1. Introduction

A circle domain is an open connected set in the Riemann sphere Ĉ whose boundary components are either circles or points. In 1908, P. Koebe [21] made the circle domain conjecture that any domain (i.e., open connected set) in the plane is conformally diffeomorphic to a circle domain. The classical Weyl problem concerns isometric embeddings of positively curved 2-spheres into 3-space. The Weyl problem in the hyperbolic space asks for isometric embeddings of genus-zero surfaces with complete metrics of curvature at least -1 into hyperbolic 3-space H 3 . This paper shows that the Koebe conjecture is equivalent to a special form of the Weyl problem in H 3 .

Let S 2 be the unit 2-sphere, Ĉ be identified with S 2 by the stereographic projection, (H 3 , d P ) or H 3 P be the Poincaré ball model of the hyperbolic 3-space whose boundary is S 2 , and C P (Y ) be the convex hull of a closed set Y in H 3 ∪ S 2 in hyperbolic 3-space. A circle type closed set Y ⊂ S 2 or Y ⊂ Ĉ is a compact set whose complement is a circle domain, i.e., each connected component of Y is either a round disk or a point. It is well known by the work of W. Thurston (see page 185 in [35] or Theorem 1.12.1 in [13]) that if Y is a closed set in S 2 containing more than two points, then the boundary of the convex hull ∂C P (Y ) ⊂ H 3 , in the intrinsic path metric, is a genus-zero complete hyperbolic surface. The special form of the Weyl problem, which is an inverse version of Thurston's theorem, is:

Conjecture 1. ( [23], see also [24]) Every genus zero complete hyperbolic surface is isometric to ∂C P (Y ) for a circle type closed set Y in S 2 .

We remark that in the case that the convex hull C P (Y ) is 2-dimensional, as a convention in this paper, we use ∂C P (Y ) to denote the metric double of C P (Y ) across its boundary, i.e., ∂C P (Y ) := C P (Y ) ∪ id ∂ C P (Y ).

Since the circle domain conjecture can be proved easily for domains whose complements in Ĉ contain at most two points, we will consider in the rest of the paper only those domains U whose boundaries ∂U contain more than two points. For each such domain U , by the uniformization theorem, there exists a unique complete conformal hyperbolic metric d U of the form λ(z)|dz| on U . The metric d U will be called the Poincaré metric of the domain U . Using Koebe's theorem that any genus-zero Riemann surface is conformally diffeomorphic to a domain in the Riemann sphere, we see that every genuszero complete hyperbolic surface is isometric to a Poincaré metric (U, d U ) where U ⊂ S 2 with |∂U | ≥ 3. In particular, the circle domain conjecture is equivalent to the statement that every genus-zero complete hyperbolic surface is isometric to (U, d U ) for some circle domain U .

The main result of the paper shows that the circle domain conjecture of Koebe is equivalent to Conjecture 1. More, precisely, we prove: Using He-Schramm's theorem [18] that the Koebe conjecture holds for domains with countably many boundary components, we obtain: Corollary 1.2. Every genus zero complete hyperbolic surface with countably many topological ends is isometric to ∂C P (Y ) for a circle type closed set Y in S 2 .

The relationship between the Koebe conjecture and Conjecture 1 was discovered during our investigation of the discrete uniformization conjecture for polyhedral surfaces. Indeed, Conjecture 1 can be considered a generalized version of the existence part of the discrete uniformization conjecture. The above corollary implies that every non-compact simply connected polyhedral surface is discrete conformal to the complex plane C or the unit disk D. For more details, see §10 and also [16], [17], [23] and [24].

The main tool we use to show Theorem 1.1 is Schramm's transboundary extremal lengths.

The Koebe conjecture is known to be true for connected open sets U ⊂ Ĉ which have finitely many boundary components ( [21]). The best work done to date is by He-Schramm [18], where they proved the conjecture for U having countably many boundary components. Conjecture 1 is known to be true for finite area hyperbolic surfaces and hyperbolic surfaces of finite topological types whose ends are funnels by the works of Rivin [30] and Schlenker [31], respectively. F. Fillastre [14] proved that Conjecture 1 holds for many symmetric domains with countably many boundary components.

Conjecture 1 is a Weyl-type problem for convex surfaces. It is well known that a smooth convex surface in the Euclidean space (respectively the hyperbolic space) has Gaussian curvature at least zero (respectively -1). Weyl's problem asks the converse. Namely, whether any Riemannian metric of positive curvature on the 2-sphere is isometric to the boundary of a convex body in 3-space. The problem was solved affirmatively by Levy, Nirenberg and Alexandrov. The natural generalization of Weyl's problem to hyperbolic 3-space states that every complete Riemannian metric of curvature at least -1 on a genus zero surface is isometric to the boundary of a closed convex set in hyperbolic 3-space. This was established by Alexandrov in [3].

Take a simply connected domain U ⊂ C with U = C and let Y = Ĉ -U be its complement. Then, in the upper-half space model of the hyperbolic 3-space, the boundary surface ∂C P (Y ) is homeomorphic to U and hence is simply connected (see Corollary 2.2). Thurston's theorem says that there exists an isometry Φ from ∂C P (Y ) to ∂C P (D c ) ∼ = H 2 , where D is the unit disk. On the other hand, the Riemann mapping theorem says that there exists a conformal diffeomorphism φ from U to D. Thus Thurston's isometry Φ can be considered as a geometric realization of the Riemann mapping φ. The Koebe conjecture and Conjecture 1 are the corresponding Riemann mapping-Thurston's isometry picture for non-simply connected domains.

The uniqueness aspect of Conjecture 1 is the following statement.

Conjecture 2. ( [23]) Suppose X and Y are two circle type closed sets in S 2 such that ∂C P (X) is isometric to ∂C P (Y ). Then X and Y differ by a Möbius transformation.

Though it is known that the uniqueness part of the Koebe circle domain conjecture is false, it is possible that Conjecture 2 may still be true in view of Pogorelov's rigidity theorem [26] (see also Theorem 1 in [10]). Since the uniqueness part of the Koebe con-jecture holds for domains with countably many ends by [18], we believe Conjecture 2 holds for X with countably many connected components.

The strongest version of the Weyl problem in hyperbolic 3-space is the following.

Conjecture 3. Suppose (S, d) is a planar surface with a complete path metric whose curvature is at least -1. Then there exists a complete convex surface Y in H 3 isometric to (S, d) such that each end of Y is either a circle or a point in the sphere at infinity of H 3 . Furthermore, the convex surface Y is unique up to isometry of H 3 .

The paper is organized as follows. Preliminaries and an outline of the proof of the main theorem are in §2. In §3, we give a proof of the main theorem for a special case which has to be dealt with separately. In §4, we prove an area estimate theorem for convex surfaces in hyperbolic 3-space and a few results on the shortest distance projection maps. In §5, we establish some results relating to Hausdorff convergence and the convergence of the Poincaré metrics. In §6, we recall Schramm's transboundary extremal length and establish a duality theorem. Part (a) of Theorem 1.1 is proved in §7. Part (b) of Theorem 1.1 is proved in §8, assuming the key equicontinuity result, which is proved in §9. In §10, we briefly discuss the relationship between the Weyl problem, the discrete conformal geometry of polyhedral surfaces, and the discrete uniformization problem.

In the Appendix, we recall the work of Reshetnyak [29] on the complex structure of non-smooth convex surfaces, which is used in the paper. We thank Michael Freedman and Francis Bonahon for stimulating discussions. Part of the work was carried out while the first author was visiting CMSA at Harvard. We thank S.T. Yau for the invitation. We greatly appreciate the referee's meticulous reading of the paper and his/her detailed comments and suggestions which helped us to improve considerably the manuscript. The work is supported in part by NSF 1760527, NSF 1737876, NSF 1811878, NSF 1405106, NSF 1760471 and NSF 2220271.

In this section, we recall some of the basic facts on convex surfaces, surfaces of bounded curvature, and their conformal structures. We will outline the main steps in the proof of Theorem 1.1.

The strategy of proving Theorem 1.1 is to approximate an arbitrary circle domain by circle domains of finite topology. The key result that enables us to show that the limiting domain is still a circle domain is the equicontinuity of the family of approximation conformal maps, i.e., Theorems 7.2 and 8.3. To prove the equicontinuity, we use Schramm's transboundary extremal length [33] and the duality Theorem 6.4 of transboundary extremal lengths on annuli. The main result for estimating extremal lengths is Theorem 4.1 which states that the Euclidean area of any convex hyperbolic surface in the Poincaré model of the hyperbolic space H 3 P is at most 16π.

We use S 2 , C, D, H 3 P = (H 3 , d P ) and H 3 K = (H 3 , d K ) to denote the standard 2-sphere, the complex plane, the open unit disk, the Poincaré model of the hyperbolic 3-space, and the Klein model of the hyperbolic 3-space respectively. We use H 2 to denote the Poincaré disk model and consider H 2 ⊂ H 3 as a totally geodesic plane. The closure of a set X ⊂ R 3 is denoted by X. The boundary of a 3-dimensional convex set X in H 3

A metric space (X, d) is called a path metric space if the distance between two points p and q is the infimum of the lengths of paths between p and q. We call d a path metric. For instance, each Riemannian manifold is a path metric space. Suppose (X, d) is a path connected metric space such that every pair of points in X can be joined by a rectifiable path, we can define the induced path metric d * on X to be the infimum of the lengths of paths between two points, i.e., d * (p, q) = inf{l d (γ) : γ is a rectifiable path from p to q}. If X is a rectifiably path connected subset of a metric space (Z, d Z ), we use d Z X to denote the induced path metric on X. We remark that, in general, the induced path metrics d Z X and d * are only pseudo metrics. However, in the cases we encounter in this paper, the induced path metrics are true metrics. See [11] chapter 2 for more details on path metrics.

We will use d = n i=1 g ij dx i dx j and sometimes a(z)|dz| for a Riemannian metric on a manifold M . The Riemannian distance d M on M is defined by the infimum of the lengths of smooth paths between two points. It is known (Proposition 2.4.1, [11]) that if X is a connected smooth submanifold of a Riemannian manifold (M, d) with associated Riemannian distance d M , then the induced path metric on X from d M is the same as the Riemannian distance metric on X where the Riemannian metric on X is obtained by restricting the Riemannian metric d to the tangent spaces of X. To avoid excessive use of notations and if no confusion arises, we will use d to denote both the Riemannian metric on M and the associated Riemannian distance d M .

In this subsection, we collect some of the known results on convex sets and surfaces. For a convex set X in R 3 or H 3 , the dimension of X is the dimension of the smallest totally geodesic submanifold P which contains X. In particular, a convex set X has a non-empty interior in the submanifold P . We will mainly deal with 3-dimensional convex sets. A convex surface S is a topological surface such that S = ∂X for some 3-dimensional convex set in R 3 or H 3 . Note that we do not require X to be closed.

The topology of convex sets and convex surfaces in hyperbolic 3-space can be understood using the Klein model. Recall that geodesics and totally geodesic planes in the Klein model H 3 K are exactly the intersections of Euclidean lines and Euclidean planes with the open unit ball B 3 . Therefore, convex surfaces, and convex sets in H 3 K are the same (as point sets) as convex surfaces, and convex sets in the open unit ball B 3 in Euclidean geometry. This shows that all topological properties of hyperbolic convex sets and convex surfaces are the same as those of Euclidean convex sets and convex surfaces in B 3 .

The fundamental topological properties of a compact 3-dimensional convex set X in R 3 are that X is homeomorphic to the closed unit 3-ball and its boundary ∂X is homeomorphic to the unit sphere S 2 . Indeed, take a point p in the interior of X. Then the restriction of the radial projection map (x -p)/|x -p| (from the point p) to the boundary of X is a homeomorphism ψ from ∂X to S 2 . It follows that the map F (x) = |x|ψ -1 (x/|x|) from the closed unit ball {x ∈ R 3 ||x| ≤ 1} to X is a homeomorphism. Therefore, each convex surface in R 3 or H 3 is homeomorphic to an open subset of S 2 and hence has genus zero. Furthermore, if W is a compact subset of S 2 which does not lie in a Euclidean plane, then the hyperbolic convex hull (union with

The radial projection map constructed above implies the following stronger result.

Lemma 2.1. Suppose X and Y are n-dimensional compact convex sets such that X ⊂ Y . Then the radial projection from an interior point of X induces a homeomorphism h from ∂X to ∂Y . Furthermore, h is the identity map when restricted to ∂X ∩ ∂Y and sends ∂X -∂Y homeomorphically to ∂Y -∂X.

Corollary 2.2. (Theorem II. 1.4.3 (5) in [13]) If A is a compact subset of the unit sphere S 2 such that its convex hull X in R 3 is 3-dimensional, then ∂X -A is homeomorphic to S 2 -A by the radial projection from an interior point of X.

One of the important tools in convex geometry is the shortest distance projection (see page 9 of [32]). The shortest distance projection map π from a Euclidean space to a closed convex subset X sends each point p ∈ R n to the unique point π(p) ∈ X which is the point in X closest to p, i.e., |pπ(p)| = min{|p -x||x ∈ X}. Geometrically, the projection point π(p) is the intersection of X with the largest closed ball centered at p whose interior is disjoint from X. Lemma 2.3. (pages 9-10 in [32], or page 201 in [7]) Suppose X is a closed convex subset of R n and π : R n → X is the shortest distance projection. Then the following statements hold.

(a) π is distance decreasing, i.e., |π(p)π(q)| ≤ |p -q|.

(b) If p / ∈ X and q ∈ X, then the angle ∠pπ(p)q at π(p) is at least π/2. (c) If X is an n-dimensional compact convex set and Y is a compact convex set containing X, then the restriction map π| ∂Y : ∂Y → ∂X is onto.

Similar properties for the shortest distance projection in hyperbolic spaces hold. These will be discussed in detail in §4.2.

This paper deals with the geometry of the boundary of the convex hulls, i.e., convex surfaces, in hyperbolic spaces. The study is complicated by the fact that most of the convex surfaces we use are not smooth. However, these surfaces have been extensively investigated in Alexandrov geometry. A convex surface S in H 3 P or H 3 K carries two natural structures: an induced path metric and a conformal structure. The metric structure on S is the induced path metric d P S (or d K S ) on S derived from the hyperbolic metric d P (or d K ). Unlike the restriction metric d P | S which is extrinsic, the path metric d P S defines the intrinsic geometry of S. One of the basic properties of the path metric d P S is that it defines the same topology as d P does. In fact, a stronger result holds. Namely, the restriction metric d P | S is locally bi-Lipschitz with respect to the path metric d P S (see Lemma II 1.5.7 in [13]). We will also consider the induced path metric d E S on S from the Euclidean metric d E . This new metric d E S has not been studied extensively in the literature. The conformal structure on S comes from the induced path metric d P S . In general, a surface with a path metric is not known to define a complex structure. The work of Reshetnyak [29], [36] shows that if a surface with a path metric (S, d) is of bounded curvature, then the path metric d defines a complex structure. There are several equivalent definitions of surfaces of bounded curvature. See, for instance, page 12 of [36] or page 6 of [4]. Basically, each surface of bounded curvature can be approximated locally uniformly by a sequence of polyhedral surfaces with bounded curvature. Equivalently (see Theorems 2.4 and 2.6 in [36]), a path metric surface (S, d) is of bounded curvature if and only if there exists a sequence of Riemannian surfaces (S, d i ) such that (1) d i converges to d uniformly on compact subsets and (2) for any compact subsurface X of S, the integrals of the absolute values of the Gaussian curvature of d i over X are uniformly bounded. For instance all convex surfaces in R 3 and H 3 are of bounded curvature (Theorem 2.7 in [36]). Another fact that we use is that if (S, d P S ) is a convex surface in H 3 P , then (S, d E S ) is a surface of bounded curvature. This can be seen as follows. Take a sequence of smooth convex surfaces (S i , d P S i ) in H 3 P approximating (S, d P S ) uniformly on compact subsets. Then (S i , d E S i ) converges to (S, d E S ) uniformly on compact subsets such that the integrals of the absolute values of the Gaussian curvature of (S i , d E S i ) over compact sets are bounded. It is known that the 2-dimensional Hausdorff measure on a bounded curvature surface (S, d) is equal to the area element which was constructed synthetically using geodesic triangles on (S, d) by Alexandrov (page 262 [4] and proposition 1.3 of [12]). The work of Reshetnyak puts surfaces of bounded curvature in the setting of Riemannian metrics by relaxing the regularity condition on Riemannian metrics. One of the main results of Reshetnyak (Theorem 7.1.2 in [29], or Theorem 2.23 in [36]) says that if (S, d) is a surface of bounded curvature, then at each point one can find a local coordinate chart (U, z) and a function λ(z) which is the difference of two subharmonic functions such that the path metric d restricted to U coincides with the Riemannian distance associated to the Riemannian metric e λ(z) |dz|. Note that the Riemannian metric e λ(z) |dz| may not be continuous. These charts (U, z) produce the complex structure on (S, d). In our case, we need to use the complex structure to compute the extremal lengths of families of curves on bounded curvature surfaces. It requires the notations of the length of curves and area of subsets, which were constructed by Alexandrov. What Reshetnyak's theorem tells us is that these notions are the same as the ones used in the complex analysis for computing extremal lengths on surfaces of bounded curvature. The last fact we use is that for a hyperbolic convex surface S, the conformal structures associated with the path metrics d P S and d E S are the same. This is due to (1) d P and d E are conformal metrics on H 3 and (2) a theorem of Reshetnyak (Theorem 7.3.1 in [29]) on isothermal coordinates. The details are in the Appendix of this paper.

Suppose U = Ĉ -X is a circle domain in Ĉ such that X contains at least three points. Let d U be the Poincaré metric on U . The goal is to find a circle type closed set

Produce a sequence of circle domains U n = Ĉ -X (n) with ∂U n consisting of finitely many circles such that {X (n) } converges to X in Hausdorff distance. More precisely, using a Möbius transformation, we may assume

consists of a single point. We make l n small such that l n decreases to 0 and

= ∅ for i = j. This construction ensures that X (n) is a circle type closed set having finitely many connected components and X (n) converges to X in the Hausdorff distance in C.

For each X (n) , by Schlenker's work [31] (see also Theorem 7.1), we construct a circle type closed set Y (n) ⊂ S 2 such that there exists an isometry

). Using a Möbius transformation, we may assume that ∂C K (Y (n) ) contains the origin (0, 0, 0) and φ n (0) = (0, 0, 0) ∈ ∂C K (Y (n) ). By taking a subsequence if necessary, we may assume that Y (n) converges in Hausdorff distance to a closed set Y ⊂ S 2 . We will show:

(1) The sequence {φ n } contains a subsequence converging uniformly on compact subsets to a continuous map φ : U → ∂C K (Y ). This is achieved by showing {φ n : (U n , d S ) → (∂C K (Y (n) ), d E )} is an equicontinuous family. The latter is proved in §7 using transboundary extremal lengths. In computing the extremal length, we use the fact that the path metric induced by d E on a possibly non-smooth hyperbolic convex surface in H 3 P is conformal to its intrinsic path metric induced from d P .

(2) The limit map φ : (U, d U ) → ∂C K (Y ) is an isometry. This follows from Alexandrov's convergence Theorem 5.1 and convergence of Poincaré metrics (Theorem 5.2) in §5;

(3) The compact set Y is of circle type. Since the Hausdorff limit of a sequence of round disks is a round disk or a point, we will prove in §7 that each component of Y is the Hausdorff limit of a sequence of components of Y (n) 's. This is proved using the equicontinuity property established in step (1).

Part (b) of Theorem 1.1 states that for any circle type closed set Y ⊂ S 2 with |Y | ≥ 3, there exists a circle domain U = Ĉ -X with Poincaré metric d U such that Σ := ∂C P (Y ) is isometric to (U, d U ). The strategy of the proof is the same as that for part (a) of Theorem 1.1. The only technical complication is due to the estimation of modules of rings in non-smooth convex surfaces (e.g., ∂C P (Y )).

By Theorem 3.1, we may assume that the set Y is not contained in any circle, i.e., C P (Y ) is 3-dimensional. By composing with a Möbius transformation, we may assume that (0, 0, 0) ∈ Σ. Since Y is a circle type closed set, there exists a sequence

. By construction (0, 0, 0) ∈ Σ n for n ≥ 4, and the sequence {Y (n) } converges in Hausdorff distance to Y in S 2 . Now each Σ n is a genus zero Riemann surface of finite type. Koebe proved that any genus zero Riemann surface is conformally equivalent to an open domain in Ĉ (see [21]) and any finitely connected domain is conformally equivalent to a circle domain (see [22] or page 234 Theorem 1 in [15]). So there exists a circle domain U n = Ĉ -X (n) and a conformal diffeomorphism φ n : Σ n → U n . Using Möbius transformations, we normalize U n such that 0 ∈ U n , φ n (0, 0, 0) = 0 and the closed unit disk D is contained in U n . By taking a subsequence if necessary, we may assume that X (n) converges in Hausdorff distance to a compact set X in the spherical metric d S . We will show:

(1) The sequence {φ n } contains a subsequence converging uniformly on compact subsets to a continuous map φ : Σ → U := Ĉ -X. This is achieved by showing {φ n : (Σ n , d E ) → (U n , d S )} is equicontinuous. The latter is proved in §9 using transboundary extremal lengths;

(2) The limit map φ : Σ → (U, d U ) is an isometry. This is a consequence of Alexandrov's convergence theorem and the convergence of Poincaré metrics theorem in §5;

(3) The compact set X is of circle type. We will prove in §8 that each component of X is the Hausdorff limit of a sequence of components of X (n) . This is proved using the results obtained in step (1).

Theorem 1.1 for the case that the hyperbolic convex hull C P (Y ) is two-dimensional has to be dealt with separately and will be proved in this section. It is an easy consequence of the Riemann mapping theorem and Carathéodory's extension theorem of the Riemann mapping.

Theorem 3.1. Suppose X is a compact subset of the circle S 1 ⊂ Ĉ such that each connected component of X is a single point and |X| ≥ 3. Then there exist two closed sets Y 1 , Y 2 ⊂ S 1 whose connected components are points such that (a) Ĉ -X is conformal to ∂C P (Y 1 ), and

Recall that since C P (Y i ) is 2-dimensional, by our convention, ∂C P (Y i ) is the metric double of C P (Y i ) along its boundary.

Proof. To see (a), let d U be the Poincaré metric on U = Ĉ -X. Since U is invariant under the orientation reversing conformal involution τ (z) = 1 z , by the uniqueness of the Poincaré metric, we see that τ is an isometric involution of (U, d U ). Since the fixed point set of an isometric involution of a Riemannian manifold is totally geodesic (page 59, Theorem 5.1 in [20]), we see that the fixed point set S 1 ∩ U of τ is a union of geodesics in the d U metric. This implies that U 0 = {|z| ≤ 1} -X is a simply connected hyperbolic surface with a geodesic boundary in the metric d U . By the monodromy theorem, there exists an isometric immersion ψ from (U 0 , d U | U 0 ) into the hyperbolic plane H 2 . Since (U 0 , d U | U 0 ) is convex with boundary consisting of geodesics, ψ is an embedding. Let D be the image of φ, which is a closed convex domain in H 2 bounded by geodesics. We claim that D is the convex hull C P (Y 1 ) of a closed set Y 1 ⊂ S 1 . To see this, let Y 1 be the intersection of the closure of D with the circle S 1 . By convexity, we see that C P (Y 1 ) is contained in D. To see that D ⊂ C P (Y 1 ), take a point p in H 2 -C P (Y 1 ). We will show that p / ∈ D. Suppose otherwise that p ∈ D. By the separation theorem for convex sets, there exists a closed half-space P of H 2 P such that P contains p and is disjoint from C P (Y 1 ). Find a geodesic γ in P containing p. Then p ∈ γ and γ ∩ C P (Y 1 ) = ∅. Since the boundary of D consists of geodesics and p ∈ γ ∩ D, γ must either intersect some boundary geodesic β of D or γ ⊂ D. If γ ∩ β = ∅, then using β ⊂ C P (Y 1 ), we see that γ ∩ C P (Y 1 ) = ∅. This contradicts the construction of γ. If γ ⊂ D, then the endpoints of γ are in Y 1 by construction. Therefore γ ⊂ C P (Y 1 ). This again contradicts that γ ∩ C P (Y 1 ) = ∅. Therefore p is not in D, i.e., D ⊂ C P (Y 1 ). Now both int(U 0 ) and int(D) are Jordan domains, and ψ is a conformal map between them. Therefore, by Carathéodory's extension theorem, ψ extends to be a homeomorphism Φ between their closures which are U 0 ∪ S 1 and D ∪ Y 1 . This homeomorphism Φ sends X to Y 1 . In particular, each component of Y 1 is a point. By the Schwarz reflection principle, Φ can be naturally extended to a conformal homeomorphism between (U, d U ) and the metric double of C P (Y 1 ) along its boundary. The metric double, as our convention, is ∂C P (Y 1 ).

To see part (b), since X ⊂ S 1 , the hyperbolic convex hull C P (X) ⊂ H 3 is a topological disk contained in the hyperbolic plane H 2 ⊂ H 3 . Then by the Riemann mapping theorem, there exists a conformal diffeomorphism φ from int(C P (X)) to the unit disk D = {z ∈ C |z| < 1}. By Carathéodory's extension theorem, φ extends to a homeomorphism Φ from the closure C P (X) in R 3 to the closed disk D. Let Y 2 = Φ(X) whose components are all points. By the Schwarz reflection principle, Φ can be naturally extended to a conformal homeomorphism between ∂C P (X) and Ĉ -Y 2 . q.e.d.

We prove several estimates on the shortest distance projections, which will enable us to give an area estimate of the transboundary extremal lengths on ∂C P (Y ) in Section 6.4. The following is the main theorem of this section. Theorem 4.1. Suppose X is a convex set of dimensional at least 2 in the Poincaré model H 3 P of the hyperbolic 3-space. Then the Euclidean area of the convex surface ∂X is at most 16π.

Here the Euclidean area is the 2-dim Hausdorff measure associated to the induced path metric on ∂X from the Euclidean metric d E in R 3 . The above theorem holds trivially if X is 2-dimensional. Since, in this case, X lies in a sphere S perpendicular to the unit 2-sphere S 2 such that X is inside the unit ball. One sees that the Euclidean area of X is at most 2π since, by our convention, ∂X is the metric double of X. For the proof, we will assume that X is 3-dimensional. We believe the constant 16π can be improved to 8π which can be shown to be optimal.

be the open unit ball in 3-space. The Poincaré model (H 3 , d P ) or simply H 3 P of the hyperbolic space is B 3 equipped with the hyperbolic metric

It is a complete metric of constant sectional curvature -1 and is conformal to the Euclidean metric

|x -y| be the distances associated with the Poincaré and the Euclidean metrics, respectively. Then

The Klein model of the hyperbolic 3-space (H 3 , d K ) or simply H 3 K is the unit ball B 3 equipped with the Riemannian metric

(

It is known, from Equations (4.5.2) and (6.1.2) in [27], or Formula 19.6.9 in [6], that the map

is an isometry from the Poincaré model H 3 P onto the Klein model H 3 K . Furthermore, we have the following estimate. Proof. We have

The main tool we use to prove Theorem 4.1 is the shortest distance projection in hyperbolic space. The shortest distance projection to a closed convex subset in a hyperbolic space is defined in the same way and enjoys similar properties as its counterpart in Euclidean geometry. An excellent reference of the topic is [13]. We will briefly recall the relevant properties and refer to the details of the proofs to [13], pages 121-127. Recall that A denotes the closure of a set A in the Euclidean space R n . Given a non-empty closed convex set X in the n-dimensional hyperbolic space H n P , considered as the Poincaré ball model, the hyperbolic shortest distance projection (or the nearest point retraction in [13])

where L is the largest horoball whose interior is disjoint from X and p ∈ L. (Sometimes p is called the center of the horoball L. See page 122 of [13]). The convexity of X implies that L ∩ X consists of a single point. Note that if p ∈ X, then π(p) = p.

The basic properties of the shortest distance projection π are in the following lemma. Proof. The proof of part (a) is in Lemma II.1.3.2 and Lemma II.1.3.4 in [13].

Part (b) follows from part (a) since the set of all points q in H n such that the angle ∠pπ(p)q = π/2 is the codimension-1 plane W p . The condition that x and p lie in different sides of W p implies ∠pπ(p)x ≥ π/2.

To see part (c), we use the upper-half-plane model {z ∈ C|im(z) > 0} of H 2 and take X = {iy|y ∈ R >0 } and A = {x|x ∈ R >0 }. The shortest distance projection onto X when restricted to A is π(x) = ix which is the same as the reflection about the line C = {xe πi/4 |x ∈ R} on A. Clearly the line C bisects the angle formed by A and X. q.e.d.

The following lemma gives an estimate of the distortion of the shortest distance projection that appeared in Lemma 4.3 (c).

Proof. Part (a) is trivial. Now we assume that C is a circle of radius r and centered at p. Then for any

So part (b) holds. To see part (c), by composing with a rotation and a translation, we may assume that B is a circle centered at the origin 0. See Fig. 1. Let s be the point in B closest to p. By the assumption, the angle ∠0qp is 3π 4 and the angle

≤ 2 where we have used the Sine law for the triangle Δpqs, φ = ∠qsp and θ ∈ [ π 4 , 3π 4 ]. q.e.d.

To prove Theorem 4.1, we may assume the hyperbolic convex set X is closed in H 3 P . The result is obvious if X is a 2-dimensional convex set. Indeed, in this case, X is contained in a (Euclidean) sphere of radius at most one. Therefore its Euclidean area is at most 4π. Let us assume X contains an interior points. Using the fact that the area of the 2-sphere is 4π, we see that Theorem 4.1 is a consequence of the following two properties of the shortest distance projection π. Theorem 4.5. Let X be a closed convex set in the Poincaré model H 3 P and π be the hyperbolic shortest distance projection from S 2 to X. Then for all p, q ∈ S 2 ,

We remark that the above theorem also holds for high dimensional hyperbolic spaces H n .

Proof. If π(p) = π(q), then the result holds trivially. So now we assume π(p) = π(q). If p ∈ X and q ∈ X, then easily we have that In the following argument we will assume p / ∈ X and q / ∈ X, and the case p ∈ X or q ∈ X could be proved in a similar and simpler way.

Let Y be the hyperbolic geodesic joining π(p) to π(q) and W p and W q be the codimension-1 hyperbolic planes perpendicular to Y at π(p) and π(q) respectively. Note that W p ∩ W q = ∅ since π(p) = π(q) and both are perpendicular to Y . Let H p and H q be the two disjoint half spaces in H 3 bounded by W p and W q respectively. By Lemma 4.3 (b), both angles ∠pπ(p)π(q) and ∠qπ(q)π(p) are at least π/2, it follows that p ∈ U p := H p ∩ S 2 and q ∈ U q := H q ∩ S 2 . Since H p ∩ H q = ∅, we have that

Let O p and O q be the centers of spherical disks U p and U q respectively, and γ be a shortest geodesic from O p to O q on S 2 such that γ intersects ∂U p and ∂U q at p and q respectively. Then

Therefore it suffices to show that |π(p)π(q)| ≤ 2d S (p , q ). To this end, let P be a Euclidean plane passing through (0, 0, 0) and containing Y and γ and consider the unit disk

where d E ∂D is the induced Euclidean path metric on the unit circle ∂D.

Let M be the component of ∂D -Y which contains {p , q }; C be the circle or the straight line in the plane P passing through Y ∩ M and bisecting the angles between Y and M ; and f be the inversion about C in P . Then the angle between C and ∂D could be either π/4 or 3π/4. In the case of 3π/4, M lies outside of the circle C. In either case, by Lemma 4.4, |f (z)| ≤ 2 for any z ∈ M . So we only need to prove f (p ) = π(p) and f (q ) = π(q) in order to obtain |π(p)π(q)| ≤ 2d S (p , q ).

To see

The same argument shows f (q ) = π(q)). q.e.d. Theorem 4.5 implies the hyperbolic shortest distance projection π is continuous on S 2 . The following lemma shows it is an onto map. Lemma 4.6. Suppose X is a closed 3-dimensional convex set in H 3 P with non-empty interior, then the shortest distance projection π from S 2 to ∂X ∪ (X ∩ S 2 ) is surjective.

W p be a supporting plane for X at p and γ be the geodesic ray perpendicular to W p at p such that γ intersects X only at p. The endpoint of the ray γ is a point q ∈ S 2 . By definition, π(q) = p. q.e.d.

Proposition 4.7. Assume X is a hyperbolic closed convex set of dimension at least 2 in H 3 P containing origin O and q ∈ X ∩ S 2 (See Fig. 3). Let π be the shortest distance projection from S 2 to ∂X ∪ (X ∩ S 2 ).

(a) For any x ∈ S 2 ,

Proof. For part (a), |π(x) -q| ≤ 2d S (x, q) is a consequence of Theorem 4.5 using π(q) = q. It remains to prove the lower bound of |π(x) -q| for x = q. For simplicity we denote d S (x, q) = θ ∈ (0, π]. Assume L is the largest horoball such that int(L) ∩ X = ∅ and x ∈ L. Let r and p be the Euclidean radius and Euclidean center of the horoball L. Since X contains (0, 0, 0) and the interior of L is disjoint from X, the Euclidean radius

and we are done. If θ < π/2, then the Euclidean distance from p to the line through q, O is at least r, i.e.,

Since π(x) ∈ ∂X, we have,

To estimate the right-hand side of the above inequality, by the Cosine Law on the triangle qOp, we have,

This shows that f (t)t is a decreasing function of t. Note that |p -q| = f (r). Now for θ < π/2, by (5), we have r ≤ sin θ/(1 + sin θ) and

Then by part (a) and the surjectivity of π (Proposition 4.6 (b)),

Remark 4.8. We thank the referee who pointed out that the inequality |π(x) -q| ≤ 2d S (π(x), q) follows from Theorem 4.5 and improved our original estimate.

Let us begin by briefly recalling the Hausdorff distance. Suppose A is a subset of a metric space (Z, d) and r > 0. The r-neighborhood of A, denoted by

For a compact metric space (Z, d), the set of all closed subsets in the Hausdorff distance is compact. See [11].

Alexandrov convergence is used extensively on the convergence of convex surfaces in the hyperbolic and Euclidean spaces. A sequence {X n } of closed subsets in a metric space (Z, d) is Alexandrov convergent to a closed subset X if (i) for any p ∈ X, there exists a sequence {p n } with p n ∈ X n such that lim n p n = p and (ii) for any convergent sequence {p n i } with p n i ∈ X n i and lim i p n i = p, we have p ∈ X. Alexandrov convergence is independent of the choice of the distance d. Clearly if {X n } converges to X in the Hausdorff distance, then {X n } Alexandrov converges to X. In general, the converse is not true. But if the space (Z, d) is compact, then Alexandrov convergence is equivalent to the Hausdorff convergence. In particular, for a compact metric space (Z, d), the Hausdorff convergence of compact subsets is independent of the choice of metrics. See [11] and [3] for details.

By a complete convex surface S in R 3 or H 3 we mean the boundary of a closed convex set of dimension at least 2. Here, the dimension of a convex set in R 3 or H 3 is defined to be the dimension of the smallest totally geodesic submanifold that contains the convex set. The convergence theorem of Alexandrov, which will be used extensively, is the following. Theorem 5.1 (Alexandrov). Suppose {S n } is a sequence of complete connected convex surfaces in the Euclidean or hyperbolic 3-space Alexandrov converging to a complete connected convex surface S. If x, y ∈ S and x n , y n ∈ S n such that lim n x n = x and lim n y n = y, then

where d S is the induced path metric on the convex surface S.

The proof of this theorem for the Euclidean case is on pages 91-95 of [3]. The hyperbolic case was stated in section 3 of Chapter 12 of [3].

If X is a closed set in the Riemann sphere Ĉ which contains at least three points, then none of the connected components of Ĉ -X is conformal to the complex plane C or the punctured plane C -{0}. Therefore, by the uniformization theorem, each connected component of Ĉ -X carries the Poincaré metric. The Poincaré metric on Ĉ -X is defined to be the Riemannian metric whose restriction to each connected component is the Poincaré metric.

The main result in this section is the following.

Theorem 5.2. Suppose {X n } is a sequence of compact sets in the Riemann sphere converging to a compact set X in the Hausdorff distance such that X contains at least three points and X = Ĉ. Let d n = a n (z)|dz| and d U = a(z)|dz| be the Poincaré metrics on U n = Ĉ -X n and U = Ĉ -X respectively. Then a n (z) converges uniformly on compact subsets of U to a(z). Furthermore, if p, q are two points in a connected component of U and p n , q n are two points in a connected component of U n such that lim n p n = p and lim n q n = q, then lim

Note that by Hausdorff convergence, for any compact set K ⊂ U , K ⊂ U n for n large. Therefore a n is defined on K for large n. Also, by our convention, the distance d n in (6) denotes the Riemannian distance associated with the Poincaré metric d n = a n (z)|dz|.

Proof. To begin, let us recall a known theorem from Riemannian geometry about the convergence of Riemannian distance functions when the Riemannian metrics converge. See for instance Proposition 11.3.2 in [25]. We provide a short proof for completeness.

and if furthermore d ∞ is a complete Riemannian metric,

Proof. To prove (7), take any > 0 and a smooth path γ :

Since {a n (z)} converges uniformly on the image of γ, we have lim n l n (γ) = l ∞ (γ) where l n (γ) and l ∞ (γ) are the lengths of the curve γ in the Riemannian metrics d n and d ∞ respectively. By definition l n (γ) ≥ d n (p, q), we have,

Therefore ( 7) holds.

To prove (8), by (7), it suffices to show

By (7), we choose R > 0 large such that d n (p, q) ≤ R for all n. Consider the set

This shows, using d n (p, q) ≤ R and d n is complete, that the shortest geodesic γ n in (U n , d n ) joining p to q is contained in the compact set W . Since {a n (z)} converges to b(z) uniformly on W , there exists a sequence of positive numbers n converging to 0, such that

It suffices to prove the theorem for the case that all X n contain a fixed set of three points. Indeed, let {u, v, w} ⊂ X with u, v, w pairwise distinct and consider sequences u n , v n , w n ∈ X n such that lim n u n = u, lim n v n = v and lim n w n = w. There exists a Möbius transformation ψ n sending u n , v n , w n to u, v, w respectively. By the construction, ψ n converges uniformly to the identity map in the spherical metric d S on Ĉ. This implies that ψ n (X n ) converges in Hausdorff distance to X and {u, v, w} ⊂ ψ n (X n ). Now suppose the theorem has been proved for the sequence ψ n (X n ). Using the fact that ψ n induces an isometry between Poincaré metrics on the open sets Ĉ -X n and Ĉψ n (X n ), we see Theorem 5.2 holds for the general case. Now using a Möbius transformation, we may assume that X n contains {0, 1, ∞} and converges to X in Hausdorff distance.

The strategy of the proof is as follows. First, we show that for any compact set K in U , the family of functions {a n | K } contains a uniformly convergent subsequence. By the Cantor diagonal process, we see that there is a subsequence of {a n } which converges uniformly on compact subsets to a limit function b(z) on U . The limiting metric b(z)|dz| can be shown easily to have constant curvature -1. Finally, we show that the hyperbolic metric b(z)|dz| is complete. Therefore, by the uniqueness of the Poincaré metric, b(z)|dz| is the Poincaré metric d U = a(z)|dz|. Since all limits of convergent subsequences are the same, it follows that the sequence {a n (z)} converges to a(z).

To show that {a n | K } contains a convergent subsequence in the L ∞ -norm, write the open set U as a union of open Euclidean round disks B j such that the Euclidean closure B j is still in U . Then each compact set K in U is contained in a finite union of these closed disks B j . By using the Cantor diagonal process, it suffices to prove the statement for K to be a compact ball {z||z -p| ≤ r} in U . We will use the following well-known consequence of the Schwarz-Pick lemma (see Theorem 10.5 in [5] for proof). For simplicity, we assume that the subsequence is {f n }. We claim that |f n (p)| is bounded away from 0. Indeed, consider the standard tangent vector v = ∂ ∂x at p. By ( 9), the length of v in d n is at least the length δ of v in d 0,1,∞ . It follows that the length of f n (p) in the Poincaré metric on D is at least δ. This shows that h (p) = 0 and, therefore, Proof. Take a connected component U of U and a Cauchy sequence

In particular, there is a point p ∈ Ĉ -{0, 1, ∞} so that lim n x n = p. We claim that p ∈ U .

Assuming the claim and using the fact that the topology determined by d ∞ and the Euclidean metric d E on U are the same, we see that p is in U and x n converges to p in the d ∞ metric.

To see the claim, suppose otherwise that p ∈ X. Then U ⊂ Ĉ -{0, 1, p}. By the definition of Hausdorff convergence, there exists a sequence p n ∈ X n such that lim n p n = p and p n = 0, 1, ∞. Let d n be the Poincaré metric on Ĉ -{0, 1, p n }. By Lemma 5.4 and Ĉ -X n ⊂ Ĉ -{0, 1, p n }, we have d n ≤ d n . Let d 0,1,p be the Poincaré metric on Ĉ -{0, 1, p} and ρ n be the Möbius transformation sending the triple (0, 1, p n ) to (0, 1, p). Then ρ n converges to the identity map uniformly on ( Ĉ, d S ) and ρ n is an isometry from ( Ĉ -{0, 1, p n }, d n ) to ( Ĉ -{0, 1, p}, d 0,1,p ). In particular, d 0,1,p (x, y) = lim n d 0,1,p (ρ n (x), ρ n (y)). For all x, y ∈ U , by (7) in Lemma (5.3), we have

Hence {x n } is a Cauchy sequence in the d 0,1,p metric on Ĉ -{0, 1, p}. Since, d 0,1,p is a complete metric, it follows that there is q = p in Ĉ -{0, 1, ∞} such that lim n x n = q in Ĉ. This contradicts the assumption that lim n x n = p in Ĉ. q.e.d.

Finally, let us prove (6). Note that

By Lemmas 5.3 and 5.6, we have |d n (p, q)d U (p, q)| → 0. It remains to show that both d n (p n , p) and d n (q n , q) converge to zero. To see this, since in a small neighborhood of p, |a n (z)| is uniformly bounded and the Euclidean distance |p n -p| goes to 0, we see that d n (p n , p) is bounded by C|p n -p| for some constant C independent of n. Therefore, lim n d n (p n , p) = 0 and similarly, lim n d n (q n , q) = 0. q.e.d.

Remark 5.7. We thank the referee for suggesting Lemma 5.3 and the estimate (10) which drastically simplifies our original proof.

For our proof, we need a slightly more general form of the Arzela-Ascoli theorem. Recall that a family of maps f n : (X n , d n ) → (Y n , d n ) between metrics spaces is called equicontinuous if for any > 0, there exists δ > 0 such that for all n and x n ,

Theorem 5.8. Suppose (Z, d) and (Y, d ) are compact metric spaces and {f n : X n → Y } is an equicontinuous family where X n ⊂ Z are compact. Let X ⊂ Z be a compact subset such that for any x ∈ X there exists a sequence x n ∈ X n converging to x. Then there exists a subsequence {f n i } converging uniformly to some continuous function f : X → Y , i.e., for any > 0 there exist δ > 0 and N > 0 such that for any i ≥ N ,

Proof. Since X is compact, we can find a countable subset A ⊂ X such that its closure A = X. Then for any > 0 there exists a finite subset A ⊂ A such that

For any a ∈ A, by the assumption, there are a n ∈ X n such that lim n a n = a. By the standard diagonal method, we find a subsequence of {f n } which, for simplicity, we may assume is {f n } itself, such that {f n (a n )} converges for all a ∈ A. Define

We first claim that f : A → Y is uniformly continuous. Indeed, for any > 0 there exists

then d(a n , a n ) < δ for n sufficiently large, and

Since f is uniformly continuous on A, we can extend f to a uniformly continuous function, still denoted by f , to X. Now to see the uniform convergence of f n to f , take any > 0. There exists δ > 0 such that (1) for any n and x, y ∈ X n with d(x, y) ≤ δ, d (f n (x), f n (y)) < /3; and

(2) for any x, y ∈ X with d(x, y) ≤ δ, d (f (x), f (y)) < /3.

Find N = N ( ) such that for any n ≥ N and any a ∈ A δ/3 , there exists a n ∈ X n such that d(a n , a) < δ/3 and d (f n (a n ), f (a)) < /3. This is possible since A δ/3 is a finite set. Then for any x ∈ X, find an a ∈ A δ/3 such that d(x, a) < δ/3.

≤ /3 + /3 + /3 = . q.e.d.

The transboundary extremal length introduced by O. Schramm ([33]) is a powerful conformal invariant and has been used in many works (see [8], [9] and others).

Suppose Σ is a Riemann surface homeomorphic to an annulus and F and F * are two families of curves in Σ such that F consists of closed curves separating two ends of Σ and F * consisting of paths joining different ends of Σ. Then a well-known duality theorem states that the extremal lengths satisfy EL(F)EL(F * ) = 1. The goal of this section is to show that the duality theorem still holds for transboundary extremal lengths. The latter result (Theorem 6.4) is the key tool for estimating the modules of rings on non-smooth convex surfaces.

Suppose Σ is a Riemann surface and E is a set of ends of Σ (E may not be the set of all ends). Note that Σ ∪ E is naturally a topological space with the end topology. Schramm's transboundary extremal length for any family of curves in Σ ∪ E is defined as follows. Take a conformal Riemannian metric g on Σ. An extended metric m on Σ ∪ E is a pair (ρg, μ) such that ρ : Σ → R ≥0 is a Borel measurable function and μ : E → R ≥0 . The area of the extended metric m is defined to be

where dA g is the area form of the Riemannian metric g. By a curve in Σ ∪ E we mean a continuous map γ from an interval to Σ ∪ E. The length of γ in the extended metric m is defined to be

where ds is the length element in the metric g. If Γ is a family of curves in Σ ∪ E, its length in the extended metric m is defined to be

Schramm's transboundary extremal length [33] of Γ is

where the supremum is over all finite positive area extended metrics m. We will drop the adjective "transboundary" when we refer to extremal lengths below. Some of the basic properties of extremal lengths follow from the definition (see [1] for a proof). Lemma 6.1. (a) Suppose Γ 1 and Γ 2 are two families of curves in Σ ∪ E such that for any

(b) Suppose Γ 1 and Γ 2 are two families of curves in Σ ∪E such that they are supported in two disjoint Borel measurable subsets

Another property of extremal lengths is the following conformal invariance. Lemma 6.2 (Schramm [33] Lemma 1.1). Suppose φ : Σ → Σ is a conformal diffeomorphism sending E onto E . Then for any curve family

where φ(Γ) = {φ(γ) : γ ∈ Γ}.

We will apply transboundary extremal lengths in the following special situation in this paper. Take a topological surface S and a compact set X ⊂ S such that Σ = S -X is connected and is equipped with a complex structure. Then each component X i of X corresponds to an end, denoted by [X i ] of Σ. Define [X] to be the set of ends of the form [X i ] for components X i of X. We will apply transboundary extremal lengths to curves in the space Σ ∪ [X] which will be denoted by S X .

A flat cylinder is a Riemannian surface isometric to S = S 1 × (0, h) equipped with the product metric g = dx 2 + dy 2 where (e √ -1x , y) are points in S. A square in S is a compact subset of the form I 1 × I 2 where I 1 and I 2 are two closed intervals of the same length. We consider a point in S as a (degenerated) square. The following lemma is known to Schramm [34]. Lemma 6.3. Suppose X is a finite disjoint union of squares in a flat cylinder S. Let Γ * be the family of curves in S X joining the two boundary components of S and Γ be the family of all simple loops in S X separating the two boundary components of S. Then

Proof. We will show that EL(Γ) = 2π h and EL(Γ * ) = h 2π . Since the computations are similar, we only compute EL(Γ * ). Let the components of X be X 1 ,..., X n of edge lengths h 1 , ..., h n with h i ≥ 0. Let the coordinate in S be (e √ -1x , y). Construct an extended metric m = (ρ(dx 2 +dy 2 ), μ) on S X such that ρ = 1 on S -X and μ([X j ]) = h j . Then the area A(m) of m is 2πh. For any curve γ ∈ Γ * , we have l m (γ) ≥ h by definition. Therefore,

To see that EL(Γ * ) ≤ h 2π , take any extended metric m = (ρ(dx 2 + dy 2 ), μ) and for each e

Then

By Cauchy inequality we have

This shows l m (Γ * )/A(m) ≤ h 2π and the result follows. q.e.d.

The main tool that enables us to estimate the module of rings in convex surfaces ∂C(W ) is the following theorem. A version of it for quadrilaterals was proved by Schramm [34] (Theorem 10.1). Recall that a doubly connected Riemann surface is a topological ring with a complex structure. Theorem 6.4. Suppose R is doubly connected Riemann surface without boundary, and X is a compact subset with finitely many components such that R\X is connected. Let Γ * be the family of curves in R X connecting the two topological ends of R and Γ be the family of all simple loops in R X separating the two topological ends of R. Then the transboundary extremal lengths satisfy

Proof. By Lemmas 6.2 and 6.3, it suffices to prove that there exists a finite collection of disjoint squares X in a flat cylinder R such that (1) R -X is conformal to R -X by a conformal map φ and (2) φ : [X ] → [X] is a bijection. The latter result was established by Jenkins (the corollary of Theorem 2 in [19]). q.e.d.

The following is a quantified version of a result of Schramm on transboundary extremal length of curves in cofat domains. Proposition 6.5. Let S be a topological annulus in Ĉ -{0}, W ⊂ S be a finite disjoint union of round closed disks and points and Γ be the family of all simple loops in S W separating the two ends of S. If S contains N disjoint rings R i = {r i < |z| < 2r i } for i = 1, 2, ..., N such that (a) each ring R i separates the two ends of S and (b) no component in W intersects two R i and R j , then

Proof. We begin with the case of N = 1.

Lemma 6.6. Assume that S, W and Γ are as in Proposition 6.5. If S contains one annulus Ω r := {r < |z| < 2r} separating two ends of S, then

Proof. Let Γ * be the family of all simple paths in S W joining two boundary components of S. Then by Theorem 6.4, we have EL(Γ) -1 = EL(Γ * ). Thus it suffices to show EL(Γ * ) ≥ 1 72π . Suppose {W 1 , ..., W n } is the set of all components of W . Consider the extended metric m = (ρ|dz|, λ) on S W where λ([W i ]) = diam(W i ∩ Ω r ) is the diameter in the Euclidean metric and ρ : S -W → R is the function which is 1 on Ω r -W and zero otherwise. Let B 2r and B 6r be the Euclidean balls of radii 2r and 6r centered at 0 and μ be the Lebesgue measure on C. We have

). Hence the result follows. If W i is not inside B 6r and W i intersects B 2r , then the diameter of W i is at least 4r and W i ∩ B 6r contains a disk of radius 2r. Then diam

) and ( 15) holds again. By (15), the area satisfies

This shows

For each path γ in Γ * joining two boundary components of S, we claim that l m (γ) =

To see this, let W be the set of all components W i of W which intersect Ω r . Construct a path γ in S W -W by gluing to γ ∩ (S -W ) a line segment of length at most λ(

. By the construction, the length of γ is at least the length of γ. Since γ is a path S W -W joining two ends of S, it contains an arc α ⊂ Ω r joining {|z| = r} to {|z| = 2r}. In particular, the Euclidean length of α is at least r. This shows the length of γ is at least r. Therefore l m (γ) ≥ r and l m (Γ) ≥ r. This shows

. q.e.d. Now back to the proof of (13). By Lemma 6.6, we may assume that N ≥ 4. Without loss of generality, we may assume that 2r j ≤ r j-1 for j = 2, ..., N . For j = 2, 3, ..., N -1, let A j be the annulus in S defined by {2r j+1 < |z| < r j-1 } -∪ k∈I j W k where the index set I j is {k|W k ∩ ({|z| = r j-1 } ∪ {|z| = 2r j+1 }) = ∅}. Note that A j is topologically an annulus since each component of W is a disk or a point and no component of W intersects two rings R i and R l . Furthermore, by construction Z j := W ∩ A j ⊂ A j is the union of all components of W which lie in A j . See Fig. 4. Let Γ j be the set of simple loops in A Z j j separating the two ends of A j . By construction, A Z j j ⊂ S W and Γ j is a subset of Γ. Furthermore, since no component of W intersect two rings R i and R h , we see that

2[N/2] respectively. By Lemma 6.6 for S = A 2j and W = Z 2j , EL(Γ 2j ) ≤ 72π, i.e., EL(Γ 2j ) -1 ≥ 1 72π . By Lemma 6.1 and N ≥ 4,

Combining these, we see inequality (13) follows. q.e.d.

The counterpart of Proposition 6.5 for non-smooth convex surfaces will be established in this section. Given a compact set Z ⊂ ∂H 3 , by the work of Thurston, the surface ∂C P (Z) is hyperbolic and therefore naturally a Riemann surface whose conformal structure is induced by the path metric d P S . Let W be a finite disjoint union of round disks and points in S 2 such that (0, 0, 0) ∈ ∂C P (W ) and C P (W ) has a non-empty interior. We also fix a point q ∈ W . The goal is to estimate the modules of rings separating q and (0, 0, 0) in the convex surface ∂C P (W ) ∪ W . Let D r = {z ∈ S 2 |d S (z, q) < r} be the ball of radius r at q and E r = {z ∈ S 2 |d S (z, q) ≥ r} be the complement of int(D r ). The disk D r is convex if r < π/2. The module of the ring {z ∈ S 2 |a < d S (z, q) < b} is 1 2π ln tan(b/2) tan(a/2) . To see this, we may assume that z is the south pole of S 2 and then the stereographic projection conformally maps the ring to a Euclidean ring {z ∈ C : tan a 2 < |z| < tan b 2 } whose module is wellknown to be 1 2π ln tan(b/2) tan(a/2) (see page 53 in [2]). So for small r > 0, the module of the

which is uniformly bounded away from 0 and infinity. The goal is to show that the image of Ω r under the shortest distance projection is a ring whose module is uniformly bounded away from zero. Let π : S 2 → ∂C P (W ) ∪ W be the shortest distance projection onto the convex hull. Note that π| W = id and π is in general not injective. We claim that for the ring Ω r = D 65r -E r in S 2 , there is a well-defined ring in ∂C P (W ) ∪ W corresponding to it under the projection π. Indeed, by Propositions 4.6 and 4.7, we see that π(D r ) and π(E 65r ) are disjoint connected compact sets in the topological 2-sphere ∂C P (W ) ∪W . By a well-known fact from surface topology, there exists a unique component of ∂C P (W ) ∪ Wπ(D r ) ∪ π(E 65r ) which is a topological annulus separating π(D r ) and π(E 65r ). (All other components, if they exist, are simply connected). We will denote this ring component by π(Ω r ). Since the projection π is onto, we see that π(Ω r ) contains π(Ω r ). (See Fig. 5.) Proposition 6.7. Let W be a finite disjoint union of round disks and points in S 2 such that (0, 0, 0) ∈ ∂C P (W ) and C P (W ) has a non-empty interior. Fix a point q ∈ W . Suppose W is a finite union of connected components of W and S is an annulus in ∂C P (W ) ∪ W -{q} such that S contains W . Let Γ be the set of all simple loops in S W separating the two ends of S. If S contains N pairwise disjoint rings π(Ω r i ) = π({z ∈ S 2 |r i < d S (z, q) < 65r i }) for i = 1, 2, ..., N such that (a) each π(Ω r i ) separates the two ends of S, (b) no component of W intersects two π(Ω r i ) and π(Ω r j ), and (c) 65r i < π/2 for all i = 1, 2, ..., N , then

The key step in the proof is to establish the counterpart of Lemma 6.6. The rest will be the same as the argument used in the proof of Proposition 6.5.

Proof. We begin with the case of N = 1. Lemma 6.8. Assume that W , W , S and Γ are as in Proposition 6.7. If S contains a ring π(Ω r ) separating two ends of S, then EL(Γ) ≤ 10 8 .

Proof. Let Γ * be the family of paths in S W joining two boundary components of S. By Theorem 6.4, we have EL(Γ) -1 = EL(Γ * ). Hence it suffices to prove EL(Γ * ) ≥ 1 10 8 . To this end, consider the extended metric m = (ρd E S , λ) on S W where ρ(z) = 1 for z ∈ π(D 65r ) ∩ S -W and is zero otherwise, and for each component

Recall that μ(S) denotes the Euclidean area, or the 2-dim Hausdorff measure, of a set S in R 3 . We claim that

To see this, we will use the well-known fact that for a spherical ball

) and ( 16) holds again. The area

) and ( 16), we have

For each path γ in Γ * joining the two boundary components of S W , let γ be the path on S obtained by gluing to γ ∩ (S -W ) the shortest geodesic path in each component

By the separation assumption, the path γ contains an arc joining a point p 1 = π(p 1 ) with p 1 ∈ D r to another point p 2 = π(p 2 ) with p 2 ∈ E 65r . By Proposition 4.7,

In particular,

8 since it contains p 1 and p 2 . Therefore, we have l m (γ) ≥ r 8 for all γ ∈ Γ * and hence l m (Γ * ) ≥ r 8 . This implies

The rest of the proof of Proposition 6.7 is the same as that of Proposition 6.5. Note that the counterpart of the annulus A j is the annulus component of the Sπ(D r j +1 ) ∪ π(E r j-1 ) ∪ ∪ k∈I j W k where the index set I j is {k|W k ∩ (π(D r j +1 ) ∪ π(E r j-1 )) = ∅}. The verification that A j and A j+2 are disjoint and there are no components of W intersecting both A j and A j+2 follows from the assumption on W . We omit the details. q.e.d. Using a Möbius transformation, we may assume that

consists of a single point. We make l n small such that l n decreases to 0 and

This shows that X (n) is a circle-type closed set having finitely many connected components and X (n) converges to X in the Hausdorff distance in Ĉ.

We will use the following theorem of Schlenker [31].

Theorem 7.1 (Schlenker). Suppose g is a complete hyperbolic metric on a genus zero surface Σ of finite topological type such that each end of (Σ, g) is of funnel type. Then there exists a closed set Y ⊂ S 2 such that Y is a disjoint union of finitely many round disks and ∂C K (Y ) with the induced path metric from d K is isometric to (Σ, g).

Let d n be the Poincaré metric on U n = Ĉ -X (n) . By the above theorem, there exists a circle type closed set Y (n) ⊂ S 2 with (0, 0, 0) ∈ ∂C K (Y (n) ) and an isometry

) such that φ n (0) = (0, 0, 0). By taking a subsequence if necessary, we may assume that {Y (n) } converges in Hausdorff distance to a compact set Y .

A key step in the proof Theorem 1.1 is to show the following equicontinuity property.

Theorem 7.2. The sequence

for any > 0, there exists a δ > 0 such that for all n and all x, y ∈ U n with d S (x, y) < δ,

This section is organized as follows. In §7.1, we prove that Y contains at least three points and is not equal to S 2 . In §7.2, we prove Theorem 7.2. In §7.3, we show, using Theorem 7.2, that φ n 's can be extended to continuous functions, still denoted by φ n from ( Ĉ, d S ) to (∂C K (Y (n) ) ∪Y (n) , d E ) and the extended family {φ n } remains equicontinuous. Finally, in §7.4, we prove that each component of Y is the Hausdorff limit of components of Y (n) . Therefore Y is a circle type closed set. On the other hand, using Theorems 5.1, 5.2 and 5.8, we know that φ n converges uniformly on compact subsets to an isometry from (U, d U ) and ∂C P (Y ). This ends the proof Theorem 1.1(a). Proof. To see part (a), take three points {p 1 , p 2 , p 3 } in X (n) for n large and let d W be the Poincaré metric on

for any r > 0. Let r 0 > 0 be a small positive number such that r 0 is less than the injectivity radius of d W at 0 and B r 0 (0, d W ) is contained in U n for all n. Then B r 0 (0, d n ) is contained in the simply connected subset B r 0 (0, d W ) in U n . Therefore, B r 0 (0, d n ) can be isometrically lifted to the universal cover of (U n , d n ). This implies B r 0 (0, d n ) is isometric to the standard radius r 0 ball in the hyperbolic plane. Therefore, r 0 is a lower bound for the injectivity radius of (U n , d n ) at 0 for all n.

To see part (b), since Y (n) converges to Y and (0, 0, 0) ∈ ∂C K (Y (n) ), we see that (0, 0, 0) ∈ ∂C K (Y ) and hence Y contains at least two points and Y = S 2 . Now if Y contains only two points, say Y = {p 1 , p 2 }, then there exists a sequence of positive numbers 2 where B r (p) is the ball of radius r centered at p in the spherical metric. Therefore, there exists a sequence of homotopically non-trivial loops γ n ⊂ ∂C K (Y (n) ) through (0, 0, 0) whose hyperbolic lengths tend to 0. The loops γ n can be constructed as follows. Let V n be the hyperbolic surface ∂C K (B r n (p 1 ) ∪ B r n (p 2 )), q n ∈ V n be a point converging to (0, 0, 0) such that the shortest distance projection from V n to ∂C K (Y (n) )) sends q n to (0, 0, 0). Note that V n is topologically a ring and there is only one simple closed geodesic separating the two ends of V n . Let δ n be the simple geodesic loop in V n based at q n such that (1) δ n separates the two ends of V n and (2) away from the based point q n , δ -{q n } is a geodesic. Let γ n be the image of δ n under the shortest distance projection from V n to ∂C K (Y (n) ). Since r n tends to zero, the length of the curve δ n tends to zero. On the other hand, the shortest distance projection decreases the distances. Therefore the length of γ n tends to zero. Finally, since δ n separates the two ends of V n , we claim that γ n is homotopically non-trivial, i.e., essential, in ∂C K (Y (n) ). To this end, let us assume without loss of generality that

→ M be the radial projection map induced from the point. Note that Θ is a homeomorphism map from a topological 2-sphere to the topological 2-sphere M , is the identity map on Y (n) , and Θ -1 (Y (n) ) = Y (n) . We will show that Θ(γ n ) is essential on M -{p

in the topological 2-sphere M and Θ • π : M → M is a degree one continuous map (by Lemma 2.1) such that Θ • π sends the annulus M -{p

2 . Thus the claim follows. Using the isometry φ n , we see that the homotopically non-trivial loops γ n = φ -1 n (γ n ) in (U n , d n ) pass through 0 such that their lengths in the Poincaré metrics d n tend to zero, i.e., l d n (γ n ) → 0. But this contradicts part (a) that the injectivity radii of d n at 0 are bounded away from 0.

To see part (c), choose the radius r 0 in the proof of part (a) to be small such that B r 0 (0,

We will prove a stronger result that 4πdiam

. Indeed, by Proposition 2.1.3 [7] and Lemma 1.2.3 in [32], the shortest distance projection map P : A → A is distance decreasing and is surjective. Therefore, P decreases the area. This shows

) with the induced metric from d K is isometric to the standard hyperbolic disk of radius r 0 in H 2 . In particular, the hyperbolic area of Z n is 4π sinh 2 (r 0 /2). On the other hand, since φ n is an isometry, we see that

Hence, there is a compact set Q ⊂ H 3 , which contains all Z n for n large. On the compact set Q, there exists a constant

1 sinh 2 (r 0 /2) and the result follows. q.e.d.

Let Σ n = ∂C K (Y (n) ) and Σn be ∂C K (Y (n) ) ∪ Y (n) which is a topological 2-sphere. By the work of Thurston, the surface Σ n is naturally a Riemann surface. The conformal map φ n : U n → Σ n implies that ĈX (n) and ΣY (n) n are conformally equivalent.

We prove Theorem 7.2 by contradiction. Suppose otherwise that there exists 0 > 0 and sequences

If the sequence {k n } is not bounded, we may assume without loss of generality (after taking a subsequence), that k n = n. If the sequence {k n } is bounded, we may assume after taking a subsequence that k n is a constant. Below we will focus on the main case that k n = n. The same proof also works for the simpler case that k n is a constant. We omit the details.

Remark 7.4. The case k n being a constant is equivalent to the statement that each φ n : (U n , d S ) → (Σ n , d E ) is uniformly continuous. We can see uniform continuity by using Carathéodory's extension theorem. By compositing φ n with the inverse of the homeomorphism Ψ(X) = 2x 1+|x| 2 :

) can be extended continuously to their compact closures in R 3 . Let us recall Carathéodory's extension theorem. Suppose A 1 and A 2 are two Riemann surfaces and B i is a subsurface of A i bounded by finitely many disjoint Jordan curves c i in A i for i = 1, 2. Then Carathéodory's extension theorem says that any biholomorphism from B 1 to B 2 extends continuously to

The standard form of Carathéodory's theorem applies to Jordan domains B 1 and B 2 . However, the proof of this theorem is inherently local, relying on the standard length-area estimate, which allows it to hold in the more general context described above. Now in our case, we take

) and A 2 to be the metric double of the bounded curvature surface (∂C P (Y (n) ), d E ∂C P (Y (n) ) ) across its boundary. By the gluing theorem of Alexandrov-Zalgaller (Theorem 8.3.1 in [29]), the metric double A 2 is again a surface of bounded curvature. Thus A 2 is a Riemann surface containing B 2 . Since the conformal structures on ∂C P (Y (n) ) induced by d P and d E are the same, we see that B 2 is conformally embedded in A 2 whose boundary consists of Jordan curves. By the Carathéodory extension theorem, we see that ρ extends to a continuous map from the compact closure U n to ∂C P (Y (n) ). Therefore, ρ is uniformly continuous.

Going forward, we'll assume that d S (x n , x n ) → 0 and |φ n (x n )φ n (x n )| ≥ 0 . By taking a subsequence if necessary, we may further assume that x n , x n → p ∈ Ĉ. (See Fig. 6.) Lemma 7.5. The limit point p is in the set X.

Proof. Suppose otherwise that p / ∈ X, i.e., p ∈ U . Then by the Hausdorff convergence, there exists an open connected neighborhood W of p such that W ⊂ U n for n large. Let d W be the Poincareé metric on W . Then the Schwarz-Pick lemma shows d n (x, y) ≤ d W (x, y) for all x, y ∈ W . But we also have

Fig. 6. Equicontinuity and X n , X * and Y (n) .

This is contradictory to d n (x n , x n ) → 0. q.e.d. By Lemma 7.5 and x n ∈ U n , we see that p ∈ X ∩ ∂U . Let X * be the connected component of X which contains p. Due to the normalization condition on U n , the closed unit disk D is contained in U and U n for all n.

Recall that if Z is a compact subset of a surface S, we define S Z to be (S -Z) ∪ [Z] which is the surface S -Z by adding the ends of S -Z corresponding to connected components of Z in the end topology. Construct two families of paths Γ n and Γ n as follows. If X * is a single point, the family Γ n is defined to be the set of simple loops in ĈX (n) separating {x n , x n } and D and Γ n is defined to be the set of simple loops in ΣY (n) n separating {φ n (x n ), φ n (x n )} and φ n (D). If X * is a round disk, then by the construction of X (n) , X * is a connected component of X (n) for n sufficiently large. Therefore p ∈ ∂X (n) for n large. We define Γ n as the union of two families: simple loops γ 1 and simple arcs γ 2 . Here γ 1 are simple loops in ĈX

. More precisely, Γ n consists of two families of simple loops γ 1 and simple arcs γ 2 such that γ 1 are in ΣY

The conformal invariance of extremal length implies that EL(Γ n ) = EL(Γ n ). We will derive a contradiction by showing that lim inf n EL(Γ n ) > 0 and lim EL(Γ n ) = 0.

Recall that by the normalization condition

Lemma 7.6. For any r > 0, there exist r < r/2 and N such that for n > N, no component of X

Proof. Suppose otherwise, there exists a sequence of components

By taking a subsequence if necessary, we may assume that Z n converges in Hausdorff distance to a disk Z of positive diameter which intersects {z ∈ C||zx k n | = r} and contains p. Since the sequence {X (n) } converges in Hausdorff distance to X, there exists a component X j of X such that Z ⊂ X j . Since p ∈ Z , we have p ∈ X j . On the other hand, X * is the component of X containing p. Therefore X j = X * . This shows that Z ⊂ X * . This is impossible if X * is a one-point component of X. Hence X * is a disk component of X. Since Z n and X * are different components of X (k n ) , we see the distance from the center of Z n to X * is bounded away from zero. This shows that the center of Z is outside of X * and contradicts Z ⊂ X * . q.e.d. By Lemma 7.6 and x n , x n → p, we construct a sequence of positive numbers {r i } and a sequence of integers {N i } increasing to infinity such that (1)

if n ≥ N i and X * is a disk, then X * ∩ {z ∈ C|r j < |zx n | < 2r j } = ∅, for all j = 1, 2, ..., i.

For each n, construct an annulus S n as follows. Since for any round disk B = {z ∈ C||z -a| < r}, the intersection B ∩ (C -X (n) ) is path connected, we can join x n to x n by a path α n in the ball {z ∈ C||z -

For any large i, take n ≥ N i . If X * consists of one point, the disjoint annuli {r j < |zx n | < 2r j } for j = 1, 2, ..., i in S n satisfy conditions in Proposition 6.5 where W = X (n) ∩ S n . Therefore, by (13),

where Γ * n is the set of all simple loops in S X (n) n separating the two ends of S n . On the other hand by construction, Γ n consists of all simple loops in ĈX (n) n) for n large. The disjoint annuli {r j < |zx n | < 2r j } for j = 1, 2, ..., i in S n satisfy conditions in Proposition 6.5 where W = X (n) ∩ S n -X * . Therefore, by (13),

where Γ * n is the set of all simple loops α in S W n separating the two ends of S n . By definition, Γ n consists of all simple arcs β in ĈX (n) such that β or β ∪ [X * ] are simple loops separating {|z| ≤ 1} from {x n , x n }. By condition (4) of the choices of r j 's, we see that each simple loop α in Γ * n contains an arc β which is in Γ n . By Lemma 6.1(a) on the monotonicity of the extremal lengths, we have

We use the work of Reshetnyak [29] on conformal geometry of surfaces of bounded curvature to justify some of computations in this subsection. See Appendix §11 for details. Suppose a convex surface S is ∂C P (Z) for a compact set Z ⊂ ∂H 3 . Then Thurston's theorem says (S, d P S ) is hyperbolic and therefore naturally a Riemann surface. However, the surface S may not be smooth in H 3 . Now consider the two conformally equivalent Riemannian metrics d E and d P on H 3 . If S is smooth, then clearly the induced Riemannian metrics d E S and d P S are conformally equivalent on S. In our case, the surface S may well be non-smooth and the induced path metric d E S may not be Riemannian. But (S, d E S ) is a surface of bounded curvature. Reshetnyak's work implies that d E S and d P S are conformally equivalent in the sense of Ahlfors-Beurling definition in extremal lengths. More precisely, the path metric d E S can be written as λd P S for some non-negative Borel measurable function λ on S. In particular, for a convex surface Σ = ∂C K (Z) in the Klein model (H 3 , d K ), the induced path metrics d K Σ and d Σ on Σ are conformal where

. By Proposition 4.2 and Theorem 4.1, d (x, y) ≥ 1 2 |x -y| and the area of a convex surface Σ in d metric is at most 16π. Define an extended metric m n on ΣY (n) n to be the pair (d Σ n , ν n ) where d Σ n is the induced path metric from d on Σ n and ν n on a connected component of Y (n) is the spherical diameter of the component. Hence the area of m n is uniformly bounded from above by 16π + π 2 • 4π 2 < 100 since the square of the diameter of a spherical disk is at most π/2 times its area. Since

) and the result follows by showing the following

Note that by assumption (0, 0, 0)

is a spherical ball of radius at most π/2 and hence is convex on S 2 . We now prove (17) by contradiction. Suppose otherwise that lim inf n l m n (Γ n ) = 0. After taking a subsequence, we may assume that there exists a sequence of simple loops γ n ∈ Γ n such that l m n (γ n ) → 0. For each γ n , construct a new path γn obtained by gluing to ) ] such that the end points of δ are the end points of γ n -Y (n) . By the construction of m n , we have

If X * is a single point, by construction, γn is a simple loop in Σn separating two compact sets A n := φ n (D) and B n = {φ n (x n ), φ n (x n )}. By Lemma 7.3 and the assumption on B n , both Euclidean diameters of A n and B n are bounded away from 0. After taking a subsequence, we may assume that A n and B n converge in Hausdorff distances to two compact sets A and B of positive Euclidean diameter and γn converges uniformly to γ in ∂C K (Y ) ∪ Y as Lipschitz maps. Then γ separates A, B in ∂C K (Y ) ∪ Y and by the well known fact on path metrics, l E (γ) ≤ lim inf n l E (γ n ). Therefore l E (γ) = 0, i.e. γ is a single point. However, a single point cannot separate A, B in ∂C K (Y ) ∪ Y , which is a topological sphere since Y contains at least 3 points by Lemma 7.3.

n by gluing to γn the shortest geodesic segment

* . If γn is already a closed loop, just let γ * n be γn . By the construction γ * n separates A n from B n in Σn . We claim that lim l E (γ * n ) = 0 and therefore reduce this case to the case just proved above. To this end, let β n be the Euclidean line segment having the same endpoints as

On the other hand, l E (β n ) ≤ l E (γ n ) since they have the same end points. It follows that

Therefore lim n l E (γ * n ) = 0.

Let Σ n = ∂C K (Y (n) ). By Theorem 7.2, each map φ n is uniformly continuous and hence can be extended continuously to a continuous map, still denoted by φ n , from (U n , d S ) to (Σ n , d E ). Here Z is the closure of a set Z in R 3 or Ĉ. Furthermore, the family of the extended maps {φ n : (U n , d S ) → (Σ n , d E )} is again equicontinuous.

Our next goal is to extend φ n continuously to a map from Ĉ = U n ∪ X (n) to Σ n ∪ Y (n) such that the extended family is still equicontinuous with respect to the spherical metric on Ĉ and the Euclidean metric. In the spherical metric d S = 2|dz| 1+|z| 2 on the Riemann sphere Ĉ, all Euclidean disks and half spaces are spherical closed balls. We extend each homeomorphism φ n to a continuous map from ( Ĉ, d S ) to Σ n ∪ Y (n) by coning from the centers of disks. More precisely, let D r = {z ∈ C||z| ≤ r} and S 1 r = ∂D r be the disk of radius r and its boundary in C. Given any homeomorphism f : S 1 r → S 1 R , its Euclidean central extension F : D r → D R is the homeomorphism defined by the formula

For a round disk W = B r (p, d S ) in the 2-sphere S 2 ⊂ R 3 of radius r ≤ π/2, let Ŵ be the 2-dimensional Euclidean disk Ŵ ⊂ R 3 such that ∂ Ŵ = ∂W . The projection ρ : Ŵ → W from -p sends each point x ∈ Ŵ to the intersection of the ray from -p to x with S 2 . It is a bi-Lipschitz homeomorphism whose bi-Lipschitz constant is at most π. We extend a homeomorphism f from the boundary of a spherical disk to the boundary of a spherical disk by the formula f =

where F is the central extension to the Euclidean disk and ρ 1 and ρ 2 are bi-Lipschitz homeomorphisms produced above. For simplicity, we still call f the central extension of f with respect to the spherical metrics. Take a disk component Z of X (n) . Then φ n (∂Z) is the boundary of a disk component Z of Y (n) . Both Z and Z have spherical radii at most π/2 by the normalization condition that X (n) ⊂ {z ∈ C|2 < |z| < 3} and (0, 0, 0)

). Extending φ n to Z by the spherical central extension produces a homeomorphism, still denote it by φ n which is now defined on Ĉ with image in Σ n ∪ Y (n) . Proposition 7.11 below shows that the family of extended continuous maps {φ n : ( Ĉ, d S ) → (Σ n ∪ Y (n) , d E )} are equicontinuous. It is proved in two steps. In the first step, we show that spherical central extensions of functions in an equicontinuous family of maps between circles form an equicontinuous family. Due to the bi-Lipschitz property of projections ρ's, it suffices to show that the Euclidean central extensions of members of an equicontinuous family of maps between circles are still equicontinuous. This is in Proposition 7.7. In the second step, we show that the extended maps {φ n } on ( Ĉ, d S ) are equicontinuous. Suppose g : (X, d) → (Y, d ) is a map between two non-empty metric spaces. Its modulus of continuity function is ω(g

where diam d (A) is the diameter of a set A in a metric space (X, d). We consider the standard 2-dim Euclidean metric in the following proposition.

Proposition 7.7. Suppose F :

given by (18). Here d E denotes the standard 2-dim Euclidean metric and d E ∂D r denotes the natural length metric on ∂D r induced by

Proof. The proof is based on several lemmas.

Proof. We divide the proof into two cases. In the first case cos(θ 2θ 1 ) ≤ 0. Then

Here Re(z) is the real part of a complex number z. In the second case cos(θ

Here Im(z) is the imaginary part of a complex number z. q.e.d.

Proof. It follows from the definition and triangle inequality that if k ∈ Z >0 is a natural number, then

Therefore, for x ∈ (0, 1],

.d. Now we prove Proposition 7.7. Assume that r 1 e √ -1θ 1 , r 2 e √ -1θ 2 are two points in D r such that |r 1 e √ -1θ 1r 2 e √ -1θ 2 | ≤ δ. This implies |r 1r 2 | ≤ δ since |r 1r 2 | ≤ |r 1 e √ -1θ 1r 2 e √ -1θ 2 |. Also, by Lemma 7.8,

Then by Lemmas 7.9, 7.8, ( 20) and ( 19), we have

As a consequence, we have Proof. To see (a), by the normalization conditions that (0, 0, 0) ∈ ∂C K (Y (n) ) and X (n) ⊂ {z ∈ C : 1 < |z| < 2}, each component of Y (n) and X (n) has radius at most π/2 in d S . Thus Corollary 7.10(b) applies. Take any > 0, by Theorem 7.2 and Corollary 7.10, there exists δ > 0 such that if d S (x, y) ≤ δ and either (i) x, y ∈ Ĉ -X (n) or (ii) x, y are both in a connected component of X (n) , then |f n (x)f n (y)| ≤ . It remains to prove the cases where the pair x, y with d S (x, y) ≤ δ satisfy that (1) one of them is in X (n) and the other is in Ĉ -X (n) , or (2) x, y are in different connected components of X (n) . Consider a shortest geodesic γ joining x to y in ( Ĉ, d S ). In the first case (1), we may assume that x ∈ X (n) and y ∈ Ĉ -X (n) . Let z be an intersection point of γ ∩ ∂X (n) such that x and z are in the same component of X (n) . Then d S (x, z) ≤ δ and d S (z, y) ≤ δ. Therefore,

In the second case (2) that x, y are in different components of X (n) , then the geodesic segment γ contains a point z / ∈ X (n) To see part (b), we begin with, Lemma 7.12. Let X n (resp. X n ) be compact subsets of a metric space W (resp. W ) such that X n (resp. X n ) converges in Hausdorff distance to a compact subspace X (resp. X ). Suppose f n : X n → X n is a sequence of continuous functions converging uniformly to f : X → X . If A n ⊂ X n is a sequence of compact sets converging in Hausdorff metric to a compact set A, then f n (A n ) converges in Hausdorff metric to f (A). In particular, if f n is onto for all n, f is onto.

Proof. Take any point f (a) in f (A) with a ∈ A. By definition that A n converges in Hausdorff distance to A, there exists p n ∈ A n such that p n → a. By uniform convergence,

) is a converging sequence whose limit is q. By taking a subsequence if necessary, we may assume that p n i → a ∈ A. Therefore, by uniform convergence, q = lim i f n i (p

This shows that {f n (A n )} Alexandrov converges to f (A). Since f n (A n ) and f (A) are compact, we see f n (A n ) converges in Hausdorff distance to f (A). q.e.d.

Lemma 7.12 implies that φ( Ĉ) = ∂C K (Y ) ∪ Y and φ(X) = Y since φ n and φ n | X (n) are onto maps. By the work of Alexandrov and convergence of Poincaré metrics (Theorems 5.1 and Theorem 5.2), we see that the restriction φ| U is an isometry from (U, d U ) into the component of ∂C K (Y ) which contains (0, 0, 0). Using φ(U ∪ X) = ∂C K (Y ) ∪ Y , φ(X) = Y and φ(U ) ⊂ ∂C K (Y ), we see that φ(U ) = ∂C K (Y ). In particular, the map φ| U is an isometry from (U, d U ) onto ∂C K (Y ). Since U is connected, we see ∂C K (Y ) is connected. q.e.d.

Now take a connected component Y k of Y . To show it is a round disk or a point, we use Proposition 7.11 to find Y k =φ(X k ) for some connected component X k of X. Indeed since φ is onto, there exists a connected component, say X k of X, which is mapped by φ into Y k . Since φ| U is a homeomorphism from U to ∂C K (Y ) and φ| U induces bijection on spaces of ends (see [28]), we have φ(X k ) = Y k . Since X is a circle type closed set, there exists a sequence X (n) k n of components of X (n) converging in Hausdorff distance to X k . By the uniform convergence of φ n to φ and Lemma 7.12, φ n (X U, d U ). The basic strategy of the proof is the same as that in Theorem 1.1 (a). In this section, we prove Theorem 1.1 (b) using an equicontinuity property which will be established in §9.

We will use the Poincaré ball model (H 3 , d P ) of the hyperbolic 3-space in the rest of the section unless mentioned otherwise. By Theorem 3.1, we may assume that the set Y is not contained in any circle, i.e., C P (Y ) is 3-dimensional. Composing with a Möbius transformation of ∂H 3 , we may assume that (0, 0, 0) ∈ ∂C P (Y ). By Carathéodory's theorem on convex hull (Proposition B.6 in [7]), there exist four components Y

Since Σ n is a genus zero Riemann surface of a finite topological type, by Koebe's circle domain theorem, there exists a circle domain U n = Ĉ -X (n) and a conformal diffeomorphism φ n : Σ n → U n . Using Möbius transformations, we normalize U n such that 0 ∈ U n , φ n (0, 0, 0) = 0 and the open unit disk D is a maximum disk contained in U n , i.e., X (n) ⊂ Ĉ -D and X (n) ∩ ∂D = ∅. By taking a subsequence if necessary, we may assume that X (n) converges in Hausdorff distance to a compact set X in Ĉ such that X ⊂ Ĉ -D and X ∩ ∂D = ∅. Our goal is to prove that U = Ĉ -X is a circle domain and (U, d U ) is isometric to ∂C P (Y ) Lemma 8.1. The hyperbolic injectivity radii of the surfaces (Σ n , d P

Σ n ) at (0,0,0) are bounded away from zero.

Proof. Let S n = ∂C K (Y (n) ) be the corresponding surface in the Klein model and d K S n be the induced path metric on S n . The isometry Ψ(x) = 2x 1+|x| 2 from d P to d K induces an isometry from (Σ n , d P Σ n ) to (S n , d K S n ). Since Ψ(0) = 0, we will prove the result for (S n , d K S n ) at (0, 0, 0). In the Klein model, both C K (Y (n) ) and S n are a Euclidean convex body and an Euclidean convex surface, respectively. By the assumption that Y is not in a circle, the convex set C K (Y ) is 3-dimensional and contains an Euclidean ball. It follows that there exists an Euclidean ball B which is contained in C K (Y (n) ) for all large n. This implies that the Euclidean injectivity radii of S n = ∂C K (Y (n) ) in the path metric d E S n at (0, 0, 0) are bounded away from zero. Indeed, if otherwise, we find a sequence of homotopically non-trivial loops δ n k based at (0, 0, 0) in ∂C K (Y (n k ) ) such that the lengths of δ n k tend to zero. Then the 3-dimensional convex bodies C K (Y (n k ) ) will converge in Hausdorff distance to a 1-dimensional convex set. This contradicts the fact that C

the injectivity radii of (S n , d K S n ) are bounded away from 0. q.e.d. Lemma 8.2. The closed set X contains at least three points, i.e., |X| ≥ 3. Furthermore, for any r > 0, there exists r > 0 such that the spherical ball B r (0, d S ) is contained in B r (0, d n ) in U n for all n where d n = a n (z)|dz| is the Poincaré metric on U n Proof. If X contains at most 2 points, say X ⊂ {a, b} ⊂ Ĉ -D. Then for any > 0, X (n) ⊂ B (a, d S ) ∪ B (b, d S ) for sufficiently large n. We claim that lim n a n

On the other hand, since (Σ n , d P Σ n ) and (U n , d n ) are isometric, by Lemma 8.1, there exists a hyperbolic ball B r (0, d n ) of radius r > 0 centered at 0 in U n such that B r (0, d n ) is isometric to the standard ball B r (0, d P ) for all n. Let f n be an orientation preserving isometry from the ball B r (0, d P ) in the Poincaré disk to B r (0, d n ) in (U n , d n ). Since f n is an isometry, we have 2|dz| 1-|z| 2 = a n (f n (z))|f n (z)|. This shows |f n (0)| = 2/a n (0) and by (21), |f n (0)| → ∞. Recall Koebe's quarter theorem says that if g : B r (0, d E ) → C is an injective analytic map, then its image g(B r (0, d E )) contains the Euclidean ball of radius |g (0)|r 4 centered at g(0). Applying it to the injective analytic maps f n defined on B r (0, d P ), we see that f n (B r (0, d P )) contains the Euclidean disk of radius 2 centered at 0 for n large. This contradicts the assumption that ∂D ∩ ∂U n = ∅ and shows |X| ≥ 3.

Finally to see the second part of the Lemma, by the Schwarz-Pick lemma applied to D ⊂ U n , we have 2|dz| 1-|z| 2 ≥ a n (z)|dz| and in particular a n (0) ≤ 2. Therefore, |f n (0)| ≥ 1. Then Koebe's quarter Theorem implies that f n (B r (0, d P )) contains B r (0, d E ) for some r independent of n. Since d E and d S are bi-Lipschitz equivalent when restricted to D, the result follows. q.e.d. Now we prove Theorem 1.1(b). The key result used in the proof is the following equicontinuity theorem to be proved in §9.

Assuming the theorem, by equicontinuity, each map φ n : Σ n → U n extends to a continuous map, still denoted by φ n : Σ n → U n between their closures in R 3 and Ĉ. Furthermore, the extended family {φ n : (Σ n , d E ) → (U n , d S )} is still equicontinuous. Now each boundary component of Σ n and U n is a round circle or a point. Use central extension to extend φ n to be a continuous map, still denoted by φ n , from Σ n ∪ Y (n) to U n ∪ X (n) = Ĉ. By the normalization condition that (0, 0, 0) ∈ ∂C P (Y (n) ) and X (n) ⊂ Ĉ-D, each component of Y (n) and X (n) has spherical radius at most π/2 for n > 4. Using Corollary 7.10 and a similar argument for Proposition 7.11, the extended family {φ n : (Σ n ∪Y (n) , d E ) → ( Ĉ, d S )} is equicontinuous. Since Y (n) and ∂C P (Y (n) ) and converge to Y and ∂C P (Y ) in Hausdorff metrics respectively, it follows that their union ∂C P (Y (n) ) ∪ Y (n) converges in Hausdorff metric to ∂C P (Y ) ∪ Y . By the generalized Arzela-Ascoli Theorem 5.8, we may assume, after taking a subsequence, that φ n converges uniformly to a continuous map φ : n) , by Lemma 7.12, φ(Y ) = X and φ is onto.

By Lemma 8.2, Alexandrov's convergence Theorem 5.1 and convergence theorem of Poincaré metrics (Theorem 5.2), we see that φ| ∂C P (Y ) is an isometric embedding of ∂C P (Y ) into a component Ω of Ĉ -X. In particular, φ(∂C P (Y )) ⊂ Ĉ -X. Together with φ(Y ) = X and that φ is onto, we see that φ(∂C P (Y )) = Ĉ -X. Therefore, φ is an isometry from ∂C P (Y ) to Ĉ -X. Now we claim that X is a circle type closed set. Indeed, by the same argument as in §7.4, each component X k of X is of the form φ(Y k ) for some component Y k of Y . By Lemma 7.12 and that Y is of circle type, each φ(Y k ) is the Hausdorff limit of a sequence of components φ n (Y

k n of X (n) . Therefore the result follows.

We prove the above theorem by deriving a contradiction. Suppose otherwise, there exist > 0 and two sequences

If the sequence {k n } is not bounded, we may assume, after taking a subsequence, that k n = n. If the sequence {k n } is bounded, we may assume after taking a subsequence that k n is a constant. Below we will focus on the main case that k n = n. The same proof also works for the simpler case that k n is a constant. We omit the details. Since the case k n is equivalent to the uniform continuity of φ n , we can also prove the uniform continuity of φ n by using Carathéodory's extension theorem as discussed in §7.2.

Moving forward, we assume that |y ny n | → 0,

and {y n } converges to some point p ∈ ∂C P (Y ) ∪ Y . We claim that q ∈ Y . To see this, we need the following lemma.

Lemma 9.1. Let d n be the Poincaré metric on U n . There exists a constant C 0 > 0 independent of n such that

Proof. Since the sequence {X (n) } Hausdorff converges to X and |X| ≥ 3 (Lemma 8.2), we can choose a 3-point set {w 1 , w 2 , w 3 } ⊂ X. Let W = Ĉ -{w 1 , w 2 , w 2 } and d W = a(z)|dz| be the Poincaré metric on W . Note that a(z) tends to infinity as z approaches ∂W since each w i corresponds to a cusp end of W . Therefore, there exists a constant

where

3 }. To see this, let M n be the Moebius transformation sending u (n) i to w i for i = 1, 2, 3. Then M n converges uniformly to the identity map in ( Ĉ, d S ). Then there exists a positive C 0 such that Lemma 9.3. For any r > 0, there exists r < r/65 such that no component of Y -Y * intersects both ∂D r and ∂D r .

Proof. By a simple spherical area estimate, we see that there exist only finitely many components Z 1 , ..., Z m of Y -Y * such that Z i intersects both ∂D r and ∂D r/65 . Let r be a positive number with r < r/65 and r < min 1≤i≤m d S (Y * , Z i ). The result follows. q.e.d. By Lemma 9.3 and q = lim n y n = lim n q n , we construct a sequence of positive numbers {r i } and a sequence of integers {N i } increasing to infinity such that (1) r j+1 < r j /65 for all j, (2) each component of Y -Y * intersects at most one of D 65r j -D r j for all j, (3) if n ≥ N i , then {y n , y n } ⊂ π(D n r i ). Note that since Y (n) ⊂ Y , we see that each component of Y (n) -Y * intersects at most one of D 65r j -D r j for all j.

We now prove lim n EL(Γ n ) = 0 by showing that EL(Γ n ) ≤ 3 •10 8 /i for all n ≥ N i . We will use Proposition 6.7 by constructing a family of annuli S n as follows. For simplicity, we will use the following notation. If Z is a closed subset of S 2 , then Z * denotes the union of Z and all components Y k of Y -Y * which intersect Z, i.e.,

Note that if Z is connected, then so is Z * . Now fix i ≥ 1 and n ≥ N i . Then by Proposition 4.7 and condition (2) above, π( Dn * r i ) ∩ π(E n * 65r 0 ) = ∅. It follows that π( Dn * r i ) and π(E n * 65r 0 ) are disjoint connected compact sets in Σn . Hence there is a unique component of Σn -(π( Dn * r i ) ∪ π(E n * 65r 0 )) which is an annulus. We denote this annulus by S n . By condition (3), the annulus S n separates {y n , y n } and B n in Σn . See Fig. 7. Let W = Y (n) ∩ S n -Y * , Ω r j = {z ∈ S 2 |r j < d S (z, q n ) < 65r j } for j = 0, 1, ..., i -1, and Γ * n be the set of all simple loops in S W n separating the two ends of S n . Applying Proposition 6.7 to S = S n , W = Y (n) , and Ω r j for j = 0, 1, ..., i, we obtain,

The rest of the proof is similar to that of §7.2.1. If Y * is a one-point set, then by the construction Γ * n ⊂ Γ n . Therefore, the monotonicity of the extremal length implies

The same proof used in §7.2.1 shows that lim n EL(Γ n ) = 0 if Y * is a disk.

A polyhedral surface is a triple (S, V, d) where S is connected surface, V ⊂ S is a discrete subset, and d is a flat cone metric on S with cone points contained in V . We call d a PL or polyhedral metric on (S, V ). The discrete curvature K of the polyhedral surface is a function defined on the vertices, and k(v) is 2π less the cone angle at v. Usually, these metrics are obtained by isometric gluing of Euclidean triangles along pairs of edges by isometries. Thus, a polyhedral surface (S, V, d) can be represented by a triangulated PL surface (S, T , l) where T is a triangulation with vertex set V and l : E(T ) → R >0 is the edge length function, i.e., l(e) is the length of the edge e. Here E(T ) is the set of all edges in T . One of the goals of the discrete conformal geometry is to define discrete conformal equivalence among PL metrics on (S, V ) and establish the corresponding discrete uniformization theorem. In our recent work [16], we are able to introduce a discrete conformality for polyhedral metrics and establish a discrete uniformization theorem for compact surfaces. We will briefly recall the related results and their relationship to the Weyl problem and discuss one application of the main result in discrete conformal geometry.

A basic tool in computational geometry is the Delaunay triangulation. In the 2dimensional case, a triangulated PL surface (S, T , l) is called Delaunay if the circumdisk of each triangle contains no vertices in its interior. This is equivalent to the condition that if e is an edge adjacent to two triangles t and t , then the sum of the two angles in t and t which are opposite to e is at most π. A fundamental theorem in computational geometry says that for any closed PL surface (S, V, d), there is always a Delaunay triangulation T of (S, V, d) with vertex set equal to V . In general, Delaunay triangulations of (S, V, d) may not be unique. Discrete conformal geometry tries to define discrete conformal equivalence between to polyhedral surfaces (S, V, d) and (S, V, d ) such that (i) discrete conformal equivalence is computable, (2) discrete conformal maps converge to smooth conformal maps as meshes tend to zero, and (3) there exists a discrete uniformization theorem within each discrete conformal class.

Given a PL metric d on (S, V ), construct a Delaunay triangulation T of (S, V, d). For each Euclidean triangle τ = ΔABC in T (considered as a triangle in C), replace τ by the ideal hyperbolic triangle τ * in the upper-half-space model C × R >0 of the hyperbolic 3-space such that τ * and τ have the same set of vertices {A, B, C} in C. If τ and σ are two Euclidean triangles in T glued along a pair of edges by a Euclidean isometry f , then one glues τ * and σ * along their corresponding edges by the same isometry f , considered as a rigid motion of H 3 . In this way, one produces a complete finite area hyperbolic metric d * on S -V . From the definition of Delaunay triangulation, one sees that d * is independent of the choices of the Delaunay triangulation T . (See Fig. 8.) Definition 10.1. (Discrete conformality of PL metrics) [16] Two PL metrics d 1 and d 2 on a marked surface (S, V ) are discrete conformal if there exists an isometry φ : (S -V, d * 1 ) → (S -V, d *

2 ) such that φ is homotopic to the identity map relative to V .

The main theorem proved in [16] is, Theorem 10.2. Suppose (S, V ) is a closed connected marked surface and d is a PL metric on (S, V ). Then for any K * : V → (-∞, 2π) with v∈V K * (v) = 2πχ(S), there exists a PL metric d , unique up to scaling and isometry homotopic to the identity on (S, V ), such that d is discrete conformal to d and the discrete curvature of d is K * .

For the constant function K * = 2πχ(S)/|V | in Theorem 10.2, we obtain a constant curvature PL metric d , unique up to scaling, discrete conformal to d. This is a discrete version of the uniformization theorem.

Theorem 10.2 takes care of compact polyhedral surfaces. For non-compact simply connected polyhedral surface (S, V, d), the discrete uniformization problem asks if it is discrete conformal to the following two types of surfaces: (C, V , d st ) or (D, V , d st ). Here d st is the standard Euclidean metric on C and V is a discrete set in C or D. It is easy to see that if V is the set of vertices of a Delaunay triangulation in C or D, then the associated hyperbolic metric to (C, V , d st ) and (D, V , d st ) are exactly the boundary of the convex hulls ∂C P (V ) and ∂C P (V ∪ D c ). It is easy to see that the hyperbolic metric associated with (S, V, d) is a complete hyperbolic metric with cusp ends at points in V . Thus the discrete uniformization problem for non-compact surfaces proposed in [23] is the following, Conjecture 4. Suppose (Σ, d) is a complete hyperbolic surface with countably many ends, and all but at most one are the cusp ends. Then there exists, unique up to Möbius transformations, a circle type closed set X such that (Σ, d) is isometric to ∂C P (X). where w d n is the curvature measure and κ d n is the total geodesic curvature of a path. Conversely, if (S, d n ) is a sequence of polyhedral surfaces converging uniformly to a path metric (S, d) such that (25) holds on S for some constant C, then (S, d) is a surface of bounded curvature.

The construction of conformal charts for surfaces of bounded curvature by Reshetnyak goes as follows. Suppose U is an open disk in the plane and w is a signed Borel measure on U . Then the function ln λ(z) = 1 π U 1 |z-ζ| w(dζ) + h(z) is the difference of two subharmonic functions on U where h is a harmonic function. Since the Hausdorff dimension of the set of points where ln λ(z) = -∞ is zero, for an arbitrary Euclidean rectifiable path L in U , the integral L λ(z(s))ds is well defined (could be ∞). One defines the distance d U on U between two points to be the infimum of the lengths of paths between them in λ(z)|dz| 2 . There may be some points whose d U -distance to any other point is infinite. These are called points at infinity and they form a discrete subset in U . Let Ũ be the complement of the set of points at infinity. It is proved by Reshetnyak that ( Ũ, d U ) is a surface of bounded curvature whose curvature measure is w. The main theorem of Reshetnyak's conformal geometry of surfaces of bounded curvature is Theorem 7.1.2 in [29].

Theorem A.2 (Reshetnyak). Let (S, d) be a surface of bounded curvature. Then for any point p ∈ S, there exists a neighborhood U of p and an open disk U in the plane together with a Borel measure w such that (U , d| U ) is isometric to ( Ũ, d U ).

Let φ : (U , d| U ) → ( Ũ, d U ) be an orientation preserving isometry produced in the above theorem. Then {(U , φ)} forms the analytic charts on the surface (S, d).

In conclusion, the metrics in surfaces with bounded curvature can be treated as Riemannian distance derived from Riemannian metrics by relaxing the smoothness condition.

Therefore, in a surface of bounded curvature (S, d) whose area measure is m and the underlying conformal structure is C, we can use the path metric d and area measure m to compute the extremal length of a curve family Γ. In particular, we have the estimate

This estimate has been used extensively in previous sections on (∂C H (Y ), d P ∂C H (Y ) ). Finally, for a compact set Y ⊂ S 2 and Σ = ∂C H (Y ), we claim that the two induced path metrics d P Σ and d E Σ on Σ produce the same complex structure. In particular, this shows for any curve family Γ in Σ, EL(Γ, Σ, d P Σ ) = EL(Γ, Σ, d E Σ ). To see the claim, following Alexandrov [3], one constructs a sequence of convex hyperbolic polyhedral surfaces converging uniformly on compact sets to (∂C H (Y ), d P ∂C H (Y ) ). The convexity implies that (25) holds. Now on polyhedral surfaces, the conformal structures induced from d P and d E are the same since these two metrics are conformal in H 3 . The work of Reshetnyak ([29], Theorems 7.3.1, p112) shows that if a sequence of bounded curvature surfaces converge uniformly to a bounded curvature surface such that (25) holds, then the isothermal coordinates (with appropriate normalization) converge to the isothermal coordinate of the limit surface. Therefore the conformal structures on ∂C H (Y ) are the same.