Hyperbolic Metrics on Surfaces with Boundary

We discuss an alternative approach to the uniformisation problem on surfaces with boundary by representing conformal structures on surfaces M of general type by hyperbolic metrics with boundary curves of constant positive geodesic curvature. In contrast to existing approaches to this problem, the boundary curves of our surfaces (M, g) cannot collapse as the conformal structure degenerates which is important in applications in which (M, g) serves as domain of a PDE with boundary conditions.


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 ≡ 1, 0, −1, while for complete surfaces this B Melanie Rupflin rupflin@maths.ox.ac.uk 1 Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK 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 formulâ 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 ≡K and k g ≡k, where the signs ofK andk both agree with the sign of χ(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. (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 i and γ 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 ( 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 M d −1 . 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 | 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.

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. 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 ≤ 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.

Lemma 2.1 Let M be an oriented surface with boundary
We recall that under a conformal change g = e 2u g 0 the Gauss curvature transforms by while, denoting by n g 0 the outer unit normal of (M, g 0 ), the geodesic curvature k g is characterised by ∂u 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 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.
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 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 < ∞ A well-known consequence of this estimate is that for every 1 < p < ∞, the maps 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 tr ∂ M (e u n ) hence converge in measure and, thanks to (2.6) (applied, e.g. for q = 2 p), 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.
Proof of Lemma 2. 3 We derive this estimate from the corresponding trace estimatê on Euclidean cylinders [0, X ] × 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 C( i ) of the boundary curves which are isometric to the cylinders with the boundary curve i corresponding to {0} × S 1 . We hence obtain from (2.9) thatˆ As the collars are disjoint, this implies the claim of the lemma.
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 setc := max{c i } < 1 and apply Lemma 2.3 to bound where all integrals are computed over M unless specified otherwise. Writing −2u = 2|u| − 4u + for u + = max{u, 0}, we can thus estimate for a constant C that is allowed to depend onc ∈ [0, 1), χ(M), and hence Area(M, g) = −2πχ(M), and an upper boundL on the length of the boundary curves of (M, g 0 ).
Coercivity of I c now easily follows: 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 ∈ L p ( i ), p > 1, for which k i ≤c for somec < 1, there exists a hyperbolic metric g compatible to c with k g = k i on i , i = 1, . . . , 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.
where ρ i and X ( i ) are given by (2.10) In these coordinates i corresponds to Proof We note that since our surface is hyperbolic, the Dirichlet energy of maps u : S 1 → (M, g) has a unique minimiser in the homotopy class of i , c.f. [10], which coincides with i if c i = 0. Otherwise, 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 C + ( i ) be the connected component of M −1 \ i γ i that is bounded by γ i and i and set M 0 := M\ i C + ( i ). As (M 0 , g) is hyperbolic with geodesic boundary, the Collar lemma [18] gives disjoint neighbourhoods C − (γ i ) of γ i in M 0 that are isometric to (−X ( i ), 0]×S 1 , ρ i (s)(ds 2 +dθ 2 ) , with ρ and X ( ) given by (2.10). The resulting disjoint neighbourhoods C( i ) := C − (γ i )∪C + ( i ) of 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 (− π 2 i , π 2 i )× S 1 , ρ 2 i (ds 2 +dθ 2 ) around a geodesic of length i , where such an isometry can, e.g. be obtained by using the fibration of C( i ) by the geodesics that cross γ i orthogonally. We note that the only closed curves of constant geodesic curvature in such a cylinder are circles {s}×S 1 , whose curvature is compare (2.3); indeed, comparing the curvature of any other closed curve σ with the one of the circles {s ± } × S 1 through points P ± = (s ± , θ ± ) of σ with extremal s coordinate, we get The collar neighbourhood C( i ) obtained above must hence be isometric to a cylinder ((−X , Y ] × S 1 , ρ 2 i (ds 2 + dθ 2 )) where, by (2.12), X and Y are as described in the lemma.
We also use the following standard property of Riemann surfaces.

Remark 2.6
For any given oriented Riemann surface (M, c) with boundary curves 1 , . . . k , there exists a numberZ so that the following holds true. Let U be any neighbourhood of one of the boundary curves i which is conformal to a cylinder (0, Z ] × S 1 . Then Z ≤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 c on M there exists δ > 0 so that the following holds true. Let g be any hyperbolic metric on M for which k g | i ≡ c i ∈ [0, 1), i = 1 . . . k, and let γ i be the geodesics in (M, g) that are homotopic to the boundary curves i . Then 13) in particular L g c ( i ) → ∞ as c i ↑ 1.
The bound on i follows directly from Lemma 2.5 and Remark 2.6, applied for Z = X ( i ) → ∞ as i → 0, while the expression for L g c ( i ) follows from (2.10) and (2.11).
For these surfaces, we can now prove the following version of the trace theorem. (M, g) be an oriented hyperbolic surface with boundary curves of constant geodesic curvature k g | i ≡ c i ∈ [0, 1) and let C + ( i ) be the subset of the collar C( i ) described in Lemma 2.5 that is bounded by i and the corresponding geodesic γ i . Then

Lemma 2.8 Let
holds true for any w ∈ W 1,1 (M, g). Furthermore, there exists ε > 0, allowed to depend on both the lengths i of the geodesics γ i and the curvatures c i , so that for every w ∈ W 1,1 (M, g) We note that the above lemma assures in particular that if w ∈ H 1 (M, g), then
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 1 , . . . , k and let g be a metric on M which satisfies (2.1) for some c ∈ [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 ∈ L 2 (M, g) and h ∈ L 2 (∂ M, g). and we have that ,g) ).

(i) Suppose that w ∈ H 1 (M, g) is a weak solution of
(2.23)

Proof of Lemma 2.9
Let w be as in the first part of the lemma and let ε > 0 be as in Lemma 2.8. Testing (2.20) with e w − 1 ∈ H 1 (M, g), we may estimate By (2.15), the first term on the right is bounded by (2.24) Hence F f ,h has a minimiser v which is of course a solution of (2.22), and satisfies 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 ∈ [0, 1) k . Then there exist numbers 0 < ε 0 < 1 − max c i and C 6 < ∞ so that for any b ∈ [−ε 0 , ε 0 ] k and any hyperbolic metricg = e 2w g with In particular, the solution of (2.1) is unique.
where ε 0 > 0 is determined later, setb = max |b i | and suppose thatg = e 2w g is as in the lemma. From (2.2) and (2.3), we obtain that w solves i.e. satisfies (2.20) for f ≡ 0 and h| i = b i e w . We note that e w ∈ 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 :=´M |dw| 2 g e w + (e w + 1)(e w − 1) 2 dv g +´∂ M (e w − 1) 2 d S g using the first part of Lemma 2.9 by and testing this equation with w −w M ,w M := ffl M wdv g , thus allows us to bound Having thus shown that dw L 2 (M) ≤ Cb, it now remains to show that also |w M | ≤ Cb, which of course follows if we prove that´|w|dv g ≤ Cb. As |x| ≤ 2e − min(x/2,0) |e x/2 − 1|, x ∈ R, we already obtain from (2.27) that so it remains to bound the corresponding integral over {w < −4}. To this end we note that as e w/2 L 4 (M) ≤ C, we obtain from (2.27) that, after reducing ε 0 if necessary, . valid for such functions v implies that alsô which completes the proof of the lemma. 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 → u c ∈ H 1 (M, g 0 ) be the map that assigns to each c ∈ [0, 1) k the unique solution u c of (2.4). We claim that S is Here and in the following g c = e 2u c g 0 is the unique metric satisfying (2.1). Given c ∈ [0, 1) k , b ∈ R k , say with |b| = 1, and |ε| ≤ 1−max c i , we let c ε = c+εb and set w ε := S(c ε )− S(c). As g c ε = e 2S(c ε ) g 0 = e 2w ε g c is hyperbolic with k g cε = c ε , we have We recall from Lemma 2.10 that the H 1 norms of e w ε , |ε| ≤ ε 0 , are uniformly bounded, and hence so are e 2w ε L 4 (M) and e w ε L 4 (∂ M) . Using (2.23) as well as that w ε = β ε + εv c,b we thus get 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 We finally remark that v c,b depends continuously on c as can be readily seen by using that gc = e 2(S(c)−S(c)) to view vc ,b as solution of (2.22) for g = g c , f = 2v(1 − e 2(S(c)−S(c)) ) and and applying Lemmas 2.9 and 2.10.

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, c) be a compact oriented Riemann surface with boundary curves 1 , . . . , k and denote by g c , c ∈ [0, 1) k , the unique metric compatible to c for which (2.1) holds. Then the map f : c → (c i · L g c ( i )) i is a diffeomorphism from (0, 1) k to (R + ) k . In particular, for every d > 0 there exists a unique hyperbolic metric g that is compatible with c and that satisfies (1.1).
We now claim that f : (0, 1) k → (R + ) k is proper: To see this we first recall that Corollary 2.7 assures that L g c ( i ) → ∞ as c i → 1 and hence that the preimage f −1 (K ) of any compact set K ⊂ (R + 0 ) k is a compact set in [0, 1) k . As c → L g c ( i ) is continuous on [0, 1) k we furthermore have a uniform upper bound on each L g c ( i ) for c ∈ f −1 (K ). For compact subsets K of (R + ) k we hence obtain that the components c i of c ∈ 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.
So suppose that there exists c ∈ (0, 1) k so that det(d f (c)) = 0. Hence there must be some non-trivial element b of the kernel of d f (c), i.e. b ∈ R k \ {0} so that for every i = 1, . . . , k Since v c,b cannot vanish identically as b = 0, there hence must be at least one i ∈ {1, . . . k} with which contradicts (3.1) as c i > 0.
Having thus proven that each conformal class is represented by a unique metric g ∈ M d −1 , 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.

2) and i is surrounded by a collar neighbourhood that is isometric to
2) while X ( ) and ρ are as in (2.10).

Proof
The existence of such a geodesic was proven in Lemma 2.5 and the relation between i = L g (γ i ) and L i = L g ( i ) follows from Corollary 2.7 which implies that 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 X d must be so that k g | i = sin( 2π X d ( i )). Combined with (2.13) this yields the , so X d ( i ) must be given by (3.2).

Remark 3.3
We remark that while X ( ) and X d ( ) have a similar asymptotic behaviour as → 0 the behaviour of X ( ) and X d ( ) as → ∞ is very different, with X ( ) decaying exponentially, X ( ) ≤ C −1 e − /2 , while X d ( ) is of order −2 for large . 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 on cylinders (which were considered in [23] and are isometric to ([−X d ( ), X d ( )] × S 1 , ρ 2 (ds 2 + dθ 2 )) the change of the length of the central geodesic is controlled by d dt ≤ C ∂ t G (t) L 2 if is large, which excludes the possibility that → ∞ 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 → 0, it does not include any limiting object corresponding to becoming unbounded.

Remark 3.4
We note that the quantity L g ( i ) 2 − L g (γ 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 ∂ tĝ = Re( ) for holomorphic quadratic differentials on (M,ĝ). Given a simple closed geodesic γ ⊂ (M,ĝ) and a closed curve ⊂ (C(γ ),ĝ) with constant geodesic curvature, which is hence described by some {s} × S 1 in collar coordinates, we can use the Fourier expansion of = j∈Z b j e j(s+iθ) dz 2 to obtain that d dt 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, θ) instead of collar coordinates (s, θ) on a collar C(σ ) around a simple closed geodesic σ . As indicated, the angular components of these two different sets of coordinates agree, while the x coordinate of a point p ∈ C(σ ) is given as the signed distance x = x(s) = dist g ( p, σ ) to σ . (M, g ( j) ) as in the theorem we denote by γ ( j) i , i = 1, . . . , k, the (unique) geodesics in (M, g ( j) ) that are homotopic to i , and note that their lengths are bounded from above by
In slight abuse of notation we now denote by C + ( ), ≥ 0, the unique hyperbolic half-collar which has one boundary curve of constant geodesic curvature and length L, where L 2 − 2 = d 2 , while the other boundary curve is a geodesic of length if > 0, respectively, degenerated to a hyperbolic cusp if = 0. We then construct the limit surface ( , g ∞ ) out of the limiting surface (ˆ ,ĝ ∞ ) with geodesic boundary obtained above and the half-collars C + ( ∞ i ), i = 1, . . . , k, by gluing the non-degenerate halfcollars C + ( ∞ i ), i ≥ κ 1 + 1, toˆ along the corresponding non-collapsed boundary curves of (ˆ ,ĝ ∞ ), and adding the degenerate collars C + ( ∞ i ), i ≤ κ 1 , as additional connected components of ( , g ∞ ). As the connected components of M \ i )} κ 1 i=1 of degenerating halfcollars, we can now extend the diffeomorphismsf j obtained above to the required diffeomorphisms δ > 0 so that E( f ) ≥ δ for all functions f : M → R which are equal to 1 on and for which there is a curve γ homotopic to γ so that f | γ ≤ 0. As in the first case, this will then imply that Z ≤ π δ . To prove the claim, we fix a neighbourhood V of the fixed curve σ that is diffeomorphic (but not necessarily conformal) to some rectangle R = [−c, c]× [−b, b], say with σ corresponding to {0}×[−b, b] and with V ∩ corresponding to [−c, c]×{−b}∪[−c, c]×{b} for the chosen diffeomorphism φ : R → V and fix some smooth metric g on M that is compatible to c. Using that φ * g is equivalent to the Euclidean metric on R we obtain that there exists c 0 > 0 (allowed to depend on the above construction) so that for any f : M → R as considered above andf := f •φ As the curves φ({a} × [−b, b]), a ∈ [−c, c] are homotopic to σ and thus intersect the curve γ for which f | γ ≤ 0, whilef (a, ±b) = 1, we thus get E( f ) ≥ δ := 4c 0 cb −1 > 0 as claimed.