{(j)})\) in \((M,g^{(j)})\) which are bounded by \(\gamma _i^{(j)}\) and \(\Gamma _i^{(j)}\) with \(f_j\) chosen so that it is given in collar coordinates by a bijection from \([0,\infty )\times S^1\) to \((0,\overline{X}_d(\ell _i^{(j)})]\times S^1\) with \(f_j(s,\theta )=(\overline{X}_d(\ell _i^{(j)})-s,\theta )\) on domains which exhaust \([0,\infty )\times S^1\), say for \(s\in [0,\frac{1}{2}\overline{X}_d(\ell _i^{(j)})]\). As \(\rho _{\ell }(\bar{X}_{d}(\ell _i^{(j)})-\cdot )\rightarrow \rho _{0}(\cdot ) \) locally uniformly on \([0,\infty )\) as \(\ell \rightarrow 0\), this ensures that the pulled back metrics converge on every compact subset of these connected components of the limit surface as required.

Finally we extend \(f_j\) to the collars that we glued to \(\hat{\Sigma }\) as follows: We let \(w_i^{+,\infty }\) and \(w_i^{+,(j)}\), \(i\ge \kappa _1+1\), be the width of the half-collars \(\mathcal {C}^+(\ell _i^{\infty })\) and \( \mathcal {C}^+(\ell _i^{(j)})\), i.e. the geodesic distance between the two boundary curves, and note that \(w_i^{+,(j)}\rightarrow w_i^{+,\infty }\) since \(\ell _i^{(j)}\rightarrow \ell _i^\infty \). We may thus choose smooth bijections \(\phi _i^{(j)}:[0,w_i^\infty ]\rightarrow [0,w_i^{(j)}]\) which agree with the identity in a neighbourhood of 0 and converge to the identity as \(j\rightarrow \infty \). Since \(\hat{f}_j:\hat{\Sigma }\rightarrow \hat{\Sigma }^{(j)}\) is given by the identity in Fermi coordinates near the boundary curves, we may extend \(\hat{f}_j\) to a smooth diffeomorphism \(f_j\) on \(\Sigma \) for which \(f_j^*g^{(j)}\) converges as claimed in Theorem 1.2 by defining \(f_j\) on \(\mathcal {C}^+(\ell _i^{\infty })\) by \(f_j(x,\theta )=(\phi _i^{(j)}(x),\theta )\) in Fermi coordinates. \(\square \)