On the Weyl problem for complete surfaces in the hyperbolic and anti-de Sitter spaces
aa r X i v : . [ m a t h . DG ] D ec ON THE WEYL PROBLEM FOR COMPLETE SURFACES IN THE HYPERBOLICAND ANTI-DE SITTER SPACES
JEAN-MARC SCHLENKER
Abstract.
The classical Weyl problem (solved by Lewy, Alexandrov, Pogorelov, and others) askswhether any metric of curvature K ≥ R . The answer was extended to surfaces in hyperbolic space by Alexandrov in the 1950s, anda “dual” statement, describing convex bodies in terms of the third fundamental form of their boundary(e.g. their dihedral angles, for an ideal polyhedron) was later proved.We describe three conjectural generalizations of the Weyl problem in H and its dual to unboundedconvex subsets and convex surfaces, in ways that are relevant to contemporary geometry since a numberof recent results and well-known open problems can be considered as special cases. One focus is on convexdomain having a “thin” asymptotic boundary, for instance a quasicircle – this part of the problem isstrongly related to the theory of Kleinian groups. A second direction is towards convex subsets with a“thick” ideal boundary, for instance a disjoint union of disks – here one find connections to problems incomplex analysis, such as the Koebe circle domain conjecture. A third direction is towards complete,convex disks of infinite area in H and surfaces in hyperbolic ends – with connections to questions oncircle packings or grafting on the hyperbolic disk. Similar statements are proposed in anti-de Sittergeometry, a Lorentzian cousin of hyperbolic geometry where interesting new phenomena can occur, andin Minkowski and Half-pipe geometry. We also collect some partial new results mostly based on recentworks.Keywords: Weyl problem, hyperbolic geometry, convex surface, isometric embedding. Contents
1. The classical Weyl problem in Euclidean and hyperbolic space 21.1. Convex surfaces in Euclidean space 21.2. The Weyl problem and its dual for closed convex surfaces in hyperbolic space 21.3. Convex domains with a “thin” ideal boundary in H H H Date : v1, January 1, 2021.Partially supported by FNR projects INTER/ANR/15/11211745, OPEN/16/11405402 and O20/14766753. The au-thor also acknowledges support from U.S. National Science Foundation grants DMS-1107452, 1107263, 1107367 “RNMS:GEometric structures And Representation varieties” (the GEAR Network).
5. Immersed locally convex surfaces 145.1. Pleated surfaces and grafting 155.2. Circle patterns and circle packings 165.3. Parameterization of locally convex equivariant embeddings 175.4. Similar statements for K -surfaces 186. The Weyl problem for unbounded surfaces in anti-de Sitter and Minkowski geometry 196.1. Basic information on AdS geometry 196.2. Convex domains with thin boundary 216.3. Spacelike disks in ADS K -surfaces in higher Teichm¨uller theory 27References 271. The classical Weyl problem in Euclidean and hyperbolic space
Convex surfaces in Euclidean space.
Smooth or polyhedral convex surfaces in the 3-dimensionalEuclidean space R have been a staple of geometry throughout the ages. One of the first “modern” resultson this theme was the proof by Legendre [LegII] and Cauchy [Cau13] of the rigidity of convex polyhedra:if two polyhedra have the same combinatorics and corresponding faces are isometric, then they arecongruent.Much later, Alexandrov [Ale05] realized that Cauchy’s rigidity result is actually a consequence ofa much deeper statement. The induced metric on the boundary of a convex polyhedron in R is anEuclidean metric with cone singularities with cone angle less than 2 π at the vertices. Alexandrov provedthat each metric of this type on the sphere can be realized as the induced metric on the boundary of aunique polyhedron. This result was later extended by Alexandrov [Ale05] to bounded convex subset of R , thus solving the “Weyl problem”, proposed by H. Weyl in 1915 and for which substantial progresshad already been made by H. Lewy [Lew35] and others. Theorem 1.1 (Lewy, Alexandrov) . Any smooth metric of positive curvature on the sphere is induced onthe boundary of a unique smooth strictly convex subset of R . Here (and elsewhere) by “strictly convex” we mean that the shape operator of the boundary is positivedefinite. By “smooth” we mean C ∞ , although different degrees of regularity could be considered.This result was then further extended by Pogorelov [Pog73] to non-smooth metrics: the induced metricon the boundary of any bounded convex subset of R , even if non-smooth, is of curvature K ≥ The Weyl problem and its dual for closed convex surfaces in hyperbolic space.
Alexan-drov and Pogorelov extended Theorem 1.1 to hyperbolic space.
Theorem 1.2 (Alexandrov, Pogorelov) . Any smooth metric of curvature
K > − on the sphere isinduced on the boundary of a unique convex subset of H with smooth boundary. Note that Theorem 1.1 can be seen as a limit case of Theorem 1.2. If h is a smooth metric on S ofcurvature K >
0, one can apply Theorem 1.2 to the sequence of metrics h n = (1 /n ) h , which are alsometrics of curvature K > > −
1, yielding a sequence of surfaces S n ⊂ H , with the induced metric on S n isometric to h n . The diameter of S n converges to 0 as n →
0, so, after applying a sequence of isometrices,the sequence ( S n ) n ∈ N converges to a point x ∈ H . After applying to H a squence of homotheties(multiplication of the metric by n ) and extracting a subsequence, ( S n ) n ∈ N converges to a convex surfacesin R , with induced metric h . HE WEYL PROBLEM FOR COMPLETE SURFACES 3
Versions of Theorem 1.2 also hold for ideal polyhedra [Riv92] and for hyperideal polyhedra [Sch98a].The existence of those extensions hint at the possibility of a wider extension to more general unboundedconvex domains in H as considered here.In addition, a new phenomenon appears, which was present in the Euclidean situation only in a ratherdegenerate form. Recall that if S is a surface in H with induced metric I and shape operator B , itsthird fundamental form is defined by III ( x, y ) = I ( Bx, By ) , for any two vector fields x, y tangent to S . The third fundamental form therefore measures how “curved”the surface S is. The following statement is from [Sch96]. Theorem 1.3.
Let Ω be a subset of H with smooth strictly convex boundary. The third fundamentalform of the boundary has K < , and its closed geodesics have length L > π . Conversely, any metric ofthis type on the sphere can be realized as the third fundamental form of the boundary of a unique subsetof H with smooth, strictly convex boundary. A version of Theorem 1.3 was obtained earlier by Hodgson and Rivin [HR93] for convex polyhedra,following work of Andreev [And71]. Note that taken together, the condition that
K <
L > π is (almost) equivalent to asking that the metric is (globally) “strongly” CAT (1), and it is this strong
CAT (1) condition that we will use below for simplicity.Similar statements exist for the dihedral angles (or more precisely “dual metrics”, which are essentiallypolyhedral versions of the third fundamental form) of compact hyperbolic polyhedra [And70, HR93], forthe dihedral angles of ideal polyhedra [And71,Riv96] and of hyperideal polyhedra [BB02,Rou04,Sch98a].In the next three subsection, we introduce three different questions, each stated for the induced metricand, in a dual version, for the third fundamental form. In the next sections we will describe special casesof those problems corresponding either to noted open problems, or to relatively recent results.There are in fact two separate points of views on the classical Weyl problem, leading to two differentkinds of generalizations. • It can be considered as a statement on the induced metric on the boundary of convex subsets of R (or H ). This point of view leads to Questions W thin H and W thick H , as well as their commongeneralization W gen H , and their duals. • Or, it can be seen as a statement on isometric embeddings in R (or H ) of surfaces with Riemann-ian metrics, under curvature conditions implying that the image surfaces are locally convex. Thispoint of view leads to Question W imm H , which concerns immersed surfaces in H with prescribedinduced metrics (or its dual concerning immersed surfaces with prescribed third fundamentalform).We will see that the “Weyl problem” point of view suggests a number of extensions of classical or recentresults that would be special case of three more general results (each for induced metrics or their duals).1.3. Convex domains with a “thin” ideal boundary in H . Given an (unbounded) convex subsetΩ ⊂ H , we denote by ∂ ∞ Ω its ideal boundary, which can be defined for instance using the projectivemodel of H , as the intersection of ∂ ∞ H with the closure of Ω in the projective model. Definition 1.4.
A convex subset Ω ⊂ H has thin boundary if ∂ ∞ Ω is the disjoint union of a finitefamily of quasi-circles, each bounding a disk in C P \ ∂ ∞ Ω . Some remarks can be made here.(1) Interesting examples arise already when ∂ ∞ Ω is only one quasicircle. (See below.)(2) On the other hand, interesting questions arise for more general boundaries, that is, when ∂ ∞ Ω isthe disjoint union of an infinite set of Jordan curves and points.(3) The condition that ∂ ∞ Ω is a union of quasicircles, rather than more general Jordan curves, mightnot be necessary in all cases. We do believe however that this condition is relevant and probablynecessary for at least some statements.(4) The “correct” condition on ∂ ∞ Ω might/should be more general than a union of quasicircle, andinclude for instance the limit set of a convex co-compact hyperbolic manifold. See Section 1.5below for a possible extension that covers this case.
JEAN-MARC SCHLENKER
Let Ω ⊂ H be a convex subset with thin boundary. Then ∂ Ω is the disjoint union of a finite numberof regions ∂ Ω , ∂ Ω , · · · , ∂ n Ω: one disk ∂ i Ω bounded by each quasicircle c i in ∂ ∞ Ω, 1 ≤ i ≤ n , plus onesurface ∂ Ω of genus 0 with one boundary component for each of the c i . Each of the disks is conformalto the disk D , while ∂ Ω is conformal by Koebe’s theorem [Koe18] to a unique domain in C P which isthe complement of the union of a finite number of disjoint round disks. (If ∂ ∞ Ω is composed of only onequasicircle, we choose ∂ Ω to be one of the connected components of its complement.)For each i ∈ { , · · · , n } , we denote by ∂ i ∂ Ω the connected component of ∂ Ω corresponding to the disk D i , so that ∂ i ∂ Ω can be identified with c i . For each i ∈ { , · · · , n } , both ∂ ( ∂ i Ω) and ∂ i ∂ Ω are equippedwith a real projective structure through the identifications with RP above, and the identification of bothwith c i determines a one-to-one map: σ i : ∂ ( ∂ i Ω) → ∂ i ∂ Ω. We let σ = ( σ , · · · , σ n ), and call σ the induced metric gluing map of Ω.One can define the third fundamental form gluing map in the same manner, but with the induced metricreplaced by the third fundamental form in the application of the Koebe theorem and of the Riemannmapping theorem. Since we will only assume that the curvatures of the metrics under consideration areless than 1 − ǫ , we will need to add in the hypothesis that the each connected component is of hyperbolictype, so that the gluing maps are well-defined.We will see in Lemma 2.1 that, if ∂ Ω has pinched curvature with lower bound larger than −
1, and ∂ ∞ Ω is a union of quasicircles, then the induced metric and third fundamental form gluing maps of eachboundary components are quasi-symmetric.We can now state, in a voluntarily rather restricted setting, a first version of the Weyl problem forunbounded convex subsets in H . This version concerns convex subsets with “thin” ideal boundary. Question W thin H . Let c , c , · · · , c n be circles bounding disjoint closed disks D , · · · , D n in C P , let D = C P \ ( ¯ D ∪ · · · ∪ ¯ D n , let h be a complete metric of curvature [ − ǫ, − ǫ ] on C P \ c ∪ · · · ∪ c n ,and for each i ∈ { , · · · , n } , let σ i : ∂D i → ∂ i D be a quasi-symmetric homeomorphism, where ∂ i D isthe connected component of ∂D corresponding to c i . Is there a unique convex domain Ω ⊂ H with thinideal boundary, such that the induced metric on ∂ Ω is isometric to h , with induced metric gluing map at c i equal to σ i ? Question W ∗ thin H . Let c , c , · · · , c n be circles bounding disjoint closed disks D , · · · , D n in C P , let D = C P \ ( ¯ D ∪ · · · ∪ ¯ D n , let h be a complete metric of curvature [ − /ǫ, − ǫ ] on C P \ c ∪ · · · ∪ c n ,and for each i ∈ { , · · · , n } , let σ ∗ i : ∂D i → ∂ i D be a quasi-symmmetric homeomorphism. Assume thateach connected component of C P \ c ∪ · · · ∪ c n , equiped with the conformal class of h , is of hyperbolictype, and that all closed contractible geodesics in ( C P \ c ∪ · · · ∪ c n , h )) have length larger than π . Isthere a unique convex domain Ω ⊂ H with thin ideal boundary, such that the third fundamental form on ∂ Ω is isometric to h , with third fundamental form gluing map at the c i equal to σ i ? Note that, here and in the other statements below, the uniqueness is up to the action of the group ofisometries of the ambient space, here H .We will see in Section 3 a number of relations between questions W thin H and W ∗ thin H on the one hand,and open questions or known results concerning quasifuchsian hyperbolic manifolds. We will also mentionrecent result which can be considered as answers to the existence part of those questions, including twoconjectures of Thurston on the geometric data on the convex core of quasifuchsian (or more generallyconvex co-compact) hyperbolic manifolds.1.4. Convex domains with a “thick” ideal boundary in H . The second setting that we wouldlike to consider is that of convex subsets with “thick” ideal boundary, that is, an ideal boundary that hasnon-empty interior. One can think for instance of a convex domain Ω ⊂ H such that ∂ ∞ Ω is a disjointunion of topological disks D , · · · , D n ⊂ C P . Then ∂ ∞ ( ∂ Ω) = ∂ ( ∂ Ω) ∪ · · · ∪ ∂ n ( ∂ Ω), where ∂ i ( ∂ Ω) isequal to ∂D i . One particularly relevant example of course is when ∂ ∞ Ω is a single topological disk.As in the case of domains with thin boundary, we can associate to each connected component D i of ∂ ∞ Ωa homeomorphism ρ i : ∂ i ( ∂ Ω) → RP , well-defined up to left composition by a M¨obius transformation,using the identification of ∂D i with RP coming from the Riemann mapping theorem applied to D i .When defined by applying the Koebe theorem to ∂ Ω equipped with its induced metric, we call it the induced metric gluing map , while we call it the third fundamental gluing map when ∂ Ω is equipped withits third fundamental form.Again, for simplicity, we state a question and its dual in a relatively limited setting.
HE WEYL PROBLEM FOR COMPLETE SURFACES 5
Question W thick H . Let Ω ⊂ S be the complement of a disjoint union of disks D i , i ∈ I . Let g be acomplete metric of curvature K ∈ [ − ǫ, − ǫ ] on Ω and, for each i ∈ I , let σ i : ∂ i (Ω , g ) → RP be aquasisymmetric homeomorphism (considered up to composition with a M¨obius transformation). Is therea unique convex domain K ⊂ H such that: • The induced metric on ∂K is isometric to (Ω , g ) , • for each i ∈ I , the induced metric gluing map between the corresponding connected component of ∂ ∞ ( ∂K ) and RP , described above, is equal to σ i (up to left composition by M¨obius transforma-tions)? Question W ∗ thick H . Let Ω ⊂ S be the complement of a disjoint union of disks D i , i ∈ I . Let g be acomplete metric of curvature K ∈ [ − /ǫ, − ǫ ] on Ω with closed geodesics of length L > π and, for each i ∈ I , let σ i : ∂ i (Ω , g ) → RP be a quasisymmetric homeomorphism (considered up to composition with aM¨obius transformation). Is there a unique convex domain K ⊂ H such that: • The third fundamental form of ∂K is isometric to (Ω , g ) , • for each i ∈ I , the third fundamental form gluing map between the corresponding connectedcomponent of ∂ ∞ ( ∂K ) and RP , described above, is equal to σ i (up to left composition by M¨obiustransformations)? Note that, here again, the curvature conditions are quite restrictive, and it appears quite possible thatmore general curvature conditions could be considered. (It should be necessary however to have curvatureat least − W thick H , and curvature at most 1 in Question W ∗ thick H , since this follows from theconvexity of the surfaces considered.)We will see in Section 4 that those questions can be related to the geometric Koebe conjecture recentlyintroduced by Luo and Wu, see [LW20].1.5. More general statements.
The questions considered above, for convex domains with thin andwith thick ideal boundary, are not entirely satisfactory since even positive answers would not cover somestatements of interest, for instance concerning the induced metrics or third fundamental forms on theboundary of bounded convex subsets in convex co-compact hyperbolic manifolds.It would moreover be quite satisfactory to have a statement covering convex subsets in H having anidea boundary containing both quasicircles and open domains.The following questions would, if answered positively, bring suitably general results. To state it, weintroduce a simple notation: given a convex subset Ω ⊂ H , we denote by ∂ Ω the boundary of the imageof Ω in the projective model of H , so that ∂ Ω contains both ∂ Ω and ∂ ∞ Ω. Note that ∂ Ω is equippednaturally with two conformal structures, • one, denoted by c I , equal on ∂ ∞ Ω to the conformal structure at infinity of ∂ ∞ H , and on ∂ Ω tothe conformal structure of the induced metric, • the other, denoted by c III , equal on ∂ ∞ Ω to the conformal structure at infinity of ∂ ∞ H , and on ∂ Ω to the conformal structure of the third fundamental form.We consider on ∂ Ω the topology induced by that of the projective model of H . Question W gen H . Let U ⊂ C P , and let h be a complete conformal metric of curvature K ≥ − on U . Is there a unique closed convex domain Ω ⊂ H such that there exists a conformal homeomorphism u : C P → ( ∂ Ω , c I ) such that u ( U ) = ∂ Ω and the pull-back by u of the induced metric on ∂ Ω is h ? Question W ∗ gen H . Let U ⊂ C P , and let h be a complete conformal metric of curvature K ≤ on U ,with closed geodesics of length L > π . Is there a unique closed convex domain Ω ⊂ H such that thereexists a conformal homeomorphism u : C P → ( ∂ Ω , c III ) such that u ( U ) = ∂ Ω and the pull-back by u ofthe third fundamental form on ∂ Ω is h ? It should be noted that, in those statements, the dependence of Ω on the gluing map is “hidden” inthe choice of U . For instance, Question W thin H is equivalent to the special case of Question W gen H when U is the complement of a quasicircle, and the gluing map occuring W thin H is then determined by the choiceof the quasicircle in Question W gen H .We would like to emphasize that(1) we don’t have much evidence for Questions W gen H and W ∗ gen H (compared to the other questionsmentioned above, for which a number of special cases are known),(2) it might be necessary to assume some kind of regularity on the boundary of U , for instance thateach connected component of each connected component of U is a quasicircle. JEAN-MARC SCHLENKER
In fact we will not provide here much specific evidence for those two questions, beyond what comes fromresults on the induced metric an third fundamental forms on convex co-compact hyperbolic manifolds.Section 3.4 contains examples of statements that would follow from a positive answer to Question W gen H and to its dual, some of which could serve as “tests”.1.6. Complete locally convex surfaces in H . So far, we have considered Theorem 1.2 and its dualTheorem 1.3 as statements on the induced metric or third fundamental forms on the boundary of convexsubsets in H . There is, however, another way to consider them, namely, as statements on isometricimmersions of locally convex surfaces (resp. on embeddings inducing a given third fundamental form).For closed surfaces in H , the two points of view are equivalent, since any closed locally convex immersedsurface bounds a convex domain. For unbounded surfaces, however, the two points of view are quitedifferent, since a locally convex immersion of a surface, even if proper, does not need to be an embedding,and its image might not bound a convex subset. (A simple example can be obtained by deforming theuniversal cover of the set of points equidistant from a geodesic in H .)When considering a locally convex immersed surface S ⊂ H , however, there is no meaningful wayto define a gluing map with a domain in C P that, together with the surface, bounds a convex subset.However, local convexity means that the hyperbolic Gauss map — the map sending a point x ∈ S to theendpoint at infinity of the geodesic starting from x in the direction of the normal vector pointing towardsthe concave side of S — is a local homeomorphism.One can therefore consider on S the pull-back on S of the conformal structure at infinity c ∞ by thishyperbolic Gauss map G : S → C P . One can then consider, as for convex domains with thick boundary,the gluing map between the induced metric (resp. third fundamental form) on S and this pull-backconformal structure. Specifically, given a proper locally convex embedding φ : D → H , let U : D → S the Riemann uniformization map for the pull-back ( φ ◦ G ) ∗ c ∞ by the Gauss map of the conformal metricat infinity, and let U : D → S be the Riemann uniformization map for the induced metric I on S . Wecall ∂ ( U − ◦ U ) : RP → RP the induced metric gluing map for φ . We can of course define in a similarmanner the third fundamental gluing map . Both are well-defined up to pre- and post-composition byM¨obius transformations.This definition of a gluing map at infinity can be extended to a wider setting, where it will be usedbelow. Let g , g be two Riemannian metrics (or conformal structures) on a surface S homeomorphicto the disk, both conformal to the disk D (i.e. of hyperbolic type). Let U i : D → ( S, g i ), i = 0 ,
1, bethe Riemann uniformization map. The gluing map at infinity between g and g is the homeomorphism ∂ ( U − ◦ U ) : RP → RP . It is well-defined up to pre- and post-composition by M¨obius transformations. Question W imm H . Let h be a complete metric of curvature K ∈ [ − , − ǫ ] on the disk D , for some ǫ > , and let σ : RP → RP be a quasi-symmetric homeomorphism. Is there a unique locally convexisometric immersion φ : ( S, h ) → H such that the induced metric gluing map of φ is σ (up to M¨obiustransformations)? Question W ∗ imm H . Let h be a complete metric of curvature K ∈ [ − /ǫ, − ǫ ] on the disk D , for some ǫ > ,with closed geodesics of length L > π , and let σ : RP → RP be a quasi-symmetric homeomorphism. Isthere a unique immersion φ : ( S, h ) → H such that the pull-back by φ of the third fundamental form is h , and that the third fundamental form gluing map of φ is σ (up to M¨obius transformations)? Note that the flavor of those questions is quite different from those of Question W thick H and its dual,since the conformal metric at infinity that is considered here is on the concave side of the surface, ratherthan on the convex side as for Question W thick H and its dual.There is a similarity between those two questions and the “hyperbolic Plateau problems” consideredfor K -surfaces in H , see e.g. [Lab00, Lab05, Smi15, Smi06].We will see in Section 5 that answers to special cases of Questions W imm H and W ∗ imm H are directlyrelated to recent results on the grafting map, while a beautiful conjecture on circle packings can be seenas another special case of Question W ∗ imm H .There are also other statements, similar to Question W imm H , that can be proved for K -surfaces butmight extend to surfaces of variable curvature, see Section 5.4. One concerns the gluing map at infinitybetween the induced metric on a surface and its third fundamental form, while the other is about thegluing map at infinity between the induced metric (or third fundamental form) of a surface and theconformal class of (1 + K ) I + III . HE WEYL PROBLEM FOR COMPLETE SURFACES 7
Theorem 1.5.
Let K ∈ ( − , , and let σ : RP → RP be a quasi-symmetric homeomorphism. Thereexists a unique immersion u : D → H with bounded principal curvatures such that the induced metrichas constant curvature K and that the gluing map at infinity between I and III is equal to σ . Theorem 1.6.
Let K ∈ ( − , , and let σ : RP → RP be a quasi-symmetric homeomorphism. Thereexists a unique immersion u : D → H with bounded principal curvatures such that the induced metrichas constant curvature K and that the gluing map at infinity between I and [(1 + K ) I + III ] is equal to σ . Theorem 1.7.
Let K ∈ ( − , , and let σ : RP → RP be a quasi-symmetric homeomorphism. Thereexists a unique immersion u : D → H with bounded principal curvatures such that the induced metrichas constant curvature K and that the gluing map at infinity between III and [(1 + K ) I + III ] is equal to σ . Note by comparison that Question W imm H and Question W ∗ imm H can be stated in the same way asTheorem 1.6 and Theorem 1.7 but with [(1 + K ) I + III ] replaced by [ I + 2 II + III ], where II denotes thesecond fundamental form, since this conformal class is the pull-back by the Gauss map of the conformalclass on ∂ ∞ H .1.7. Anti-de Sitter geometry.
The 3-dimensional anti-de Sitter space
ADS is the model 3-dimensionalLorentzian space of constant curvature −
1. It can be considered as the Lorentzian cousin of H . We referto Section 6.1, or to [Mes07, ABB +
07, BS20a] for the basic properties of AdS geometry, and of space-likesurfaces in
ADS .It is quite natural to consider convex subsets of ADS with “thin” boundary in ∂ ADS . We arespecifically interested in convex subsets with space-like boundary having a boundary “at infinity” whichis a “quasicircle”. (See Section 6.1 for the definition of a quasicircle in ∂ ADS .) Given such a convexsubset Ω, its boundary is the disjoint union of two disks D − and D + , each equipped with an inducedmetric which we require to be complete. Moreover there is a natural identification between the conformalboundaries of D − and D + – each equipped with its induced metric – given by the gluing at infinity ofthe two, we call this map the gluing map at infinity between D − and D + , each equipped with its inducedmetric. Question W thin ADS . Let g − , g + be two complete Riemannian metrics on the disk D , of curvature − M ≤ K ≤ − for some M > . Let σ : ∂ ( D , g − ) → ∂ ( D , g + ) be a quasisymmetric homeomorphism. Is there aunique convex subset Ω ⊂ H , with boundary at infinity ∂ ∞ Ω a quasicircle, such that the induced metricon the past (resp. future) boundary component of ∂ Ω is isometric to g − (resp. g + ), with induced metricgluing map equal to σ ? Note that the boundaries ∂ ( D , g ± ) appearing here are the conformal boundaries of ( D , g − ) and ( D , g + ).Each is equipped with a real projective structure through Riemann uniformization, so that the notion ofquasi-symmetric homeomorphism makes sense.One could of course state a dual question, concerning the third fundamental form on the past andfuture boundary components of a convex subset, and the corresponding gluing map. This dual question,however, is equivalent to Question W thin ADS through a duality between convex subsets of ADS , see Section6.1.6.We will see in Section 6.2.2 that the existence part of Question W thin ADS has a positive answer formetrics of constant curvature K ≤ −
1, as well as the existence question concerning the measured bendinglamination on the boundary of the convex hull of a quasicircle in ∂ ADS .We will recall in Section 6.1 how the left (or right) metric can be defined for any space-like surface ofnegative curvature in ADS . The case of globally hyperbolic AdS spacetimes (see below), and in particularthe Mess analog of the Bers Simultaneous Uniformization for those spacetimes [Mes07], suggests that theleft and right metrics play the role of the conformal metrics at infinity in hyperbolic geometry. We cantherefore state an AdS analog of Questions W thick H and W imm H — there is no significant difference betweenthe two questions in the AdS context, because the left and right metrics play symmetric roles and anylocally convex, properly immersed space-like surface in ADS is necessarily embedded.Given a space-like, complete surface S ⊂ ADS , one can consider on S the induced metric I , as well asthe left metric m L (using the definition of the left metric in [KS07]). Again, we can consider the Riemannuniformization map U : D → S for the left metric m L , and the Riemann uniformization map U : D → S for the induced metric. The boundary map ∂ ( U ◦ U − ) is then the induced metric gluing map of S . JEAN-MARC SCHLENKER
Question W imm ADS . Let h be a Riemannian metric on D of bounded, uniformly negative curvature, and let σ : RP → RP be a quasisymmetric homeomorphism. Is there a unique locally convex properly embeddedsurface S ⊂ ADS such that the induced metric on S is isometric to h , while its induced metric gluingmap is σ ? As above, the curvature conditions here could conceivably be relaxed. Another remark is that there isa dual statement, but it is equivalent to Question W imm ADS through the duality described in Section 6.1.6.The same question can also of course be asked for the right metric.We will see in Section 6 some cases where a positive answer can be given to Question W imm ADS . • For pleated surfaces, that is, when the metric h has constant curvature −
1, Question W imm ADS hasa positive answer, equivalent to Thurston’s Earthquake Theorem [Thu06, FLP91]. • For K -surfaces (when the metric h has constant curvature K < −
1) the answer is also positive,and follows from a recent result of Bonsante and Seppi [BS18]. For K -surfaces invariant under acocompact surface group, the positive answer follows from earlier work on “landslides” [BMS13,BMS15].There is in the physics literature a notion of globally hyperbolic maximal compact (GHMC) AdSspacetime. Mess discovered some remarkable analogies between quasifuchsian hyperbolic manifolds andthose GHMC AdS spacetimes [Mes07, ABB +
07] which for this reason are now sometimes called “quasi-fuchsian AdS spacetimes”. In this analogy, the role of the conformal metrics at infinity of a quasifuchsianhyperbolic manifold is played, for quasifuchsian AdS spacetimes, by the “left” and “right” hyperbolicmetrics. Considering closed surfaces in quasifuchsian AdS spacetimes is equivalent to considering surfacesinvariant under a co-compact surface group action in Question W thin ADS . In that case, existence results areknown. The results can then be stated in terms of the induced metrics or third fundamental forms onthe boundary of convex domains in quasifuchsian AdS spacetimes. One can mention in particular: • The result of Diallo (see [Dia13] or [BDMS19a, Appendix]): any pair of hyperbolic metrics on asurface of genus at least 2 can be realized as the induced metric on the boundary of the convexcore of a GHMC AdS spacetime. • Its extension by Tamburelli [Tam18] to prescribing metrics of curvature less than − • A dual result [BS12] on prescribing the measured bending laminations on the boundary of theconvex core of GHMC AdS spacetimes.More details can be found in Section 6.1.8.
Minkowski geometry.
There is another setting where an analog of the Weyl problem can beconsidered for unbounded surfaces: the Minkowski space R , . There is a well-understood way to associatea surface in R , to a first-order deformation of a hyperbolic plane in H – in fact one can associate tosuch a first-order deformation a surface in half-pipe geometry, and then by duality a surface in R , , seeSection 7.Through this correspondence, the following question turns out to be an infinitesimal version of Question W ∗ thin H near the simplest instance of that statement, namely, when Ω is a totally geodesic plane in H .We will see in Section 7 how Question W thin R , is related to Question W thin H and to Question W thin ADS , seeSection 7.2. Question W thin R , . Let g + , g − be two complete metrics of curvature K ∈ [ − /ǫ, − ǫ ] on D , for some ǫ > ,and let σ : ∂ Ω − → ∂ Ω + . Is there a unique pair of matching convex domains (Ω − , Ω + ) such that theinduced metric on ∂ Ω ± is isometric to ( D , g ± ) and that the identification map between the boundaries of ( D , g − ) and ( D , g + ) is σ ? Note that this question is equivalent to Question W ∗ thin HP appearing in Section 7.3.The statement will be explained in more details in Section 7. When a complete, space-like disk isembedded in a domain of dependence, each ideal point of the disk corresponds to a light-like line in theboundary of the domain. This isotropic line is also contained in the boundary of the matching domainof dependence, and this defines a natural identification between ( D , g − ) and ( D , g + ).The equivariant version of this question – for corresponding pairs of globally hyperbolic maximalcompact Minkowski manifolds – was recently answered positivey by Graham Smith [Smi20].2. Background and underlying results
Notations.
We start by fixing some notations used throughout the paper.
HE WEYL PROBLEM FOR COMPLETE SURFACES 9 • QS is the space of quasi-symmetric, orientation-preserving homeomorphisms from RP to RP . • T is the universal Teichm¨uller space, the quotient of QS by the group of M¨obius transformations. • ML is the space of bounded measured laminations on the hyperbolic disk. • S denotes a closed, oriented surface of genus at least 2. • T S is the Teichm¨uller space of S , that is, the space of complex structures on S considered up toisotopy. • ML S is the space of measured laminations on S . • CP S is the space of complex projective structures on S , considered up to isotopy.2.2. Quasi-symmetric regularity of gluing maps.
We first state a lemma that clarifies the relationbetween the regularity of the ideal boundary, on one hand, and the regularity of the gluing maps, on theother.
Lemma 2.1.
For all ǫ, M, k > , there exists c > as follows. Suppose that D ⊂ H is a completeembedded disk of curvature K ∈ [ − ǫ, − M ] , and assume that ∂ ∞ D is a k -quasicircle. Denote by Ω the domain bounded by ∂ ∞ D facing the convex side of D . Then there exists a c -quasiconformal mapfrom D to Ω that is the identity on ∂ ∞ D . The proof is a slight expansion of the proof appearing [BDMS19a, Lemma 3.8], so we only outline ithere with a special focus on the differences.
Proof.
The statement will follow from the existence of constant k min , k max > D satisfying the hypothesis of the lemma has principal curvatures in [ k min , k max ]. The hyperbolicGauss map, sending a point x ∈ D to the endpoint on Ω of the geodesic ray starting from x orthogonallyto D , is then uniformally quasi-conformal, thus proving the lemma.The proof of this bound on the principal curvatures proceeds by contradiction. We suppose that thereis no such bound, and therefore that there exists a sequence of (cid:3) The duality between hyperbolic and de Sitter space.
The “duality” between the questionsconcerning the induced metrics and those concerning the third fundamental form is related to the po-lar duality between the hyperbolic and de Sitter space, which we briefly recall here. A similar polarduality also occurs in anti-de Sitter geometry, see Section 6.1.6. We briefly recall this duality here forcompleteness, refering the reader to e.g. [HR93, Sch98a, FS19].The de Sitter space can be defined as a quadric in the 4-dimensional Minkowski space: DS = { x ∈ R , | h x, x i = 1 } , equipped with the induced metric. It is simply connected, geodesically complete Lorentzian space ofconstant curvature 1.The geodesics (resp. totally geodesic planes) in DS are its intersections with the 2-dimensional planes(resp. hyperplanes) of R , containing 0. The space-like planes are isometric to the standard sphere,while the time-like planes are topologically cylinders.The de Sitter space has a projective model, similar to the Klein model of hyperbolic space. in fact onlyone “hemisphere” of DS – the future of a totally geodesic space-like plane is represented as the exteriorof the unit ball in R , while the interior of this ball contains a projective model of H . This model canalso be considered in RP , while a projective model of the whole of DS can be obtained in the sphere S , considered as the double cover of RP , as the complement of the union of two balls.The “polar” duality is defined between points in H and totally geodesic space-like planes in DS , orbetween points in DS and oriented totally geodesic planes in H . Given a point x ∈ H , we denoteby x ⊥ the orthogonal hyperplane in R , , which is space-like since h x, x i = − H . Asa consequence, the intersection x ⊥ ∩ DS is a totally geodesic space-like plane, which we denote by x ∗ .Similarly, given a point y ∈ DS , its orthogonal in R , is an oriented hyperplane, which is time-like since h y, y i = 1. The dual y ∗ of y is then the intersection of y ⊥ with H , considered as a totally geodesic plane.The same definitions can be used to define the dual of a space-like plane in DS , which is a point in H ,and the dual of an oriented plane in H , which is a point in DS .Consider now a convex subset Ω ⊂ H . Its polar dual is the subset Ω ∗ ⊂ DS defined as the set ofpoints dual to the oriented planes in H bounding a half-plane disjoint from Ω. It is a geodesically convexsubset of DS . The boundary of Ω ∗ is the dual of ∂ Ω, in the sense that it is the set of points dual to theoriented support planes of Ω. If ∂ Ω is smooth and strictly convex (meaning that its second fundamentalform is positive definite) then ∂ Ω ∗ is also smooth and strictly convex. This definition of polar duality extends to unbounded convex subsets. One can check that the dualof a convex subset of H with ideal boundary a quasicircle C ⊂ ∂ ∞ H is a convex subset of DS withideal boundary the same curve C . (Note that the ideal boundary of H is naturally identified with oneconnected component of the ideal boundary of DS , for instance through the projective models describedabove.) However, the dual of a convex subset with ideal boundary a topological disks D ⊂ ∂ ∞ H is aconvex subset in DS whose ideal boundary is the complement of D .A key feature of this polar duality (see e.g. [Sch98a,FS19]) is that if Σ ⊂ H is a strictly convex surfacein H , then the dual surface Σ ∗ ⊂ DS is also strictly convex, with induced metric corresponding to thethird fundamental form of Σ, and third fundamental form corresponding to the induced metric of Σ.As a consequence, Question W ∗ thin H can be considered as a direct analog of Question W thin H but in thede Sitter space rather than in H . Moreover, the behavior of the duality on subsets with “thick” idealboundary suggests that Question W ∗ thick H can be considered as (close to) an analog of Question W imm H but in the de Sitter space, and conversely.2.4. The third fundamental form and the bending lamination of pleated surfaces.
Anotherpoint that is worth clarifying is the relation between the third fundamental form of smooth surfaces andthe measured bending lamination of pleated surfaces, since we claim in the introduction that the questionsconcerning the bending lamination forms are limit cases of questions concerning the third fundamentalform.It is helpful here to use the polar duality between H and DS . Let Σ be a locally convex pleatedsurface in H , with measured bending lamination l . Then the dual of Σ is a real tree Σ ∗ in DS , withone vertex corresponding to each totally geodesic region in Σ, see [Bel14, Bel17]. Moreover the distancebetween two vertices in this real tree is equal to the transverse measure for the bending lamination of atransverse segment connecting a point of one of the totally geodesic region to a point of the other.Now if Σ r is the equidistant surface at constant distance r > ∗ r converges as r → DS which contains Σ ∗ and is light-like outside of l ∗ . Moreover, Σ r is a C , surface, and its third fundamental form is well-defined. It is then provedin [Bel14, Bel17] that the distance for the third fundamental form on Σ r converges as r → l . More specifically, if x, y ⊂ Σ are containedin totally geodesic open subsets, if x r , y r ⊂ Σ r and x r → x, y r → y as r →
0, then d III r ( x r , y r ) → d l ( x, y ) , where d III r is the distance associated to the third fundamental form on Σ r , while d l is the pseudo-distanceassociated to the measured lamination l on Σ. We refer to [Bel14, Bel17] for details.3. Convex domains with thin ideal boundary
We develop in this section the questions and results already mentioned more briefly in Section 1.3, andgive more precise statements of both known results and specific questions that appear natural in light ofthe general questions stated in the introduction.We first focus on convex domains with ideal boundary a quasi-circle, and then explain the relationswith geodesically convex subsets in quasifuchsian, or more generally geometrically finite, hyperbolic 3-manifolds.3.1.
Convex domains with ideal boundary a quasi-circle.
The simplest case to consider, topologi-cally speaking, is a convex subset Ω ⊂ H such that the ideal boundary ∂ ∞ Ω is just one quasicircle. Then ∂ Ω – the boundary of Ω – is the disjoint union of two complete surfaces ∂ − Ω and ∂ + Ω, each diffeomorphicto the disk D and having ∂ ∞ Ω as its asymptotic boundary.Both ∂ − Ω and ∂ + Ω, equipped with its induced metric, is then conformal to D , the unit disk in R , sothat they can both be equipped with a conformal boundary ∂ c ( ∂ − Ω) and ∂ c ( ∂ + Ω), both equipped with areal projective structure. Both ∂ c ( ∂ ± Ω) are identified homeomorphically to ∂ ∞ Ω, and Lemma 2.1 showsthat the map σ : ∂ c ( ∂ − Ω) → ∂ c ( ∂ + Ω) is quasi-symmetric.We can now ask whether Question W thin H has a positive answer in this restricted setting. Question 3.1.
Let h − , h + be two complete metrics of curvature K ∈ [ − ǫ, − ǫ ] on the disk D , and let σ : RP → RP be an orientation-reversing quasi-symmetric homeomorphism. Is there a unique convexsubset Ω ⊂ R , with ∂ ∞ Ω a quasi-circle, such that the induced metric on ∂ ± Ω is h ± , with gluing map atinfinity corresponding to σ ? HE WEYL PROBLEM FOR COMPLETE SURFACES 11
The existence is proved in [BDMS19a] when h − and h + have constant curvature K ∈ [ − , K = − W ∗ thin H can be stated as follows. Question 3.2.
Let h − , h + be two complete metrics of curvature K ∈ [ − /ǫ, − ǫ ] on the disk D , withclosed geodesics of length larger than π , and let σ : RP → RP be an orientation-reversing quasi-symmetric homeomorphism. Is there a unique convex subset Ω ⊂ R , with ∂ ∞ Ω a quasi-circle, such thatthe third fundamental form on ∂ ± Ω is h ± , with gluing map at infinity corresponding to σ ? Again, the existence part is shown in [BDMS19a] in the special case of metrics of constant curvature K ∈ ( − , ∂ ∞ H . A possible statement is suggested by [MS20, TheoremB], which gives an existence result for the corresponding problem in anti-de Sitter space, see Section 6.We will say that two measured laminations λ − , λ + on the hyperbolic disk D strongly fill if, for every ǫ >
0, there exists c > γ in D of hyperbolic length at least c , i ( λ − , γ ) + i ( λ + , γ ) ≥ ǫ . We also need to remember that a measured lamination on D can be considered asa measure on RP × RP \ ∆, where ∆ ⊂ RP × RP is the diagonal, and notice that given a quasicircle C ⊂ ∂ ∞ H , the measured bending laminations on the two connected components of boundary the convexhull of C can be considered as measures on C × C \ ∆, where again ∆ ⊂ C × C is the diagonal. Question 3.3.
Let λ − , λ + be two bounded measured laminations on the hyperbolic disk D that stronglyfill, without leaf of weight at least π . Is there a unique parameterized quasicircle u : RP → ∂ ∞ H such that u ∗ ( λ − ) and u ∗ ( λ + ) are the measured bending laminations on the connected components of theboundary of the convex hull of u ( RP ) ? Here a parameterized quasi-circle u : RP → RP is a continuous injective map such that if v ± : D → ∂ ∞ H are the Riemann uniformization maps of the two connected component of ∂ ∞ H \ u ( RP ), then v − ± ◦ v : RP → RP are both quasi-symmetric.It appears possible that the type of arguments used in [MS20] could lead to a proof of the existence partof Question 3.3, but a serious technical question (concerning a compactness issue) needs to be resolved.3.2. Bounded convex domains in quasifuchsian manifolds.
The questions in Section 3.1 take asomewhat simpler form when considered for quasicircles that are invariant under a quasifuchsian groupaction. Let Ω ⊂ H be a convex subset invariant under such an action ρ : Γ → P SL (2 , C ), where Γ is thefundamental group of a closed surface of genus at least 2. The quotient of Ω by ρ (Γ) is then a geodesicallyconvex subset in M = H /ρ (Γ), which by our hypothesis on ρ is a quasifuchsian hyperbolic manifold.Question 3.1 corresponds in this restricted context to the following statement, proved in [Lab92a] (forthe existence part) and [Sch06] (for the uniqueness): given a closed surface S of genus at least 2 andtwo smooth Riemannian metrics h − , h + of curvature K > − S , there exists a unique quasifuchsianmanifold M homeomorphic to S × R and a unique geodesically convex subset Ω ⊂ M such that theinduced metrics on the two boundary components of Ω are isotopic to h − and h + . Similarly, [Sch06]contains an answer to Question 3.2 in the quasifuchsian setting: given two smooth metrics h − , h + on S ofcurvature K <
L > π , there exists a unique quasifuchsianmanifold and a unique geodesically convex subset Ω ⊂ M such that the third fundamental forms on thetwo boundary components of Ω are isotopic to h − and h + .However the uniqueness part of the same questions remains open for pleated surfaces, for both theinduced metrics and for the third fundamental form (which in this case appears under the form of ameasured pleating lamination, as seen in Section 2.4). The only closed convex pleated surfaces in a quasi-fuchsian manifold are the boundary components of the convex core, the smallest non-empty geodesicallyconvex subset that they contain. Questions W thin H and W ∗ thin H therefore take, in this quite specific setting,the following form. Conjecture 3.4 (Thurston) . Let m − , m + be two hyperbolic metrics on a closed surface S of genus atleast . Is there a unique quasifuchsian hyperbolic structure on S × R such that the induced metrics onthe boundary components of the convex core are isotopic to m − and m + ? Conjecture 3.5 (Thurston) . Let l − , l + be two measured laminations that fill on a closed surface S of genus at least . Assume that l − and l + have no leaf of weight larger than π . Is there a unique quasifuchsian hyperbolic structure on S × R such that the measured bending laminations on the boundaryof the convex core are l − and l + ? The existence part of both conjectures is known. For Conjecture 3.4 it is a consequence of resultsof Epstein-Marden [EM86] or Labourie [Lab92a], while for Conjecture 3.5 it is a result of Bonahon-Otal [Bor77].Other results can be found in [Sch01,Sch02] for geodesically convex domains in quasifuchsian manifoldswhich are locally like an ideal or hyperideal hyperbolic polyhedron.3.3.
Fuchsian results.
Special cases of the questions considered in the previous section are known tohold for surfaces that are invariant under a Fuchsian group action — that is, a group action that leavesinvariant a plane. This happens when the two metrics h − and h + considered above are identical. Existenceand uniqueness results of this type were obtained for the third fundamental form of smooth surfaces in H in [LS00], and for the induced metrics on polyhedral surfaces in e.g. [Fil07, Fil11, Fil08, Lei02].Another relatively simple setting is that of surfaces invariant under a parabolic group action. Thiscorresponds geometrically to the universal cover of a surface embedded in a “cusp”, a hyperbolic manifoldhomeomorphic to T × R , where T is the torus. See for instance [FI09].In view of those results for surfaces invariant under a Fuchsian group action — and therefore with anideal boundary that is a round circle — it seems natural to ask whether Question W thin H has a positiveanswer assuming only that the ideal boundary is a circle (but without assuming invariance under a groupaction). Question 3.6.
Let h be a complete hyperbolic metric on the disk D , with curvature K > − . Assumethat ( D , h ) is conformal to a disk. Is there a unique isometric embedding of ( D , h ) in H such that theideal boundary of the image is a round circle? We can also ask the dual question.
Question 3.7.
Let h be a complete hyperbolic metric on the disk D , with curvature K < and closedgeodesics of length L > π . Assume that ( D , h ) is conformal to a disk. Is there a unique embedding of ( D , h ) in H with third fundamental form h such that the ideal boundary of the image is a round circle? In addition, those two questions can also be considered for “parabolic” embeddings, that is, when theimage surface has only one point at infinity.
Question 3.8.
Let h be a complete hyperbolic metric on the disk D , with curvature K > − . Assumethat ( D , h ) is conformal to C . Is there a unique isometric embedding of ( D , h ) in H such that the idealboundary of the image is a point? We can also ask the dual question.
Question 3.9.
Let h be a complete hyperbolic metric on the disk D , with curvature K < and closedgeodesics of length L > π . Assume that ( D , h ) is conformal to C . Is there a unique embedding of ( D , h ) in H with third fundamental form h such that the ideal boundary of the image is a point? From a technical point of view, the most interesting feature of those questions is that the infinitesimalrigidity, which is arguably the most challenging aspect of the questions considered here, might be amenableto relatively simple geometric arguments.3.4.
Bounded convex domains in convex co-compact manifolds.
The same questions can beconsidered more generally in geometrically finite hyperbolic manifolds. The questions mentioned inSection 3.2 on the induced metrics and third fundamental forms on the boundary of geodesically convexsubsets of quasifuchsian manifolds can be stated, basically in the same manner, for Kleinian manifolds.For smooth metrics of curvature
K > − K <
L > π ) the results of [Lab92a, Sch06] also apply for convex co-compacthyperbolic manifolds. For pleated surfaces (induced metrics of constant curvature −
1, resp. measuredbending laminations) Conjectures 3.4 and 3.5 have been stated by Thurston in this setting, and theexistence parts also hold for both conjectures.It should be pointed out that Questions W thin H and W ∗ thin H as stated do not cover the case of boundedconvex domains in Kleinian manifolds.Questions W gen H and W ∗ gen H however are much more general, and do cover those cases, in the sensethat a positive answer to those questions would provide a new proof of the existence and uniqueness ofconvex co-compact manifold containing a geodesically convex subset with prescribed induced metric (or HE WEYL PROBLEM FOR COMPLETE SURFACES 13 third fundamental form) on the boundary. It would be sufficient to have a positive answer to Questions W gen H and W ∗ gen H under the additional asumption that the boundary of each connected component of U is a quasi-circle.However Question W gen H is much more general than Question W thin H also in terms of regularity, evenfor convex subsets of H with ideal boundary a Jordan curve, and with boundary a surface of constantcurvature. In fact the following question is a very special case of Question W gen H . Question 3.10.
Let Γ be a Jordan curve in C P , and let h − and h + be complete metrics of constantcurvature K ∈ [ − , on the two connected component of C P \ Γ . Is there a unique convex subset Ω ⊂ H such that there is a conformal homeomorphism from C P to ∂ Ω (with the notations of Question W gen H ) which sends Γ to ∂ ∞ Ω and is an isometry on C P \ Γ ? It is proved in [BDMS19a] that the existence part of this statement holds when Γ is a quasicircle.However it is not clear whether this condition (Γ is a quasicircle) is necessary, whether for existence orfor uniqueness.4.
Convex domains with thick ideal boundary and the geometric Koebe conjecture
We now turn our attention to Questions W thick H and W ∗ thick H . We first give a more precise description ofthe simplest example, then explain a connection to the Geometric Koebe conjecture recently formulatedand studied by Luo and Wu [LW20].4.1. Convex domains with a disk at infinity.
Consider first a convex subset Ω ⊂ H such that theboundary of Ω in H , denoted by ∂ Ω, is topologically a disk, while the ideal boundary of Ω, denotedby ∂ ∞ Ω, is also a topological disk. In this case, ∂ ∞ Ω and ∂ Ω meet along their common boundary, ∂ ( ∂ ∞ Ω) = ∂ ∞ ( ∂ Ω). This common boundary is by definition a Jordan curve in ∂ ∞ H . Moreover, theinduced metric on ∂ Ω has curvature
K > − ∂ Ω has
K <
L > π and is also conformal to the disk.Questions W thick H and W ∗ thick H can be restricted to this special case, leading to the two simpler questionsbelow. Question 4.1.
Given a complete metric g of curvature K ∈ [ − ǫ, − ǫ ] on the disk D and a quasisym-metric homeomorphism σ : ∂ ( D , g ) → RP , is there a unique geodesically convex subset Ω as describedabove such that the induced metric on ∂ Ω is isometric to g , while the gluing map between ( ∂ Ω , I ) and ∂ ∞ Ω is equal, up to left composition by a M¨obius transformation, to σ ? Question 4.2.
Given a complete metric g of curvature K ∈ [ − /ǫ, − ǫ ] with closed geodesics of length L > π on the disk D and a quasisymmetric homeomorphism σ : ∂ ( D , g ) → RP , is there a uniquegeodesically convex subset Ω as described above such that the induced metric on ∂ Ω is isometric to g , whilethe gluing map between ( ∂ Ω , III ) and ∂ ∞ Ω is equal, up to left composition by a M¨obius transformation,to σ ? It might be tempting, in view of Questions W gen H and W ∗ gen H , to replace the asumption that K ∈ [ − ǫ, − ǫ ], by the condition that K ≥ − D , g ) is conformal to the disk, and similarly for thedual question.4.2. Unbounded geodesically convex subsets in quasifuchsian manifolds.
An even simpler in-stance occurs when considering convex domains invariant under a quasifuchsian group, and, correspond-ingly, induced metrics (or third fundamental forms) invariant under a surface group action, and gluingmaps equivariant under two actions. This is the case for the universal cover of a geodesically convexdomain Ω in a quasifuchsian manifold M , when the boundary of Ω is a connected surface. The idealboundary of Ω then corresponds to one connected component of the ideal boundary of M . Question 4.1can be stated, under those additional asumptions (but with the weaker curvature asumption), as follows. Question 4.3.
Let S be a closed surface of genus at least , let h be a metric of curvature K ≥ − on S , and let c ∈ T S be a conformal class on S . Is there a unique quasifuchsian manifold M , containing ageodesically convex subset Ω ⊂ M which is a neighborhood of one connected component of ∂ ∞ M wherethe conformal class at infinity is isotopic to c , while the induced metric on ∂ Ω is isotopic to h ? Similarly, Question 4.2 can then be stated as:
Question 4.4.
Let S be a closed surface of genus at least , let h be a metric of curvature K ≤ on S with closed, contractible geodesics of length L > π , and let c ∈ T S be a conformal class on S . Is there aunique quasifuchsian manifold M , containing a geodesically convex subset Ω ⊂ M which is a neighborhoodof one connected component of ∂ ∞ M where the conformal class at infinity is isotopic to c , while the thirdfundamental form on ∂ Ω is isotopic to h ? Unbounded convex subsets in geometrically finite manifolds.
Questions 4.3 and 4.4 canclearly be generalized to geometrically finite hyperbolic manifolds. In this setting, Question 4.3 andQuestion 4.4 can be extended to geodesically convex domain containing some of the connected componentof the ideal boundary, while the induced metrics (or third fundamental forms) are prescribed on convexsurfaces in the other ends. We leave the precise statement to the reader for brevity. A positive answerto Question W gen H and to Question W ∗ gen H would provide a positive answer to those questions.4.4. The Geometric Koebe Conjecture.
Question W thick H appears to be related to the GeometricKoebe Conjecture, recently introduced by Luo and Wu [LW19]. We first recall this conjectures, as wellas recent and less recent results concerning them. Recall that a circle domain in C P is the complementof a family of open round disks and points. Conjecture 4.5 (Geometric Koebe Conjecture, Luo–Wu) . Let Σ be a surface of genus , equipped witha complete hyperbolic metric h . Is there a unique circle domain Γ ⊂ C P such that (Σ , h ) is isometric tothe boundary of the convex hull of C P \ Γ , equipped with its induced metric? This conjecture is motivated by the Koebe circle packing conjecture (1909), which asks whether anygenus 0 Riemann surface is biholomorphic to a circle domain. He and Schramm [HS93] proved that thisconjecture holds for surfaces with countably many boundary components.Luo and Wu [LW19] proved that the Geometric Koebe Conjecture is equivalent to the Koebe CircleDomain conjecture, and as a consequence of the result of [HS93] they obtained that the Geometric KoebeConjecture holds for surfaces with countably many boundary components.It is possible to extend the Geometric Koebe conjecture to consider not only the convex hull of thecomplement of a circle domain Γ, but more generally a convex domain containing this convex hull, withboundary in H a surface with induced metric of constant curvature K ∈ [ − ,
0) – the convex hullcorresponds precisely to the case K = −
1. The existence and uniqueness of such a convex domain withprescribed ideal boundary is a result of Rosenberg and Spruck [RS94]. Specifically, we can state twoconjectures which are quite natural from the work of Luo and Wu.
Conjecture 4.6 ( K -Geometric Koebe Conjecture) . Let Σ be a surface of genus , equipped with acomplete metric h of constant curvature K ∈ ( − , . Is there a unique circle domain Γ ⊂ C P such that (Σ , h ) is isometric to the boundary of the unique convex domain with ideal boundary ∂ Γ and boundary ofconstant curvature K , equipped with its induced metric? Conjecture 4.7 (Dual K -Geometric Koebe Conjecture) . Let Σ be a surface of genus , equipped with acomplete metric h of constant curvature K ∗ < with closed geodesics of length L > π . Is there a uniquecircle domain Γ ⊂ C P such that (Σ , h ) is isometric to the boundary of the unique convex domain withideal boundary ∂ Γ and boundary of constant curvature K (with K ∗ = K/ ( K + 1) ), equipped with its thirdfundamental form? An existence and uniqueness theorem for Conjecture 4.6 was given, for complete metrics on the sphereminus a finite number of disks, in [Sch98b]. A dual statement proving Conjecture 4.7 for completemetrics on the sphere minus a finite number of disks was also given in [Sch98b]. Seen in this manner,the Geometric Koebe Conjecture appears as an instance of the Weyl problem in hyperbolic space, and itis tempting to assume that a similar statement extends to metrics of variable curvature, see [LW19, v2,Conjecture 3].There is a clear similarity between Conjecture 4.5 (and its extension to K -surfaces as in [Sch98b]) andQuestion W thick H . The relation is however not completely straightforward, since it is not clear, given acircle domain Γ ⊂ C P , what the gluing maps of the boundary of the convex hull of C P \ Γ is. It doesnot appear likely, for instance, that the gluing maps of this surface are real projective maps.5.
Immersed locally convex surfaces
We now turn to Question W imm H and its dual, Question W ∗ imm H . We focus here mostly on thosequestions when restricted to locally convex pleated surfaces, possibly with cusps, where they correspond HE WEYL PROBLEM FOR COMPLETE SURFACES 15 to either relatively recent results or to well-known open questions. Considering more general cases ofQuestions W imm H and W ∗ imm H leads to new questions on the grafting map, and on new perspectives onquestions concerning circle packings or circle patterns.5.1. Pleated surfaces and grafting.
Immersed pleated surfaces in H . Perhaps the simplest instance of Question W imm H is when h ishyperbolic, that is, of constant curvature −
1. The image of φ then needs to be a locally convex, pleatedsurface in H . The question can then be formulated in the following simpler form. Question 5.1.
Let σ : RP → RP be a quasi-symmetric homeomorphism. Is there a unique locallyconvex pleated isometric immersion φ : H → H such that the induced metric gluing map of φ is σ ? Here the gluing map that is considered is between the induced metric and the pull-back by the Gaussmap of the conformal class at infinity, see Section 1.6.Question W ∗ imm H takes for pleated surfaces a slightly different from, since the notion of gluing maphas to be formulated a bit differently between a measured lamination and a conformal disk. Recall thatgiven a locally convex immersed surface S ⊂ H , the hyperbolic Gauss map G : S → ∂ ∞ H sends a point x ∈ S to the endpoint at infinity of the geodesic ray starting from x orthogonally to S .This definition extends to locally convex pleated surfaces, but the hyperbolic Gauss map is then definedon the unit normal bundle N S of S (the space of unit vectors normal to oriented support planes of S ). Question 5.2.
Let l be a bounded measured lamination on the disk D . Is there a unique immersed locallyconvex pleated surface Σ ⊂ H together with a map u : D → N Σ which is a parameterization of N Σ such that • G ◦ u is conformal, where G is the hyperbolic Gauss map of Σ , • the pull-back by u of the measured bending lamination of Σ corresponds to l ? Note that the pull-back by u of the measured bending lamination of S is naturally a measured foliation on D , and the second point requests that this measured foliation corresponds to l under the naturalcorrespondence between measured foliations and measured laminations on D . Prescribing the measuredfoliation on the disk at infinity is similar, heuristically at least, to prescribing the “gluing” between themeasured pleating lamination and the conformal structure at infinity. (Note also that Question 5.2 isstated in a slightly indirect way, because G is not necessarily injective.)5.1.2. Pleated surfaces in hyperbolic ends.
Questions 5.1 and 5.2 can be considered in the special caseof surfaces invariant under a surface group representation ρ : π ( S ) → P SL (2 , C ). They have positiveanswers in those restricted cases, and are in fact equivalent to two questions on the grafting map.Recall that, given a closed surface S of genus at least 2, the grafting map gr : T S × ML S → T S is such that, if a quasifuchsian manifold M (or more generally a hyperbolic end E ) contains a locallyconvex pleated surface S with induced metric m and measured bending lamination l , then the conformalstructure at infinity on the ideal boundary component facing S (on the concave side of S ) is gr ( m, l ), seee.g. [Dum08].In this case, the gluing map σ : RP → RP in Question 5.1 is equivariant under a pair or representa-tions ρ , ρ : π S ( S ) → P SL (2 , R ). Then H /ρ ( π S ( S )) is the quotient of the pleated surface in H , while H /ρ ( π ( S )) is the hyperbolic metric in the pull-back by the hyperbolic Gauss map of the conformalclass at infinity of H . This conformal class is obtained by grafting the hyperbolic metric on the pleatedsurface along its measured bending lamination.Question 5.1, in this restricted context, is therefore equivalent to the following result of Dumas andWolf [DW08]. Theorem 5.3 (Dumas–Wolf) . Let h be a hyperbolic metric on a closed surface S . The grafting map gr ( h, · ) : ML S → T S is a homeomorphism. Still for pleated surfaces invariant under a surface group representation in
P SL (2 , C ), the measuredbending lamination considered above is the lift to the universal cover of a measured bending lamination l on S , while the condition in Question 5.2 is equivalent to asking that the pull-back on S by the hyperbolicGauss map of the conformal structure at infinity is prescribed. A positive anwer to Question 5.2 in thisequivariant case therefore follows from a result of Scannell and Wolf [SW02]. Theorem 5.4 (Scannell-Wolf) . Let l be a measured lamination on a closed surface S of genus at least . The grafting map gr ( · , l ) : T S → T S is an analytic diffeomorphism. Grafting and pleated disks in H . Questions 5.1 and 5.2 can be formulated in terms of graftingon the hyperbolic disk. This is particularly natural in view of Thurston’s Earthquake Theorem, whichstates that any quasi-symmetric homeomorphism of RP is the boundary map of a unique left earthquakeon the hyperbolic plane (see Section 6.1.4). Indeed, earthquakes are intimately related to grafting (seee.g. [McM98]) while we will see below that Thurston’s Earthquake Theorem gives a positive answer toquestion W immAdS in the special case of locally convex pleated disks in AdS .To state the questions in a simple manner, we introduce notations for the grafting map on the hyper-bolic disk. Let l be a bounded measured lamination on D . Let Σ ⊂ H be a pleated disk in H obtainedby pleating the hyperbolic disk along the measured bending lamination l , and let G : N Σ → C P bethe hyperbolic Gauss map. We also denote by u : D → N Σ the Riemann uniformization map for theconformal structure obtained on N Σ by pull-back of the conformal structure at infinity of H . Finallywe denote by π : N Σ → Σ the canonical projection.
Lemma 5.5.
There exists δ > such that, if l is bounded, then the map π ◦ u : D → Σ is within distance δ from a quasi-conformal map, with quasi-conformal constant depending only on the bound on l .Proof. The proof can be done following the arguments in [EM86], in a closely related context. (cid:3)
As a consequence of Lemma 5.5, the map π ◦ u : D → Σ extends so a quasi-symmetric map from RP to ∂ ∞ Σ.Moreover, l lifts to a measured foliation on N Σ. For instance, if l contains a leaf L with atomicweight w , then the inverse image of L by π is a strip in N Σ, which is foliated by parallel lines, withtotal transverse weight w . This measured foliation can be pulled back to D by u , yielding a measuredfoliation on D , which corresponds to a unique measured lamination on D . Definition 5.6.
We denote by: • GR ( l ) : RP → ∂ ∞ Σ the boundary extension of π ◦ u , • GR ( l ) the measured lamination corresponding to ( π ◦ u ) ∗ ( l ) on D , • GR ( l ) = ( G ( l ) , G ( l )) . Note that it is also possible to define a “universal grafting map”, which associates to a measuredlamination on D a complex projective structure also on D . This map however does not appear to bedirectly related to the topics considered here.It follows quite directly from the definition that Question 5.1 can be stated equivalently as follows. Question 5.7.
Is the map GR : ML → T a homeomorphism from bounded measured laminations toquasi-symmetric homeomorphisms?
Moreover, Question 5.2 can be reformulated equivalently in terms of grafting, too.
Question 5.8.
Is the map GR : ML → ML a homeomorphism from bounded measured laminations tobounded measured laminations?
The equivalence between Question 5.2 and Question 5.8 should be quite clear from the definitions.5.2.
Circle patterns and circle packings.
This section is focused on a slightly different type of pleatedsurfaces in H and in hyperbolic manifolds: those which have cusps, and therefore look locally like idealhyperbolic polyhedra. Those surfaces are closely related to circle packings and to Delaunay circle patterns(see below). Through this correspondence, an interesting question on circle patterns turns out to be aspecial case of Question W ∗ imm H .5.2.1. Convex surfaces in hyperbolic ends and circle packings.
One motivation for Question W ∗ imm H isa beautiful conjecture of Kojima, Mizushima and Tan [KMT03, KMT06b, KMT06a]. They consideredcircle packings on closed surfaces equipped with a complex projective structure, which are more flexiblethan circle packings on closed hyperbolic or Euclidean surfaces. Kojima, Mizushima and Tan considera graph Γ that is the 1-skeleton of a triangulation of a closed surface S , and the space C Γ of complexprojective structures on S admitting a circle packing with incidence graph Γ. Conjecture 5.9 (Kojima, Mizushima and Tan) . The forgetful map from C Γ to the Teichm¨uller space of S is a homeomorphism. HE WEYL PROBLEM FOR COMPLETE SURFACES 17
The Koebe circle packing theorem is the special case when S is a sphere, since there is then a uniquecomplex projective structure at infinity. Kojima, Mizushima and Tan verified the conjecture in someother simple cases, in particular on the torus for a circle packing with only one circle.In [SY18] we noticed that Conjecture 5.9 can be extended from circle packings to “Delaunay circlepatterns” – patterns of circles that occur as the Delaunay decomposition of a finite set of points on asurface equipped with a complex projective structure. We also proved a statement that could be “onehalf” of this generalized conjecture, namely, that the projection map to Teichm¨uller space is proper.A key point in [SY18] is that a Delaunay circle pattern on a surface with a complex projective structurecan be considered as a polyhedral surface (with all vertices at infinity) in a hyperbolic end. Morespecifically, Conjecture 5.9 is in fact equivalent to Question W ∗ imm H considered in the special case of equivariant convex immersions of surfaces that are locally like ideal polyhedra.Let us briefly explain how the correspondence between Delaunay circle patterns (or circle packings)and “ideal” polyhedral surfaces functions. A Delaunay circle pattern is basically the sort of circle patternthat occurs when considering the Delaunay decomposition of a discrete set of points on a surface equippedwith a complex projective structure (for instance, an open subset of C P ), see [SY18, Definition 1.9]. Thepoints in this discrete set will be the vertices of the ideal polyhedral surface. Given such a Delaunay circlepattern, each empty disk (disk containing no vertex) is the boundary at infinity of a unique half-plane.The boundary of the complement of the union of the half-planes corresponding to all the disks is the idealpolyhedral surface. This construction can be done either in a hyperbolic end, or in its universal cover, inwhich case both the circle pattern and the corresponding ideal polyhedral surface are invariant under asurface group action.Circles packings (as considered in [KMT03], that is, when faces of the incidence graph are triangles)can be considered as special cases of Delaunay circle patterns if one adds for each face of the dual graphanother circle, orthogonal to the adjacent circles. Adding those “dual” circles yields a Delaunay circlepattern for which all intersection angles are π/ C on a surface S equippedwith a complex projective structure c ∈ CP S , C and c are uniquely determined by the incidence graphand intersection angles of C , together with the complex structure underlying c . And whether, given anembedded graph Γ in S and a set of weights in (0 , π ) associated to the edges of Γ, satisfying some naturalconditions (basically corresponding to the conditions of Question W ∗ imm H , there is for each complexstructure X on S a unique complex projective structure c compatible with X , admitting a circle patternwith incidence graph Γ and intersection angles given by the prescribed weights. A positive answer wasrecently given by W. Y. Lam for the torus, see [Lam19].5.2.2. Infinite circle patterns and circle packings.
The point of view of Question W ∗ imm H suggests anextension of Conjecture 5.9 to infinite circle patterns of infinite circle packings. We state the questionhere in a relatively limited setting for simplicity. Definition 5.10.
A Delaunay circle pattern in the hyperbolic disk D is bounded if it covers the whole of D and the hyperbolic radius of the circles is bounded from above and is bounded from below by a positiveconstant. Question 5.11.
Let C be a bounded Delaunay circle pattern in the hyperbolic disk D , and let σ : RP → RP be a quasi-symmetric homeomorphism. Is there a unique local homeomorphism u : D → C P sendingeach circle to a round circle, preserving the intersection angles, and such that if v : D → ( D , u ∗ c ) is theRiemann uniformization map, then ∂v : RP → RP is equal (up to M¨obius transformation) to σ ? Again, this statement is a special case of Question W ∗ imm H . The third fundamental form becomes,for surfaces which are locally isometric to the boundary of an ideal polyhedron, a measure transverse tothe edges, with weight the exterior dihedral angle, so prescribing III in this case becomes prescribing thecombinatorics and exterior dihedral angles of an ideal polyhedral surface, which means prescribing thecombinatorics and intersection angles of a Delaunay circle pattern.5.3.
Parameterization of locally convex equivariant embeddings.
We believe that a positive an-swer to Question W imm H , and the dual Question W ∗ imm H , would provide a general description of equivariantconvex isometric embeddings of surfaces of higher genus in H .Let ( S, g ) be a closed surface of genus at least 2, equipped with a Riemannian metric of curvature
K > −
1. If we consider the Alexandrov Theorem 1.2 as a statement on locally convex isometric embeddings, it is quite natural to ask for a description of the space of all isometric immersions of the universal cover( ˜ S, ˜ h ) in H equivariant under a representation ρ of π ( S ) into P SL (2 , C ).Section 3.3 already provides us with one canonical such immersion: the one for which ρ takes valuesin P SL (2 , R ). A positive answer to Question W imm H for smooth surfaces would provide a natural param-eterization of the space of equivariant isometric immersions of a closed surface S equipped with a metric h of curvature K > − H , with the natural parameter being the pull-back on S by the Gauss map ofthe conformal class at infinity of H .5.4. Similar statements for K -surfaces. This section contains the proofs of Theorem 1.5, Theorem1.6 and Theorem 1.7. All three proofs are direct consequences of recent results on the existence anduniqueness of an extension to the hyperbolic disk of a quasi-symmetric homeomorphism from RP to RP .5.4.1. Parameterization by the gluing between I and III . The proof of Theorem 1.5 relies on the fact(observed by Labourie [Lab92b]) that if S ⊂ H is a surface of constant curvature K > −
1, then theidentity is a minimal Lagrangian map between ( S, | K | I ) and ( S, | K ∗ | III ), where K ∗ = K/ ( K + 1) is thecurvature of III .Conversely, if h, h ′ are two hyperbolic metrics on a surface S such that the identity is minimal La-grangian between ( S, h ) and (
S, h ′ ), and if K ∈ ( − ,
0) is a constant, then there is a unique immersionof S into H such that the induced metric is (1 / | K | ) h and the third fundamental form is (1 / | K ∗ | ) h ′ .Indeed, the fact that the identity is minimal Lagrangian between ( S, h ) and (
S, h ′ ) is equivalent to theexistence of a bundle morphism b : T S → T S which is self-adjoint and Codazzi for h , of determinant 1,and such that h ′ = h ( b · , b · ). One can then set B = √ K + 1 b . Then B is self-adjoint and Codazzi for (1 / | K | ) h , and of determinant K + 1, so that ( h, B ) satisfy theGauss-Codazzi equations.Thanks to this correspondence, immersed K -surfaces with bounded principal curvatures such thatthe gluing map at infinity between I and III is σ are in one-to-one correspondence with quasi-conformalminimal Lagrangian maps from H to H which extend at infinity as σ . Thanks to [BS10, Theorem 1.4],there is a unique such quasi-conformal minimal Lagrangian diffeomorphism, and Theorem 1.5 follows.5.4.2. Parameterization by the gluing between I and III . The same argument can be used to prove The-orem 1.6 and Theorem 1.7, based on the fact that if h and h ′ are two hyperbolic metrics on a surface S such that the identity is a minimal Lagrangian map between ( S, h ) and (
S, h ′ ), then the identity maps Id : ( S, h + h ′ ) → ( S, h ) and Id : ( S, h + h ′ ) → ( S, h ′ ) are harmonic maps, with opposite Hopf differential,and conversely, see e.g. [BS10].Suppose now that u : D → H is an immersion with induced metric I of constant curvature K ∈ ( − , K ∗ = K/ ( K + 1). Then the metrics | K | I and | K ∗ | III are hyperbolic, and the identity is minimalLagrangian between them. So the identity map from ( S, | K | I + | K ∗ | III ) to ( S, | K | I ) is harmonic. Scalingby a factor 1 / | K | on both side and using that K ∗ = K/ ( K + 1), we find that the identity between( S, I + (1 / (1 + K )) III ) and (
S, I ) is harmonic. In other terms (since harmonicity only depends on theconformal class in the source), the identity between ( S, (1 + K ) I + III ) and (
S, I ) is harmonic.Conversely, given a hyperbolic metric h on the disk D such that the identity between D (equipped withits standard conformal class) and h is harmonic, there is a unique hyperbolic metric h ′ on D (obtained byintegrating the Schwarzian equation, see [Wol89]) such that the identity map from D to ( D , h ′ ) is harmonic,with Hopf differential opposite to that of the harmonic map from D to ( D , h ). The identity map from( D , h ) to ( D , h ′ ) is then minimal Lagrangian. Given K ∈ ( − , h, h ′ ) an immersion from D to H with induced metric (1 / | K | ) h and thirdfundamental form (1 / | K ∗ | ) h ′ , and by construction the identity is harmonic from ( D , (1 + K ) I + III ).This construction shows that there is a one-to-one correspondence between immersed K -surfaces withbounded principal curvature and quasi-conformal harmonic diffeomorphism between D and H . Theorem1.6 follows.Theorem 1.7 is proved in the same manner, using the fact that the identity from ( D , [ h + h ′ ]) to ( D , h ′ )is also harmonic, and that it can be used to reconstruct a K -surface as above. HE WEYL PROBLEM FOR COMPLETE SURFACES 19 The Weyl problem for unbounded surfaces in anti-de Sitter and Minkowski geometry
Basic information on AdS geometry.
We review very briefly in this section some basic infor-mation on anti-de Sitter (AdS) geometry. We refer the reader to e.g. [BS20b] for more details.6.1.1.
The anti-de Sitter space.
Anti-de Sitter (AdS) geometry is a Lorentzian cousin of hyperbolic geom-etry. The space
ADS is a Lorentzian space of constant curvature −
1, originally introduced by physicistsas a cosmological model. It can be defined as a quadric in the 4-dimensional Minkowski space, equippedwith the induced metric:
ADS = { x ∈ R , | h x, x i = − } . Its fundamental group is Z . ADS admits a projective model, similar to the Klein model for H . Thismodel represents a “hemisphere” of ADS as the interior of a one-sheeted hyperboloid in R , with geodesicsof ADS corresponding to line segments. Space-light geodesics in ADS correspond to lines in R whichintersect the boundary hyperboloid at two points, light-like geodesics corresponds to line intersecting thehyperboloid tangentially in one point, while time-like geodesics corresponds to lines in the interior of thehyperboloid but not intersecting it. A full projective model of ADS can be obtained by taking a doublecover of RP , and thus in the sphere S , as the interior of a quadric of signature (1 , ADS is naturally equipped with a boundary, which can be seen in the projective model of ADS in S .This boundary is endowed with a conformal Lorentzian structure, analog to the conformal Riemannianmetric on the ideal boundary of H .It follows by its definition above that the isometry group of ADS is O (2 , SO (2 , P SL (2 , R ) × P SL (2 , R ), and ADS can in be fact identified isometricallywith P SL (2 , R ) equipped with its Killing form.6.1.2. Globally hyperbolic anti-de Sitter spacetimes.
There is a physically relevant notion of non-completeAdS spacetimes, namely, those which are globally hyperbolic : they contain a closed Cauchy surface, whichwe always assume to be of genus at least 2. Mess [Mes07,ABB +
07] discovered striking analogies betweenthe geometric properties of 3-dimensional Globally Hyperbolic Maximal Compact (GHMC) anti-de Sitterspacetimes, and those of quasifuchsian hyperbolic manifolds. For this reason, GHMC AdS spacetimesare now often called “quasifuchsian AdS spacetimes”, and we will follow this convention here.Mess [Mes07] proved that when M is a quasifuchsian AdS spacetime with Cauchy surface a closedsurface S , its holonomy representation ρ : π ( S ) → SO (2 ,
2) can be written, in the decomposition of SO (2 ,
2) as
P SL (2 , R ) × P SL (2 , R ), as ρ = ( ρ L , ρ R ), where ρ L , ρ R : π ( S ) → P SL (2 , R ) have maximalEuler number, and are therefore (by a result of Goldman [Gol88]) holonomy representations of hyperbolicmetrics on S , which can be called the left and right hyperbolic metrics of M .Mess [Mes07] discovered an analog for quasifuchsian AdS spacetimes of the Bers Simultaneous Uni-formization Theorem: given two hyperbolic metrics on a surface, there is a unique quasifuchsian AdSspacetime having them as left and right metrics.6.1.3. The left and right metrics on a space-like surface.
A different perspective is given in [KS07] onthose left and right hyperbolic metrics. Let S be a C space-like surface in ADS , one can define twometrics on S as: I L = I (( E + JB ) · , ( E + JB ) · ) , I R = I (( E − JB ) · , ( E − JB ) · )where E : T S → T S is the identity, B is the shape operator, and J is the complex structure of the inducedmetric. It can be checked by a simple computation using the Gauss formula in ADS (see [KS07, Lemma3.15]) that I L and I R are hyperbolic metrics as soon as S has induced metric of negative curvature.Those left and right metrics can be defined using a notion of “left” and “right” projection from aspace-like surface S on a fixed totally geodesic plane P in ADS . Those left and right projections canbe defined using the two foliations of the boundary quadric ∂ ADS by families of lines, which we will callthe left and right foliations. Suppose first that S is replaced by a totally geodesic plane P . Followingthe lines of the left (resp. right) foliation defines a homeomorphism between ∂P and ∂P , which can beshown to be a projective transformation, i.e. the boundary value of an isometry from P to P , and thisisometry is then the left (resp. right) projection from P to P , we denote it by π L,P (resp. π R,P ). Nowcoming back to the general case where S is a space-like surface, for each x ∈ S , we define π L ( x ) := π L,P ( x ) ( x ) , π R ( x ) := π R,P ( x ) ( x ) , where P ( x ) is the totally geodesic plane tangent to S at x . The pull-back by the left (resp. right)projections of the induced metric on P then turns out to be I L (resp. I R ), see [KS07]. Let M be a quasifuchsian AdS spacetime. It is noticed in [KS07, Lemma 3.16] that if the inducedmetric has negative curvature, then I ± are isotopic to the left and right metrics of M , respectively.6.1.4. Earthquakes and landslides.
The properties of the left and right projection above take a special(and interesting) form in two special cases: for locally convex pleated surfaces, and for K -surfaces, for K ∈ ( −∞ , − earthquakes (see e.g. [Thu86]). Moreover (see [BDMS19a, Lemma 7.7]) if S is a complete locally convex pleated surfacein ADS , then the left (resp. right) projection Π L (resp. Π R ) is an earthquake map, which extends theleft (resp. right) projection of ∂S to ∂P .A similar situation holds for K -surfaces, for K ∈ ( −∞ , − landslides ,as introduced and studied in [BMS13, BMS15]. Landslides can be considered as smoother versions ofearthquakes, which share most of their key properties. A θ -landslide between two hyperbolic ( S, h ) and( S ′ , h ′ ) can be defined as a diffeomorphism u : S → S ′ such that u ∗ h ′ = h ( b · , b ˙), where b : T S → T S isCodazzi, det( b ) = 1, tr( b ) = 2 cos( θ/ Jb ) <
0. Equivalently it can be defined by the conditionthat h ′ = h ((cos( θ/ E + sin( θ/ J ¯ b ) · , (cos( θ/ E + sin( θ/ J ¯ b ) · ) , where ¯ b is Codazzi, self-adjoint for h , and of determinant 1. (The equivalence between the two definitionscan be proved by setting b = cos( θ/ E + sin( θ/ J ¯ b .)Let now S ⊂ ADS be a K -surface, for K ∈ ( −∞ , − θ ∈ (0 , π ) be such that K = − / cos ( θ/ ( θ/ I is a hyperbolic metric on S . Moreover, by the Gauss formula in ADS , K = − − det( B ) , and therefore det( B ) = − K − ( θ/ − ( θ/ , and B can be written as B = tan( θ/ b , where ¯ b is Codazzi, self-adjoint for I , and has determinant 1.Therefore the left metric can be written as I L = I (( E + JB ) · , ( E + JB ) · )= I (( E + tan( θ/ J ¯ b ) · , ( E + tan( θ/ J ¯ b ) · )= 1cos ( θ/ I ((cos( θ/ E + sin( θ/ J ¯ b ) · , (cos( θ/ E + sin( θ/ J ¯ b ) · ) . As a consequence, the identity between the hyperbolic metrics (1 / cos ( θ )) I and I L is a θ -landslide.Similarly, the identity map between (1 / cos ( θ )) I and I R is a − θ -landslide. If S has bounded principalcurvatures, then this landslide diffeomorphism is quasi-conformal.Conversely, a quasi-conformal θ -landslide between two copies of the hyperbolic plane defines in thismanner a K -surface in ADS .The same construction works in the limit K = −
1, that is, for locally convex pleated surfaces in
ADS . The corresponding map is then an earthquake, and earthquakes can in this manner be consideredas limit cases of landslides, see [BMS13, BMS15], and an earthquake map from H to H is associatedto any locally convex complete pleated surface in ADS Quasicircles in ∂ ADS . It was already mentioned above that the “ideal” boundary of
ADS can beidentified with a quadric of signature (1 ,
1) in S , identified with the double cover of RP . This quadrichas a canonical decomoposition as RP × RP , such that for x ∈ RP , { x } × RP and RP × { x } are linesin S .Moreover, ∂ ADS can be equipped with a conformal Lorentzian metric, such that the light-like linesare also lines in S . The boundaries of complete space-like surfaces are acausal meridians, in the sensethat they are limits of graphs of homeomorphisms from RP → RP .Among those acausal meridians, those that are graphs of quasi-symmetric homeomorphisms play aparticular roles. They are often call quasi-circles in ∂ ADS , and appear to play a similar role in AdSgeometry as the “usual” quasi-circles in C P play in hyperbolic geometry, with some limited differences(see [BDMS19b] for such a difference). HE WEYL PROBLEM FOR COMPLETE SURFACES 21
Duality.
The polar duality between hyperbolic and de Sitter space, already recalled in Section 2.3,also appears for the anti-de Sitter space. However the duality is with
ADS itself. We recall briefly itsdefinition and main properties here.Let x ∈ ADS . It can be considered as point in R , , let x ⊥ be its oriented orthogonal hyperplane.Since h x, x i = − ADS ⊂ R , , x ⊥ is of signature (2 , ADS isa totally geodesic oriented space-like plane, which we denote by x ∗ .Given an oriented space-like surface Σ ⊂ ADS , we can define its dual as the set Σ ∗ of points dualto the tangent planes of Σ. If Σ is oriented and strictly convex, then Σ ∗ is also space-like, smooth andstrictly convex, and (Σ ∗ ) ∗ = Σ.As in H , the polar duality exchanges the induced metric and third fundamental form: the inducedmetric on Σ corresponds under the duality to the third fundamental form on Σ ∗ , and conversely.One point that can be noted is that the left and right metrics on a space-like, strictly convex surfaceare exchanged by duality. Indeed, under the polar duality, I is replaced by III , the shape operator B isreplace by B − , while the complex structure of the induced metric J is replaced by the complex structure¯ J of III , which is equal to ¯ J = B − JB . The left metric of the dual surface is thus equal to: III (( E + ¯ JB − ) · , (( E + ¯ JB − ) · ) = III (( E + B − J ) · , (( E + B − J ) · )= I (( B + J ) · , ( B + J ) · )= I ( J ( B + J ) · , J ( B + J ) · )= I (( E − JB ) · , ( E − JB ) · ) , which is the right metric on the primary surface, and conversely.6.2. Convex domains with thin boundary.
This section is focused on Question W thin ADS . This questiontakes a simpler form in ADS than in H since we only consider space-like surfaces in ADS . As aconsequence, if a convex domain Ω ⊂ ADS has boundary at infinity a disjoint union of quasicircles, thisfamily can only contain one quasicircle, and the boundary ∂ Ω in
ADS is composed of two disks sharingthe same “ideal” boundary.We first recall a number of known results and some open questions concerning quasifuchsian AdSspacetimes, corresponding to the special case of Question W thin ADS where the data on the boundary surfacesare invariant under a pair of cocompact actions of a surface group, and the gluing map at infinity is alsoequivariant under this pair of actions. The next section presents some recent results as well as openquestions without group actions, in particular concerning metrics of constant curvature. In the last partwe restate some of the same questions for surfaces of constant curvature in terms of fixed points ofearthquakes and of landslides.6.2.1. Quasifuchsian AdS spacetimes.
Let Ω ⊂ ADS be a convex domain invariant under a surface groupaction ρ : π S → SO (2 ,
2) which acts cocompactly on a Cauchy surface. The quotient Ω /ρ ( π S ) is thenisometric to a geodesically convex subset ¯Ω in a quasifuchsian AdS spacetime M .Such a quasifuchsian spacetime M contains a smallest non-empty geodesically convex subset, its convexcore C ( M ). If M is not Fuchsian, C ( M ) has non-emtpy interior, and its boundary is the disjoint unionof two pleated space-like surfaces ∂ ± C ( M ), both with hyperbolic induced metrics m ± pleated along ameasured lamination l ± . Mess extended to this AdS setting Conjectures 3.4 and 3.5. Conjecture 6.1 (Mess) . Let m − , m + be two hyperbolic metrics on a closed surface S of genus at least . Is there a unique quasifuchsian AdS structure on S × R such that the induced metrics on the boundarycomponents of the convex core are isotopic to m − and m + ? Conjecture 6.2 (Mess) . Let l − , l + be two measured laminations that fill on a closed surface S of genusat least . Is there a unique quasifuchsian AdS structure on S × R such that the measured bendinglaminations on the boundary of the convex core are l − and l + ? The first of those conjecture corresponds to a special case of Question W thin ADS , while the second corre-sponds to a special case of Question W ∗ thin ADS .The existence part of Conjecture 6.1 was proved by Diallo, see [Dia13] or [BDMS19a, Appendix] butthe uniqueness remains open. Similarly, the existence part of Conjecture 6.2 was proved in [BS12], butthe uniqueness is still unknown.If Ω is a geodesically convex domain with smooth, strictly convex boundary, its boundary is the disjointunion of two space-like surfaces, and each is equipped with a Riemannian metric of curvature K < − +
12, Question 3.5]). Tamburelli [Tam18] proved that any pair of metrics of curvature
K < − on a closed surface can be realized in this manner. However uniqueness remains elusive here, too. Thanksto the duality recalled in Section 6.1.6, it follows that any two metrics of curvature K < − W ∗ thin ADS .6.2.2. Existence results for domains with a quasicircle at infinity.
Without the asumption of a surfacegroup acting, there are some recent existence results but only for metrics of constant curvature.The main result of [BDMS19a] is that the existence part of Question W thin ADS has a positive answerfor metrics of constant curvature K ≤ −
1: given K ≤ − σ : RP → RP , there exists a convex domain Ω ⊂ ADS with boundary at infinity a quasi-circle, suchthat the induced metrics on the two boundary components of Ω have constant curvature K , and thegluing map at infinity between them is equal to σ .Thanks to the duality in Section 6.1.6, this also implies a similar existence result concerning the thirdfundamental form on ∂ Ω, for metrics of constant curvature
K < −
1. This does not however cover theuniversal version of Conjecture 6.2, for which a partial result is provided in [MS20]. To state it, we needtwo definitions.
Definition 6.3.
Let λ and µ be two mesured laminations on H . We say that λ and µ strongly fill if,for any ε > , there exists c > such that, if γ is a geodesic segment in H of length at least c , i ( γ, λ ) + i ( γ, µ ) > ε . Definition 6.4. A parameterized quasicircle in ∂ ADS is a map u : RP → ∂ ADS such that, underthe identification of ∂ ADS , the composition on the left of u with either the left or the right projection isquasi-symmetric. The following statement is the main result of [MS20].
Theorem 6.5.
Let λ − , λ + ∈ ML two bounded measured laminations that strongly fill. There exists aparameterized quasicircle u : RP → ∂ ADS such that the measured bending laminations on the upperand lower boundary components of CH ( u ( RP )) are u ∗ ( λ + ) and u ∗ ( λ − ) , respectively. Question 6.6.
In this setting, is u unique? Convex domains with thin boundary and fixed points of landslides.
We first recall the well-knownrelations between pleated surfaces in
ADS and earthquakes.Pleated surfaces in ADS are related to earthquakes, and the results and questions on prescribing thebending laminations on the boundary on convex hulls of quasi-circles can be stated in terms of fixed pointsof earthquakes. This equivalence was already noted (and used) in [BS12] for the bending lamination onthe boundary of the convex core of quasifuchsian AdS spacetimes. A similar equivalence holds withoutgroup actions, for convex hulls of quasi-circles in ADS , and is stated in [MS20]. The statement relies ona definition. Definition 6.7.
We denote by E l : ML × QS → QS the map defined as E l ( λ )( u ) = E l ( u ∗ λ ) ◦ u , and similarly for E r . Defined in this way, the map E l satisfies a “flow” property, which follows more or less directly fromthe definition: E l ( u ∗ ( tλ ))( E l ( sλ )( u )) = E l (( s + t ) λ )( u ) . There is of course a similar definition for right earthquake map E r , based on the right earthquakes E r .It then follows again more or less directly from the definitions that E r is the inverse of E l , in the sensethat E r ( u ∗ λ )( E l ( λ )( u )) = u . Theorem 6.5 can then be stated equivalently as follows.
Theorem 6.8.
Let λ − , λ + ∈ ML be two bounded laminations that strongly fill. There exists a quasi-symmetric homeomorphism u : RP → RP such that E l ( λ l )( u ) = E r ( λ r )( u ) . HE WEYL PROBLEM FOR COMPLETE SURFACES 23
Question 6.6 can then also be formulated equivalently as the uniqueness of this quasi-symmetric home-omorphism u .Thanks to the “cycle” properties above, this theorem can be stated as the existence of fixed point ofthe map u
7→ E l ( u ∗ λ r ) ◦ E l ( λ l )( u ), as a map from T to T . The uniqueness in Question 6.6 is equivalentto the uniqueness of this fixed point.We now consider the similar relation between K -surfaces and landslides.For K -surfaces, a similar statement can be made, with earthquakes replaced by landslides. Howevermeasured laminations have to be replaced by quasi-symmetric maps. The construction requires some care,and begins with the definition of the landslides on QS × QS , generalizing (in a not completely direct way)the definition in [BMS13] for closed surfaces. We recall the definition here, basically repeating materialfrom [BMS13, § u, u ∗ ∈ QS . Then u ∗ ◦ u − is quasi-symmetric. It follows (using the main result in [BS10])that there exists a unique quasi-conformal minimal Lagrangian diffeomorphism w : H → H such that ∂w = u ∗ ◦ u − .There is then a unique bundle morphism b : T H → T H which is self-adjoint for h , Codazzi, ofdeterminant 1, and such that w ∗ h = h ( b · , b · ).Let θ ∈ (0 , π ). We set β θ = cos( θ/ E + sin( θ/ Jb .
A direct computation (see [BMS13]) shows that β θ is Codazzi and of determinant 1, and therefore h θ = h ( β θ · , β θ · ) is hyperbolic. As a consequence, the identity map between H and ( H , h θ ) determinesa quasiconformal diffeomorphism from H to H , well-defined up to post-composition by a hyperbolicisometry. Taking its boundary value determines a quasi-symmetric map v θ .Note that composing u or u ∗ on the left by a M¨obius transformation does not change b , and thereforedoes not change v θ , which is defined up to post-composition by a hyperbolic isometry anyway. So v θ iswell-defined as an element of the universal Teichm¨uller space T , and depends on u and u ∗ considered aselements of T . Definition 6.9.
Let u, u ∗ ∈ T , we set L e iθ ( u, u ∗ ) = ( v θ u, v θ + π u ) ∈ T × T , and L e iθ ( u, u ∗ ) = v θ u ∈ T . As an example, for θ = 0, β = E , and as a consequence w = Id and L ( u, ∗ ) = u . On the otherhand, for θ = π , β π = Jb , so that h π = h ( Jb · , Jb · ) = h ( b · , b · ) = w ∗ h , v π = w , and L − ( u, u ∗ ) = u ∗ .This definition is strongly related to the definition given in [BMS13] for hyperbolic metrics on closedsurfaces. Specifically, consider a closed surface S of genus at least 2, equipped with a fixed hyperbolicmetric h ∈ T S , see [BMS13, Prop. 8.4].The arguments and computations in [BMS13, § L e iθ on T × T satisfies the flow property, that is, L e iθ ◦ L e iθ ′ = L e i ( θ + θ ′ ) for all θ, θ ′ ∈ R .We can now formalize the relation between landslides and K -surfaces in ADS . Let u L , u R : RP → RP be quasi-symmetric homeomorphism, and let C be the graph of u R ◦ u − L , seen as a quasicircle in ∂ ADS .Note that C can be identified with RP by choosing any parameterization ρ : RP → C such that ρ composed with the left projection is a quasi-symmetric homeomorphism. We use such an identificationbelow implicitly, so as to simplify notations.Let K ∈ ( −∞ , − S + (resp. S − ) be the unique past-convex (resp. future-convex) K -surfacewith boundary C . Let u L : C → RP and u R : C → RP be the left and right projections, and let u + : C → RP (resp. u ∗ + : C → RP ) be identification of C with ∂ ∞ H obtained by consideringthe developing map of the hyperbolic metric | K | I (resp. | K ∗ | III ) on S + . Finally, let u − , u ∗− be thecorresponding maps for S − . Finally, let θ ∈ (0 , π ) be such that K = − / cos ( θ/ Theorem 6.10.
Under those conditions, u L = L θ ( u + , u ∗ + ) = L − θ ( u − , u ∗− ) ,u R = L − θ ( u + , u ∗ + ) = L θ ( u − , u ∗− ) . Conversely, if u L , u R , u + , u ∗ + , u − , u ∗− are elements of T satisfying those two equations, then they arisefrom the construction above. Spacelike disks in
ADS . We now turn to Question W imm ADS and its dual. Following the organisationof the previous section, we first focus on quasifuchsian AdS spacetimes, corresponding to immersions ofthe universal cover of a surface which are equivariant under a representation of the fundamental groupof the surface in SO (2 , W imm ADS reduces to the following. Question 6.11.
Let h be a Riemannian metric of curvature K < − on a closed surface S of genus atleast , and let h ∈ T S be a hyperbolic metric on S . Is there a unique quasifuchsian AdS spacetime M containing a past-convex Cauchy surface with induced metric isotopic to h and left metric isotopic to h ? Tamburelli [Tam18, Prop. 7.1] shows that existence holds in this question. It then follows from theduality described in Section 6.1.6 that the same result holds with the third fundamental form prescribedinstead of the metric.Back to the universal case, a full answer to Question W imm ADS can be given for K -surfaces for K ≤ − K = −
1, that is, for locally convex pleated surfaces, this answer is a direct consequence of Thurston’sEarthquake Theorem.
Proposition 6.12.
Let σ : RP → RP be a quasi-symmetric homeomorphism. There is a unique past-convex pleated surface in ADS such that the gluing map at infinity between the induced metric and theleft metric is σ . The proof follows from the fact (see [BS10, Prop. 7.7]) that given a past-convex pleated surface S ⊂ ADS , the left projection from the boundary curve to a circle in ∂ ADS (the boundary of a totallygeodesic space-like plane P ) is the continous extension to the boundary of the left projection from S to P , which itself is the earthquake along the measured bending lamination on S . But any quasi-symmetrichomeomorphism from RP to RP is the continuous extension to the boundary of the earthquake alonga unique bounded measured lamination on the hyperbolic disk, and the result follows.Similarly, for K -surfaces for K < −
1, a positive answer to Question Q imm ADS follows from a recent resultof Bonsante and Seppi [BS18], who proved that any quasi-symmetric homeomorphism from RP to RP is the boundary of a unique quasi-conformal θ -landslide. Proposition 6.13.
Let σ : RP → RP be a quasi-symmetric homeomorphism, and let K < − . There isa unique past-convex K -surface in ADS such that the gluing map at infinity between the induced metricand the left metric is σ . Again the result follows from the fact following facts.(1) For such a K -surface, the gluing map between the induced metric and the left metric is given bythe left projection from the boundary of S to the boundary of a totally geodesic space-like plane P ⊂ ADS (see [BS10, Lemma 3.18]).(2) For a K -surface, the left projection is a θ -landslide, for K = − / cos ( θ ), and conversely if theleft projection is a θ -landslide, then S is a K -surface, see above.(3) It is proved in [BS18] that any quasi-symmetric homeomorphism from RP to RP extendsuniquely as a quasiconformal θ -landslide from H to H .6.4. Ideal and hyperideal polyhedra.
There are other recent results concerning versions of the Weylproblem in
ADS , but for polyhedra rather than smooth surfaces. We would like to briefly mention therecent results in [DMS20] showing that ideal polyhedra in ADS are uniquely determined by their inducedmetrics, or by their dihedral angles (which play the role of third fundamental form in this setting). Similarresults hold for hyperideal polyhedra in ADS [CS19].7. Minkowski domains of dependence and half-pipe geometry
We consider in this section analogs of the questions considered above in hyperbolic and anti-de Sitterspace, but in Minkowski space instead. Minkowski space is naturally dual to a 3-dimensional spaceintroduced by Jeff Danciger, see [Dan13, Dan14], and called half-pipe space. Half-pipe geometry can benaturally considered as a “transitional” geometry between hyperbolic and anti-de Sitter geometry, andmost of the questions considered above in the hyperbolic and anti-de Sitter settings also make sense inhalf-pipe, and by duality, in Minkowski space.Space-like surfaces in half-pipe space can be considered as first-order deformations of a totally geodesicspace-like plane in
AdS , or of a totally geodesic plane in H , see Section 7.2. HE WEYL PROBLEM FOR COMPLETE SURFACES 25
Half-pipe space as a dual of Minkowski space.
We provide here some very basic definitions,see e.g. [Dan13, BS17] for more details.Half-pipe space HP can be defined simply as H × R , equipped with the degenerate metric h + 0 dt ,where h is the standard metric on H . It is naturally the geometric structure occuring in the convexsubset bounded by a degenerate quadric in RP (or equivalently a cylinder in R — through the Hilbertmetric. In this way, HP is equipped with a natural notion of lines and planes, which are just theintersection with the cylinder of lines and planes in R . A line or planed is called space-like if it is notparallel to the generatrix of the cylinder, that is, if the metric induced from that of HP is non-degenerate.There is a natural duality between HP and Minkowski space R , . To any space-like plane p in R , one can associate its unit future-oriented normal vector n , and the oriented distance d between 0 and p along the line directed by n through 0. The pair ( n, d ) defines a point in HP , which we call the dual of p and denote by p ∗ . Conversely, given a point q ∈ R , , the set of points p ∗ ∈ HP dual to the space-likeplanes p ∋ q in R , form a space-like plane in HP , which we call the plane dual to q and denote by q ∗ .The isometry group of HP is infinite-dimensional (a fact that follows from the degeneraty of themetric). However HP has a restricted group of isometries, those that preserve the totally geodesicplanes. This restricted isometry group is finite-dimensional.The identity component of the (restricted) isometry group of both HP and R , can be identified to P SL (2 , R ) ⋉ R , , and the action of the (restricted) isometry group commutes with the duality.A surface S ⊂ HP is called space-like if the induced metric on S is everywhere non-degenerate. If S is smooth, one can then define its second fundamental form, by considering it locally as the graphof a function over its tangent plane p (each point of S being associated to the point of p on the samedegenerate line of HP ), and taking the Hessian of that function. One can then define its Weingartenoperator and third fundamental form III in the usual manner.Let now S ⊂ R , be a future-convex space-like surface with positive definite second fundamental form.One can consider the set of points in HP dual to the planes tangent to S . This turns out to be a locallystrictly convex surface S ∗ . Moreover this duality exchanges the induced metric and third fundamentalform, as seen for hyperbolic space in Section 2.3 and for the anti-de Sitter space in Section 6.1.6. (Thisduality can of course be seen as a limit case of the duality seen in Section 6.1.6, see below in Section 7.2.)7.2. Half-pipe surfaces as first-order deformations of hyperbolic planes.
We have already seenin the introduction how a surface in R can be seen as a limit of surfaces in H of diameter convergingto 0. In a similar manner, a surface in H can be seen as a first-order deformation of a totally geodesicplane in H , or of a totally geodesic space-like plane in AdS . We briefly describe the construction for atotally geodesic plane in ADS , it can be adapted almost verbatim, with some small simplifications, to atotally geodesic space-like plane in H .Let P be a totally geodesic plane in ADS , and let v = νn be a normal vector field defined along P ,corresponding to a first-order deformation of P . For each x ∈ P and t ∈ [0 , φ t ( x ) = exp x ( tv ),where exp x is the exponential map at x . This defines a one-parameter family ( φ t ) t ∈ [0 , of embeddingsof P in ADS .We then call Ω ⊂ ADS the union of geodesics orthogonal to P — this subset is the future cone ofthe point dual to P in ADS . For each x ∈ Ω, we denote by π ( x ) ∈ P its orthogonal projection on P ,and by t ( x ) the time-oriented distance from π ( x ) to x along the geodesic joining them, which is thereforeorthogonal to P at π ( x ). A direct computation then shows that the AdS metric on Ω can be written as g ( ˙ x, ˙ x ) = cos( t ( x )) h ( dπ ( ˙ x ) , dπ ( ˙ x )) − dt ( ˙ x ) , where h is the induced metric on P .We then for s ∈ (0 ,
1) perform the change of variable τ ( x ) = t ( x ) /s , which leads to a one-parameterfamily of rescaled metrics ( g s ) s ∈ (0 , defined as g s ( ˙ x, ˙ x ) = cos( sτ ( x )) h ( dπ ( ˙ x ) , dπ ( ˙ x )) − s dτ ( ˙ x ) . Clearly those metrics converge, as s →
0, to the metric on HP .For each s ∈ (0 , v the surface S s = φ s ( P ).In the coordinates ( π ( x ) , τ ( x )) defined above, S s is the graph of the function v : P → R . So as s → S s converges to the graph of this function v over P , considered as a surface S HP in HP .Note that a similar description can be used to describe the dual surface S ∗ of S in R , . Indeed, thedual of P is a point P ∗ , and as t → S ∗ t converges to the past light cone of P ∗ . To obtain the surfacedual to S HP , one needs only to apply a one-parameter family of homotheties (or scaling) centered at P ∗ . Quasifuchsian HP manifolds.
Once half-pipe surfaces are considered as a first-order deformationsof totally geodesic planes in
ADS or H , one can consider equivariant first-order deformations of a totallygeodesic plane as “Cauchy surfaces” in a half-pipe manifold. We do not elaborate on this point here,and refer to Danciger’s work [Dan13] for a detailed construction. We will call those half-pipe manifolds“quasifuchsian”, since a number of their properties are similar or analoguous to those of quasifuchsianhyperbolic manifolds or AdS spacetimes. • Their holonomy representation takes value in
P SL (2 , R ) ⋉ R , . • They contain a smallest non-empty geodesically convex subset, their convex core. Except in the“Fuchsian” case – for manifolds containing a totally geodesic closed surface – the convex corehas non-empty interior, and its boundary is the disjoint union of two hyperbolic surfaces pleatedalong a measured bending lamination. • They contain a unique minimal Cauchy surface (i.e. a Cauchy surface for which the traceof II with respect to I vanishes).There is also a notion of Minkowski dual of a quasifuchsian half-pipe manifold. However this dual isnot a single Minkowski spacetime, but rather a pair of globally hyperbolic maximal compact Minkowskispacetimes, one future-complete and one past-complete, which share the same holonomy (which is ofcourse also the holonomy of their dual half-pipe manifold).Note that the questions considered above concerning the induced metrics on surfaces do not workwell in half-pipe geometry. This is because the induced metric on any complete space-like surface in HP is isometric to the hyperbolic plane, while all Cauchy surfaces in a quasifuchsian half-pipe manifoldhave the same induced metric. However the dual questions concerning the third fundamental forms onthose surfaces do make sense. Equivalently, one can consider questions concerning the induced metricson locally convex surfaces, but not really those concerning their third fundamental forms.The natural analog of Question W ∗ thin H in this half-pipe setting, for domains with smooth boundary,is the following. Question W ∗ thin HP . Let h − , h + be two complete, conformal metrics of curvature K < on the disk D ,and let σ : RP → RP be a quasi-symmetric homeomorphism. Is there a unique geodesically convexdomain in H with boundary at infinity a quasicircle, such that the induced metric on the past and futureboundary components are isometric respectively to h − and h + , with gluing at infinity given by σ ? This question can also be stated dually in terms of the induced metrics on space-like surfaces in pairsof Minkowski domains of dependence. The statement is somewhat less visual than that of Question W ∗ thin HP . Given a pair of corresponding domains of dependence D − , D + in R , , one past-complete andone future-complete, and given two complete space-like surfaces S − ⊂ D − and S + ⊂ D + , there is anatural identication between asymptotic directions on S − and on S + , obtained by identifying points atinfinity corresponding to parallel half-lines in ∂D − and ∂D + . Question W ∗ thin HP asks whether given anytwo complete conformal metrics on D of negative curvature, and a quasi-symmetric homeomorphism σ : RP → RP , there is a unique pair of corresponding domains of dependence D − , D + containing completespace-like surfaces S − , S + with induced metrics isometric to g − and g + ,respectively, with identificationat infinity given by σ .7.4. Equivariant surfaces.
The special case of Question W ∗ thin HP was recently proved by Graham Smith[Smi20]. His result can be stated as follows. Theorem 7.1 (G. Smith) . Let S be a closed surface, and let g − , g + be two metrics of negative curvatureon S . There is then a unique quasifuchsian half-pipe manifold containing a geodesically convex subsetsuch that the induced metrics on the past and future boundary components are isotopic to g − and g + ,respectively. Smith in fact states and proves the dual statement: given a closed surface S and a pair ( g − , g + ) ofnegatively curved metrics on S , there exists a unique GHMC Minkowski spacetime into which ( S, g − )and ( S, g + ) isometrically embed as Cauchy surfaces in the past and future components respectively.Note also that the corresponding statements for pleated surfaces in half-pipe manifolds, where oneprescribes the measured pleating lamination, is proved in [BS12, Theorem B.2], see also [Bon05].7.5. Pairs of unbounded surfaces and minimizing diffeomorphisms.
The proof of Theorem 7.1in [Smi20] (translated into half-pipe geometry) is based on the fact that if S − and S + are future-convexand past-convex surfaces in a quasifuchsian HP manifold M , with third fundamental forms g − and g + , HE WEYL PROBLEM FOR COMPLETE SURFACES 27 respectively, then the natural projection π : S + → S − obtained by following the degenerate lines in M can be factored as π = π − ◦ ( π + ) − , where π ± are the identity maps from ( S ± , I ) to ( S ± , III ) and ( S − , I ) is identified isometrically to ( S + , I ).Moreover both π − and π + are minimizing in the sense of Trapani and Valli [TV95], who proved theexistence and uniqueness of such a minimizing map isotopic to the identity between two negativelycurved metrics on a surface.Question W ∗ thin HP thus leads quite naturally to the question of the existence and uniqueness of a mini-mizing map with given (quasi-symmetric) boundary behavior between two complete conformal metrics ofnegative curvature on the disk D – a result which is known [BS10] only for metrics of constant curvature − Further questions: the Weyl problem and K -surfaces in higher Teichm¨uller theory There has been in the last years considerable interest in extending elements of Teichm¨uller theoryto specific representations in Lie groups more general than SL (2). The first development in this areawas by Hitchin [Hit87], with key developments by Labourie [Lab06, Lab07, LM09] as well as Fock andGoncharov [FG06] (we can obviously not give proper references to all important contributions here).Several types of representations fit into this general setting, in particular Anosov representations andmaximal representations (depending on the Lie group being considered).AdS quasifuchsian spacetimes can be considered as a “baby case” of this higher Teichm¨uller theory,with the Lie group being SO (2 ,
2) or
P SL (2 , R ) × P SL (2 , R ).As already mentioned above, it would be particularly interesting in this area to attach to each “ad-missible” representation of a surface group a “canonical” surface, satisfying a specific property, invariantunder this surface group in the symmetric space of the Lie group, or in another associated space. Afirst candidate for the defining property of this surface is to consider minimal surfaces, but the existenceand uniqueness of minimal surfaces in the symmetric space of the Lie group is only known in somecases [Lab17], and it fails in some key cases, for instance quasifuchsian hyperbolic representations. References [ABB +
07] Lars Andersson, Thierry Barbot, Riccardo Benedetti, Francesco Bonsante, William M. Goldman, Fran¸coisLabourie, Kevin P. Scannell, and Jean-Marc Schlenker. Notes on: “Lorentz spacetimes of constant curvature”[Geom. Dedicata (2007), 3–45; mr2328921] by G. Mess.
Geom. Dedicata , 126:47–70, 2007.[Ale05] Alexander D. Alexandrov.
Convex polyhedra . Springer Monographs in Mathematics. Springer-Verlag, Berlin,2005. Translated from the 1950 Russian edition by N. S. Dairbekov, S. S. Kutateladze and A. B. Sossinsky,With comments and bibliography by V. A. Zalgaller and appendices by L. A. Shor and Yu. A. Volkov.[And70] E.M. Andreev. Convex polyhedra in Lobacevskii space.
Mat. Sb.(N.S.) , 81 (123):445–478, 1970.[And71] E.M. Andreev. On convex polyhedra of finite volume in Lobacevskii space.
Math. USSR Sbornik , 12 (3):225–259, 1971.[BB02] Xiliang Bao and Francis Bonahon. Hyperideal polyhedra in hyperbolic 3-space.
Bull. Soc. Math. France ,130(3):457–491, 2002.[BBD +
12] T. Barbot, F. Bonsante, J. Danciger, W.M. Goldman, F. Gu´eritaud, F. Kassel, K. Krasnov, J.M. Schlenker,and A. Zeghib. Some open questions on anti-de sitter geometry.
Arxiv preprint arXiv:1205.6103 , 2012.[BDMS19a] Francesco Bonsante, Jeffrey Danciger, Sara Maloni, and Jean-Marc Schlenker. The induced metric on theboundary of the convex hull of a quasicircle in hyperbolic and anti de sitter geometry. arXiv preprintarXiv:1902.04027 , 2019.
Geometry & Topology , to appear.[BDMS19b] Francesco Bonsante, Jeffrey Danciger, Sara Maloni, and Jean-Marc Schlenker. Quasicircles and width of jordancurves in CP , 2019. To appear, Bull. London Math. Soc. [Bel14] Mehdi Belraouti. Sur la g´eom´etrie de la singularit´e initiale des espaces-temps plats globalement hyperboliques.
Ann. Inst. Fourier (Grenoble) , 64(2):457–466, 2014.[Bel17] Mehdi Belraouti. Asymptotic behavior of Cauchy hypersurfaces in constant curvature space-times.
Geom.Dedicata , 190:103–133, 2017.[BMS13] Francesco Bonsante, Gabriele Mondello, and Jean-Marc Schlenker. A cyclic extension of the earthquake flowI.
Geom. Topol. , 17(1):157–234, 2013.[BMS15] Francesco Bonsante, Gabriele Mondello, and Jean-Marc Schlenker. A cyclic extension of the earthquake flowII.
Ann. Sci. ´Ec. Norm. Sup´er. (4) , 48(4):811–859, 2015.[Bon05] Francis Bonahon. Kleinian groups which are almost Fuchsian.
J. Reine Angew. Math. , 587:1–15, 2005.[Bor77] A. A. Borisenko. Complete l-dimensionnal surfaces of non-positive extrinsic curvature in a riemannian space.
Math. USSR Sb. , 33(4):485–499, 1977.[BS10] Francesco Bonsante and Jean-Marc Schlenker. Maximal surfaces and the universal Teichm¨uller space.
Invent.Math. , 182(2):279–333, 2010. [BS12] Francesco Bonsante and Jean-Marc Schlenker. Fixed points of compositions of earthquakes.
Duke Math. J. ,161(6):1011–1054, 2012.[BS17] Francesco Bonsante and Andrea Seppi. Spacelike convex surfaces with prescribed curvature in (2+1)-Minkowskispace.
Adv. Math. , 304:434–493, 2017.[BS18] Francesco Bonsante and Andrea Seppi. Area-preserving diffeomorphisms of the hyperbolic plane and K -surfacesin anti-de Sitter space. J. Topol. , 11(2):420–468, 2018.[BS20a] Francesco Bonsante and Andrea Seppi. Anti-de sitter geometry and teichm \ ” uller theory. arXiv preprintarXiv:2004.14414 , 2020.[BS20b] Francesco Bonsante and Andrea Seppi. Anti-de sitter geometry and teichm¨uller theory, 2020.[Cau13] Augustin Louis Cauchy. Sur les polygones et poly`edres, second m´emoire. Journal de l’Ecole Polytechnique ,19:87–98, 1813.[CS19] Qiyu Chen and Jean-Marc Schlenker. Hyperideal polyhedra in the 3-dimensional anti-de sitter space. arXivpreprint arXiv:1904.09592 , 2019.[Dan13] Jeffrey Danciger. A geometric transition from hyperbolic to anti-de Sitter geometry.
Geom. Topol. , 17(5):3077–3134, 2013.[Dan14] Jeffrey Danciger. Ideal triangulations and geometric transitions.
J. Topol. , 7(4):1118–1154, 2014.[Dia13] Boubacar Diallo. Prescribing metrics on the boundary of convex cores of globally hyperbolic maximal compactads 3-manifolds. arXiv preprint arXiv:1303.7406 , 2013.[DMS20] Jeffrey Danciger, Sara Maloni, and Jean-Marc Schlenker. Polyhedra inscribed in a quadric.
Invent. Math. ,221(1):237–300, 2020.[Dum08] David Dumas. Complex projective structures. In
Handbook of Teichm¨uller theory. Vol. II , volume 13 of
IRMALect. Math. Theor. Phys. , pages 455–508. Eur. Math. Soc., Z¨urich, 2008.[DW08] David Dumas and Michael Wolf. Projective structures, grafting and measured laminations.
Geom. Topol. ,12(1):351–386, 2008.[EM86] D. B. A. Epstein and A. Marden. Convex hulls in hyperbolic spaces, a theorem of Sullivan, and measuredpleated surfaces. In D. B. A. Epstein, editor,
Analytical and geometric aspects of hyperbolic space , volume 111of
L.M.S. Lecture Note Series . Cambridge University Press, 1986.[FG06] Vladimir Fock and Alexander Goncharov. Moduli spaces of local systems and higher Teichm¨uller theory.
Publ.Math. Inst. Hautes ´Etudes Sci. , (103):1–211, 2006.[FI09] Fran¸cois Fillastre and Ivan Izmestiev. Hyperbolic cusps with convex polyhedral boundary.
Geom. Topol. ,13(1):457–492, 2009.[Fil07] Fran¸cois Fillastre. Polyhedral realisation of hyperbolic metrics with conical singularities on compact surfaces.
Ann. Inst. Fourier (Grenoble) , 57(1):163–195, 2007.[Fil08] Fran¸cois Fillastre. Polyhedral hyperbolic metrics on surfaces.
Geom. Dedicata , 134:177–196, 2008.[Fil11] Fran¸cois Fillastre. Fuchsian polyhedra in Lorentzian space-forms.
Math. Ann. , 350(2):417–453, 2011.[FLP91] A. Fathi, F. Laudenbach, and V. Poenaru.
Travaux de Thurston sur les surfaces . Soci´et´e Math´ematique deFrance, Paris, 1991. S´eminaire Orsay, Reprint of
Travaux de Thurston sur les surfaces , Soc. Math. France,Paris, 1979 [MR 82m:57003], Ast´erisque No. 66-67 (1991).[FS19] Fran¸cois Fillastre and Andrea Seppi. Spherical, hyperbolic, and other projective geometries: convexity, duality,transitions. In
Eighteen essays in non-Euclidean geometry , volume 29 of
IRMA Lect. Math. Theor. Phys. ,pages 321–409. Eur. Math. Soc., Z¨urich, 2019.[Gol88] William M. Goldman. Topological components of spaces of representations.
Invent. Math. , 93(3):557–607, 1988.[Hit87] N. J. Hitchin. The self-duality equations on a Riemann surface.
Proc. London Math. Soc. (3) , 55(1):59–126,1987.[HR93] Craig D. Hodgson and Igor Rivin. A characterization of compact convex polyhedra in hyperbolic 3-space.
Invent. Math. , 111:77–111, 1993.[HS93] Zheng-Xu He and Oded Schramm. Fixed points, Koebe uniformization and circle packings.
Ann. of Math. (2) ,137(2):369–406, 1993.[KMT03] Sadayoshi Kojima, Shigeru Mizushima, and Ser Peow Tan. Circle packings on surfaces with projective struc-tures.
J. Differential Geom. , 63(3):349–397, 2003.[KMT06a] Sadayoshi Kojima, Shigeru Mizushima, and Ser Peow Tan. Circle packings on surfaces with projective struc-tures: a survey. In
Spaces of Kleinian groups , volume 329 of
London Math. Soc. Lecture Note Ser. , pages337–353. Cambridge Univ. Press, Cambridge, 2006.[KMT06b] Sadayoshi Kojima, Shigeru Mizushima, and Ser Peow Tan. Circle packings on surfaces with projective structuresand uniformization.
Pacific J. Math. , 225(2):287–300, 2006.[Koe18] Paul Koebe. Abhandlungen zur theorie der konformen abbildung.
Mathematische Zeitschrift , 2(1-2):198–236,1918.[KS07] Kirill Krasnov and Jean-Marc Schlenker. Minimal surfaces and particles in 3-manifolds.
Geom. Dedicata ,126:187–254, 2007.[Lab92a] Fran¸cois Labourie. M´etriques prescrites sur le bord des vari´et´es hyperboliques de dimension 3.
J. DifferentialGeom. , 35:609–626, 1992.[Lab92b] Fran¸cois Labourie. Surfaces convexes dans l’espace hyperbolique et CP1-structures.
J. London Math. Soc., II.Ser. , 45:549–565, 1992.[Lab00] Fran¸cois Labourie. Un lemme de Morse pour les surfaces convexes.
Invent. Math. , 141(2):239–297, 2000.[Lab05] Fran¸cois Labourie. Random k -surfaces. Ann. of Math. (2) , 161(1):105–140, 2005.[Lab06] Fran¸cois Labourie. Anosov flows, surface groups and curves in projective space.
Invent. Math. , 165(1):51–114,2006.
HE WEYL PROBLEM FOR COMPLETE SURFACES 29 [Lab07] Fran¸cois Labourie. Cross ratios, surface groups, PSL( n, R ) and diffeomorphisms of the circle. Publ. Math. Inst.Hautes ´Etudes Sci. , (106):139–213, 2007.[Lab17] Fran¸cois Labourie. Cyclic surfaces and Hitchin components in rank 2.
Ann. of Math. (2) , 185(1):1–58, 2017.[Lam19] Wai Yeung Lam. Quadratic differentials and circle patterns on complex projective tori. arXiv preprintarXiv:1909.07258 , 2019. To appear,
Geometry & Topology .[LegII] Adrien-Marie Legendre.
El´ements de g´eom´etrie . Paris, 1793 (an II). Premi`ere ´edition, note XII, pp.321-334.[Lei02] Gregory Leibon. Characterizing the Delaunay decompositions of compact hyperbolic surfaces.
Geom. Topol. ,6:361–391 (electronic), 2002.[Lew35] Hans Lewy. A priori limitations for solutions of monge-ampere equations.
Transactions of the American Math-ematical Society , 37(3):417–434, 1935.[LM09] Fran¸cois Labourie and Gregory McShane. Cross ratios and identities for higher Teichm¨uller-Thurston theory.
Duke Math. J. , 149(2):279–345, 2009.[LS00] Fran¸cois Labourie and Jean-Marc Schlenker. Surfaces convexes fuchsiennes dans les espaces lorentziens `a cour-bure constante.
Math. Ann. , 316(3):465–483, 2000.[LW19] Feng Luo and Tianqi Wu. Koebe conjecture and the weyl problem for convex surfaces in hyperbolic 3-space. arXiv preprint arXiv:1910.08001 , 2019.[LW20] Feng Luo and Tianqi Wu. Koebe conjecture and the weyl problem for convex surfaces in hyperbolic 3-space,2020.[McM98] Curtis T. McMullen. Complex earthquakes and Teichm¨uller theory.
J. Amer. Math. Soc. , 11(2):283–320, 1998.[Mes07] Geoffrey Mess. Lorentz spacetimes of constant curvature.
Geom. Dedicata , 126:3–45, 2007.[MS20] Louis Merlin and Jean-Marc Schlenker. Bending laminations on convex hulls of anti-de sitter quasicircles, 2020.arxiv:2006.13470.[Pog73] Aleksei V. Pogorelov.
Extrinsic Geometry of Convex Surfaces . American Mathematical Society, 1973. Trans-lations of Mathematical Monographs. Vol. 35.[Riv92] Igor Rivin. Intrinsic geometry of convex ideal polyhedra in hyperbolic 3-space. In M. Gyllenberg and L. E.Persson, editors,
Analysis, Algebra, and Computers in Mathematical Research , pages 275–292. Marcel Dekker,1992. (Proc. of the 21st Nordic Congress of Mathematicians).[Riv96] Igor Rivin. A characterization of ideal polyhedra in hyperbolic 3-space.
Annals of Math. , 143:51–70, 1996.[Rou04] Mathias Rousset. Sur la rigidit´e de poly`edres hyperboliques en dimension 3 : cas de volume fini, cas hyperid´eal,cas fuchsien.
Bull. Soc. Math. France , 132:233–261, 2004.[RS94] H. Rosenberg and J. Spruck. On the existence of convex hypersurfaces of constant Gauss curvature in hyperbolicspace.
J. Differential Geom. , 40:379–409, 1994.[Sch96] Jean-Marc Schlenker. Surfaces convexes dans des espaces lorentziens `a courbure constante.
Comm. Anal.Geom. , 4(1-2):285–331, 1996.[Sch98a] Jean-Marc Schlenker. M´etriques sur les poly`edres hyperboliques convexes.
J. Differential Geom. , 48(2):323–405,1998.[Sch98b] Jean-Marc Schlenker. R´ealisations de surfaces hyperboliques compl`etes dans H . Ann. Inst. Fourier (Grenoble) ,48(3):837–860, 1998.[Sch01] Jean-Marc Schlenker. Hyperbolic manifolds with polyhedral boundary. math.GT/0111136, available athttp://picard.ups-tlse.fr/˜schlenker, 2001.[Sch02] Jean-Marc Schlenker. Hyperideal polyhedra in hyperbolic manifolds. Preprint math.GT/0212355., 2002.[Sch06] Jean-Marc Schlenker. Hyperbolic manifolds with convex boundary.
Invent. Math. , 163(1):109–169, 2006.[Smi06] Graham Smith. Pointed k -surfaces. Bull. Soc. Math. France , 134(4):509–557, 2006.[Smi15] Graham Smith. Hyperbolic Plateau problems.
Geom. Dedicata , 176:31–44, 2015.[Smi20] Graham Smith. On the weyl problem in minkowski space. arXiv preprint arXiv:2005.01137 , 2020.[SW02] Kevin P. Scannell and Michael Wolf. The grafting map of Teichm¨uller space.
J. Amer. Math. Soc. , 15(4):893–927 (electronic), 2002.[SY18] J.-M. Schlenker and A. Yarmola. Properness for circle packings and Delaunay circle patterns on complexprojective structures.
ArXiv e-prints , June 2018.[Tam18] Andrea Tamburelli. Prescribing metrics on the boundary of anti–de Sitter 3-manifolds.
Int. Math. Res. Not.IMRN , (5):1281–1313, 2018.[Thu86] William P. Thurston. Earthquakes in two-dimensional hyperbolic geometry. In
Low-dimensional topology andKleinian groups (Coventry/Durham, 1984) , volume 112 of
London Math. Soc. Lecture Note Ser. , pages 91–112.Cambridge Univ. Press, Cambridge, 1986.[Thu06] W. P. Thurston. Earthquakes in 2-dimensional hyperbolic geometry. In
Fundamentals of hyperbolic geometry:selected expositions , volume 328 of
London Math. Soc. Lecture Note Ser. , pages 267–289. Cambridge Univ.Press, Cambridge, 2006.[TV95] Stefano Trapani and Giorgio Valli. One-harmonic maps on Riemann surfaces.
Comm. Anal. Geom. , 3(3-4):645–681, 1995.[Wol89] Michael Wolf. The Teichm¨uller theory of harmonic maps.
J. Differential Geom. , 29(2):449–479, 1989.
Jean-Marc Schlenker: University of Luxembourg, FSTM, Department of Mathematics, Maison du nombre,6 avenue de la Fonte, L-4364 Esch-sur-Alzette, Luxembourg
Email address ::