Discrete Conformal Geometry of Polyhedral Surfaces and Its Convergence
DDISCRETE CONFORMAL GEOMETRY OF POLYHEDRAL SURFACES AND ITSCONVERGENCE
FENG LUO, JIAN SUN, AND TIANQI WUA
BSTRACT . The paper proves a result on the convergence of discrete conformal maps to theRiemann mappings for Jordan domains. It is a counterpart of Rodin-Sullivan’s theorem on con-vergence of circle packing mappings to the Riemann mapping in the new setting of discreteconformality. The proof follows the same strategy that Rodin-Sullivan used by establishing arigidity result for regular hexagonal triangulations of the plane and estimating the quasiconfor-mal constants associated to the discrete conformal maps. C ONTENTS
1. Introduction 12. Polyhedral metrics, vertex scaling and a variational principle 43. A maximum principle, a ratio lemma and spiral hexagonal triangulations 74. Rigidity of hexagonal triangulations of the plane 135. Existence of discrete uniformization metrics on polyhedral disks with specialequilateral triangulations 186. A proof of the convergence theorem 317. A convergence conjecture on discrete uniformization maps 32References 351. I
NTRODUCTION
Thurston’s conjecture on the convergence of circle packing mappings to the Riemann map-ping is a constructive and geometric approach to the Riemann mapping theorem. The conjec-ture was solved in an important work by Rodin and Sullivan [34] in 1987. There have beenmany research works inspired by the work of Thurston and Rodin-Sullivan since then. Thispaper addresses a counterpart of Thurston’s convergence conjecture in the setting of discreteconformal change of polyhedral surfaces associated to the notion of vertex scaling (Definition1.1). We prove a weak version of Rodin-Sullivan’s theorem in this new setting. There are stillmany problems to be resolved in order to prove the full convergence conjecture.Let us begin with a recall of Thurston’s conjecture and Rodin-Sullivan’s solution. Given abounded simply connected domain Ω in the complex plane C , one constructs a sequence ofapproximating triangulated polygonal disks ( D n , T n ) whose triangles are equilateral and edgelengths of the triangles tend to zero such that D n converges to Ω . For each such polygonaldisk, by the Koebe-Andreev-Thurston’s existence theorem, there exists a circle packing of the Mathematics Subject Classification.
Key words and phrases. polyhedral metrics, discrete conformal map, Riemann mapping, triangulation, Delau-nay triangulation, convex polyhedra and discrete harmonic functions. a r X i v : . [ m a t h . G T ] S e p FENG LUO, JIAN SUN, AND TIANQI WU unit disk D such that the combinatorics (or the nerve) of circle packing is isomorphic to the1-skeleton of the triangulation T n . This produces a piecewise linear homeomorphism f n , calledthe circle packing mapping, from the polygonal disk D n to a polygonal disk inside D associatedto the circle packing. Thurston conjectured in 1985 that, under appropriate normalizations, thesequence { f n } converges uniformly on compact subsets of Ω to the Riemann mapping for Ω .Here the normalization condition is given by choosing a point p ∈ Ω , a sequence of vertices v n in ( D n , T n ) such that lim n v n = p , and f n ( v n ) = 0 such that f (cid:48) n ( v n ) > . The Riemannmapping f for Ω sends p to and f (cid:48) ( p ) > . Rodin-Sullivan’s proof of Thurston’s conjectureis elegant and goes in two steps. In the first step, they show that the circle packing mappings f n are K -quasiconformal for some constant K independent of the indices. In the second step,they show that there is only one hexagonal circle packings of the complex plane up to Moebiustransformations. This implies that the limit of the sequence { f n } is conformal.Circle packing metrics introduced by Thurston [42] can be considered as a discrete con-formal geometry of polyhedral surfaces. In recent times, there have been many works ondiscretization of 2-dimensional conformal geometry ([26], [4], [19], [13], [12], and others).In this paper, we consider the counterpart of Thurston’s conjecture in the setting of discreteconformal change defined by vertex scaling.To state our main results, let us recall some related material and notations. A compacttopological surface S together with a non-empty finite subset of points V ⊂ S will be called a marked surface . A triangulation T of a marked surface ( S, V ) is a topological triangulation of S such that the vertex set of T is V . We use E = E ( T ) , V = V ( T ) to denote the sets of alledges and vertices in T respectively. A polyhedral metric d on ( S, V ) , to be called a PL metric on ( S, V ) for simplicity, is a flat cone metric on ( S, V ) whose cone points are contained in V .We call the triple ( S, V, d ) a polyhedral surface. The discrete curvature , or simply curvature , ofa PL metric d is the function K : V → ( −∞ , π ) sending an interior vertex v to π minus thecone angle at v and a boundary vertex v to π minus the sum of angles at v . All PL metrics areobtained by isometric gluing of Euclidean triangles along pairs of edges. If T is a triangulationof a polyhedral surface ( S, V, d ) for which all edges in T are geodesic, we say T is geometric in d and d is a PL metric on ( S, T ) . In this case, we can represent d by the length function l d : E ( T ) → R > sending each edge to its length. Thus the polyhedral surface ( S, V, d ) can berepresented by ( S, T , l d ) where l d ∈ R E> . We will also call ( S, T , l d ) or l d a PL metric on T . Definition 1.1. (Vertex scaling change of PL metrics [26] ) Two PL metrics l and l ∗ on a tri-angulated surface ( S, T ) are related by a vertex scaling if there exists a map w : V ( T ) → R so that if e is an edge in T with end points v and v (cid:48) , then the edge lengths l ( e ) and l ∗ ( e ) arerelated by (1) l ∗ ( e ) = e w ( v )+ w ( v (cid:48) ) l ( e ) . We denote l ∗ by w ∗ l if (1) holds and call l ∗ obtained from l by a vertex scaling and w a discreteconformal factor. Condition (1) was proposed in [26] as a discrete conformal equivalence between PL metricson triangulated surfaces. There are three basic problems related to the vertex scaling. Thefirst is the existence problem. Namely, given a PL metric l on a triangulated closed surface ( S, T ) and a function K : V ( T ) → ( −∞ , π ) satisfying the Gauss-Bonnet condition, is therea PL metric l ∗ of the form w ∗ l whose curvature is K ? Unlike Koebe-Andreev-Thurston’stheorem which guarantees the existence of circle packing metrics, the answer to the aboveexistence problem is negative in general. This makes the convergence of discrete conformal ONVERGENCE OF DISCRETE CONFORMAL MAPS 3 mappings a difficult problem. On the other hand, the uniqueness of the vertex scaled PL metric l ∗ with prescribed curvature holds. This was established in an important paper by Bobenko-Pinkall-Springborn [4]. The third is the convergence problem. Namely, assuming the existenceof PL metrics with prescribed curvatures, can these discrete conformal polyhedral surfacesapproximate a given Riemann surface? The main result of the paper gives a solution to theconvergence problem for the simplest case of Jordan domain.The convergence theorem that we proved is the following. Let Ω be a Jordan domain withthree points p, q, r specified in the boundary. By Caratheodory’s extension theorem [32], theRiemann mapping from Ω to the unit disk D extends to a homeomorphism from the closure Ω to the closure D . Therefore, there exists a unique homeomorphism g from Ω to an equilateralEuclidean triangle ∆ ABC with vertices
A, B, C such that p, q, r are sent to
A, B, C and g isconformal in Ω . For simplicity, we call g and g − the Riemann mappings for (Ω , ( p, q, r )) .Given an oriented triangulated polygonal disk ( D, T , l ) and three boundary vertices p, q, r ∈ V , suppose there exists a PL metric l ∗ = w ∗ l on ( D, T ) for some w : V → R such that itsdiscrete curvature at all vertices except { p, q, r } are zero and the curvatures at p, q, r are π .Then the associated flat metric on ( D, T , l ∗ ) is isometric to an equilateral triangle ∆ ABC ,i.e., there is a geometric triangulation T (cid:48) of ∆ ABC such that (∆ ABC, T (cid:48) , l st ) is isometric to ( D, T , l ∗ ) . Here and below, if T is a geometric triangulation of a domain in the plane, weuse l st : E ( T ) → R to denote the length of edges e in T in the standard metric on C . Let f : D → ∆ ABC be the piecewise linear orientation preserving homeomorphism sending V tothe vertex set V ( T (cid:48) ) of T (cid:48) , and p, q, r to A, B, C respectively and being linear on each triangleof T . We call f the discrete uniformization map associated to ( D, T , l, { p, q, r } ) . Note that f may not exist due to the lacking of existence theorem. Theorem 1.2.
Suppose Ω is a Jordan domain in the complex plane with three distinct points p, q, r ⊂ ∂ Ω . Then there exists a sequence (Ω n , T n , l st , ( p n , q n , r n )) of simply connected trian-gulated polygonal disks in C where T n are triangulations by equilateral triangles and p n , q n , r n are three boundary vertices such that(a) Ω = ∪ ∞ n =1 Ω n with Ω n ⊂ Ω n +1 , and lim n p n = p , lim n q n = q and lim n r n = r ,(b) discrete uniformization maps associated to (Ω n , T n , l st , ( p n , q n , r n )) exist and convergeuniformly to the Riemann mapping for (Ω , ( p, q, r )) . In Rodin-Sullivan’s convergence theorem, any sequence of approximating circle packingmaps associated to the approximation triangulated polyhedral disks Ω n such that Ω n ⊂ int (Ω n +1 ) and Ω = ∪ n Ω n converges to the Riemann mapping. Theorem 1.2 is less robust in this aspectsince discrete conformal maps may not exist if the triangulations T n are not carefully selected.A stronger version of convergence is conjectured in § T of polyhedral surfaceis called Delaunay if the sum of two angles facing each interior edge is at most π . Delaunaytriangulations always exist for each PL metric on compact surfaces. Theorem 1.3.
Suppose T is a Delaunay geometric triangulations of the complex plane C such that its vertex set is a lattice and l st : E ( T ) → R is the edge length function of T . If ( C , T , w ∗ l st ) is a Delaunay triangulated surface isometric to an open set in the Euclideanplane C , then w is a constant function. We remark that the same result as above for the standard hexagonal lattice has been provedindependently by Dai-Ge-Ma [9] in a recent preprint.
FENG LUO, JIAN SUN, AND TIANQI WU
Using an important result in [4] that vertex scaling is closely related to hyperbolic 3-dimensionalgeometry and the work of [13], one sees that Theorem 1.4 implies the following rigidity resulton convex hyperbolic polyhedra.
Theorem 1.4.
Suppose L = Z + τ Z is a lattice in the plane C and V ⊂ C is a discrete setsuch that there exists an isometry between the boundaries of the convex hulls of L and V inthe hyperbolic 3-space H preserving cell structures. Then V and L differ by a complex affinetransformation of C . This prompts us to propose the following conjecture. A closed set X in the Riemann sphereis said to be of circle type if each connected component of X is either a point or a rounddisk. Consider the Riemann sphere C ∪ {∞} as the infinity of the (upper-half-space model of)hyperbolic 3-space H . Conjecture 1.5.
For any genus zero connected complete hyperbolic surface Ω , there exists acircle type closed set X ⊂ C ∪ {∞} such that Ω is isometric to the boundary of the convexhull of X in H . Conjecture 1.6.
Suppose X and Y are circle type closed sets in C such that boundary of theconvex hulls of X and Y in H are isometric. Then X and Y differ by a M¨obius transformation. The paper is organized as follows. § § § § § § Acknowledgement . We thank Michael Freedman for discussions which lead to the formula-tion of Conjectures 1.5, 1.6. The work is partially supported by the NSF DMS 1405106, NSFDMS 1760527, NSF DMS 1811878, NSF DMS 1760471 of USA and a grant from the NSF ofChina.2. P
OLYHEDRAL METRICS , VERTEX SCALING AND A VARIATIONAL PRINCIPLE
We begin with some notations. Let C , R , Z be the sets of complex, real, and integersrespectively. R > = { t ∈ R | t > } , Z ≥ k = { n ∈ Z | n ≥ k } and SS = { z ∈ C || z | = 1 } . Weuse D to denote the open unit disk in C and H n to denote the n -dimensional hyperbolic space.Given that X is a compact surface with boundary, its interior is denoted by int ( X ) . A graphwith vertex set V and edge set E is denoted by ( V, E ) . Two vertices i, j in a graph ( V, E ) are adjacent , denoted by i ∼ j , if they are the end points of an edge. If i ∼ j , we use [ ij ] (respectively ij ) to denote an oriented (respectively unoriented) edge from i to j . An edgepath joining i, j ∈ V is a sequence of vertices { v = i, v , ..., v m = j } such that v k ∼ v k +1 .The length of the path is m . The combinatorial distance d c ( i, j ) between two vertices in a ONVERGENCE OF DISCRETE CONFORMAL MAPS 5 connected graph ( V, E ) is the length of the shortest edge path joining i, j . Suppose ( S, T ) is atriangulated surface with possibly non-empty boundary ∂S and possibly non-compact S . Let E = E ( T ) , V = V ( T ) be the sets of edges, vertices respectively and T (1) = ( V, E ) be theassociated graph. A vertex v ∈ V ( T ) ∩ ∂S (resp. v ∈ V ∩ ( S − ∂S )) is called a boundary(resp. interior) vertex. Boundary and interior edges are defined in the same way. A PL metricon ( S, T ) or simply on T can be represented by a length function l : E ( T ) → R > so that if e i , e j , e k are three edges forming a triangle in T , then the strict triangle inequality holds,(2) l ( e i ) + l ( e j ) > l ( e k ) . We will use limits of PL metrics. To this end, we introduce the notion of generalized PLmetrics on ( S, T ) . Take three pairwise distinct points v , v , v in the plane. The convex hullof { v , v , v } is a generalized triangle with vertices v , v , v . We denoted it by ∆ v v v . If v , v , v are not in a line, then ∆ v v v is a (usual) triangle. If v , v , v lie in a line, then ∆ v v v is a degenerate triangle with the flat vertex at v i if | v j − v i | + | v k − v i | = | v j − v k | , { i, j, k } = { , , } . Let l i = | v j − v k | ∈ R > be the edge length and a i ∈ [0 , π ] be the angle at v i . Then l i + l j ≥ l k > and the angles are given by(3) a i = arccos( l j + l k − l i l j l k ) . Furthermore, the angle a i = a i ( l , l , l ) ∈ [0 , π ] is continuous in ( l , l , l ) . Degenerate tri-angles are characterized by either having an angle π or the lengths satisfying l i = l j + l k forsome i, j, k .A generalized PL metric on a triangulated surface ( S, T ) is represented by an edge lengthfunction l : E ( T ) → R > so that if e i , e j , e k are three edges forming a triangle in T , then thetriangle inequality holds,(4) l ( e i ) + l ( e j ) ≥ l ( e k ) . We will abuse the use of terminology and call l a generalized PL metric on ( S, T ) or T . The discrete curvature K : V ( T ) → ( −∞ , π ] of a generalized PL metric ( S, T , l ) is defined asfollows. If v ∈ V ( T ) is an interior vertex, K ( v ) is π minus the sum of angles (of generalizedtriangles) at v ; if v is a boundary vertex, K ( v ) is π minus the sum of angles at v . Note thatthe Gauss-Bonnet theorem (cid:80) v ∈ V ( T ) K ( v ) = 2 πχ ( S ) still holds for a compact surface S witha generalized PL metric. Clearly the curvature K and inner angles depend continuously on thelength vector l ∈ R E ( T ) > . A generalized PL metric is called flat if its curvatures are zero at allinterior vertices v . A generalized PL metric ( S, T , l ) (or sometimes written as ( T , l ) ) is called Delaunay if for each interior edge e ∈ E ( T ) the sum of the two angles facing e is at most π .If ( S, T , l ) is a Delaunay generalized PL metric such that each angle facing a boundary edge isat most π/ , then the metric double of ( S, T , l ) along its boundary is a Delaunay triangulatedgeneralized PL metric surface. Two generalized PL metrics l and ˜ l on ( S, T ) are related by a vertex scaling if there is w ∈ R V so that ˜ l ( vv (cid:48) ) = e w ( v )+ w ( v (cid:48) ) l ( vv (cid:48) ) for all edges vv (cid:48) ∈ E ( T ) . We write ˜ l = w ∗ l and call w a discrete conformal factor .Two generalized triangles ∆ v v v and ∆ u u u are equivalent if there exists an isometrysending v i to u i for i = 1 , , . The space of all equivalence classes of generalized trianglescan be identified with { ( l , l , l ) ∈ R > | l i + l j ≥ l k } . It contains the space of all equivalence FENG LUO, JIAN SUN, AND TIANQI WU classes of triangles { ( l , l , l ) ∈ R > | l i + l j > l k } . Given two generalized triangles l =( l , l , l ) and ˜ l = ( ˜ l , ˜ l , ˜ l ) there exists w = ( w , w , w ) ∈ R such that ˜ l i = l i e w j + w k . The following result was proved in [26, Theorem 2.1] for Euclidean triangles. The extensionto generalized triangles is straight forward.
Proposition 2.1 ([26]) . Let ∆ v v v be a fixed generalized triangle of edge length vector l =( l , l , l ) and w ∗ l is the edge length vector of a vertex scaled generalized triangle whose innerangle at v i is a i = a i ( w ) .(a) For any two constants c i , c j , the set { ( w , w , w ) ∈ R | w ∗ l is a generalized triangle and w i = c i , w j = c j } is either connected or empty.(b) If (∆ v v v , l ) is a non-degenerate triangle and i, j, k distinct, then (5) ∂a i ∂w i | w =0 = − sin( a i )sin( a j ) sin( a k ) < , ∂a i ∂w j | w =0 = ∂a j ∂w i | w =0 = cot( a k ) , and (cid:88) j =1 ∂a i ∂w j = 0 . The matrix − [ ∂a r ∂w s ] × is symmetric, positive semi-definite with null space spanned by (1 , , T .(c) If (∆ v v v , l ) is a degenerate triangle having v as the flat vertex, then for small t > , (∆ v v v , (0 , , t ) ∗ l ) is a non-degenerate triangle. The angle a (0 , , t ) is strictly decreasingin t for all t for which (0 , , t ) ∗ l is a generalized triangle. The angles a i (0 , , t ) , i = 1 , , arestrictly increasing in t ∈ [0 , (cid:15) ) for some (cid:15) > . F IGURE
1. Vertex scaling of a triangle
Proof.
To see part (a), without loss of generality, we may assume c and c are the givenconstants. Then the variable w is defined by three inequalities e w ( e c l + e c l ) ≥ e c + c l , e c + c l ≥ e w ( e c l − e c l ) ≥ − e c + c l . Each of these inequalities defines an interval in w variable. Therefore the solution space is either the empty set or a connected set.Part (b) is in [26, Theorem 2.1].To see (c), due to l = l + l , for small t > , we have (0 , , t ) ∗ l = ( e t l , e t l , l + l ) ∈ ∆ := { ( x , x , x ) ∈ R > | x i + x j ≥ x k } . Now by (5) and the Sine Law, ∂a ∂w (0 , , t ) = − sin( a )sin( a ) sin( a ) < . Together with part (a), the angle a (0 , , t ) as a function of t is definedon an interval and is strictly decreasing in t . Since lim t → + ∂a ∂w = lim t → + cot( a ) = ∞ dueto lim t → + a ( t, ,
0) = 0 , therefore the result holds for a . By the same argument, the resultholds for a . (cid:3) As a consequence,
ONVERGENCE OF DISCRETE CONFORMAL MAPS 7
Corollary 2.2.
Under the same assumption as in Proposition 2.1, if w ( t ) = ( w ( t ) , w ( t ) , w ( t )) ∈ R is smooth in t such that w ( t ) ∗ l is the edge length vector of a triangle with inner angle a i ( t ) = a i ( w ( t ) ∗ l ) at v i , then (6) da i ( t ) dt = (cid:88) j ∼ i cot( a k )( dw j dt − dw i dt ) where j ∼ i means v j is adjacent to v i and { i, j, k } = { , , } . Write w (cid:48) j ( t ) = dw j dt . Indeed by the chain rule and (5), we have da i ( t ) dt = ∂a i ∂w i w (cid:48) i + (cid:88) j (cid:54) = i ∂a i ∂w j w (cid:48) j = − (cid:88) j (cid:54) = i cot( a k ) w (cid:48) i + (cid:88) j (cid:54) = i cot( a k ) w (cid:48) j = (cid:88) j ∼ i cot( a k )( w (cid:48) j − w (cid:48) i ) . Suppose ( S, T , l ) is a geometrically triangulated compact polyhedral surface and w ( t ) ∈ R V is a smooth path in parameter t such that w ( t ) ∗ l is a PL metric on ( S, T ). Let K i = K i ( t ) bethe discrete curvature at i ∈ V and θ ijk = θ ijk ( t ) be the inner angle at the vertex i in ∆ ijk inthe metric w ( t ) ∗ l . For an edge [ ij ] in the triangulation T , define η ij to be cot( θ kij ) + cot( θ lij ) if [ ij ] is an interior edge facing two angles θ kij and θ lij and η ij = cot( θ kij ) if [ ij ] is a boundaryedge. If [ ij ] is an interior edge, then η ij ≥ if and only if θ kij + θ lij ≤ π , i.e., the Delaunaycondition holds at [ ij ] .The curvature variation formula is the following, Proposition 2.3. (7) dK i ( t ) dt = (cid:88) j ∼ i η ij ( dw i dt − dw j dt ) . This follows directly from the Corollary 2.2 since K i = cπ − (cid:80) r,s ∈ V θ irs where c = 1 or and θ irs are angles at i . Since dK i ( t ) dt = − (cid:80) r,s ∈ V dθ irs dt , (7) follows from (6) and the definition of η ij . 3. A MAXIMUM PRINCIPLE , A RATIO LEMMA AND SPIRAL HEXAGONALTRIANGULATIONS
Let v be an interior point of a star-shaped n -sided polygon P n having vertices v , ..., v n labelled cyclically. The triangulation T of P n with vertices v , ..., v n and triangles ∆ v v i v i +1 ( v n +1 = v ) is called a star triangulation of P n . See Figure 2. Theorem 3.1 (Maximum principle) . Let T be a star triangulation of P n and l : E ( T ) → R > be a generalized Delaunay polyhedral metric on T . If w : { v , v , ..., v n } → R satisfies(a) w ∗ l is a generalized Delaunay polyhedral metric on T ,(b) the curvatures K ( w ∗ l ) of w ∗ l and K ( l ) of l at vertex v satisfy K ( w ∗ l ) ≤ K ( l ) ,and(c) w ( v ) = max { w ( v i ) | i = 0 , , ..., n } ,then w ( v i ) = w ( v ) for all i . FENG LUO, JIAN SUN, AND TIANQI WU
As a convention, if x = ( x , ..., x m ) and y = ( y , y , ..., y m ) are in R m +1 , then x ≥ y means x i ≥ y i for all i . Given w : { v , ..., v m } → R , we use w i = w ( v i ) and identify w with ( w , w , ..., w m ) ∈ R m +1 . The cone angle of w ∗ l at v will be denoted by α ( w ) . ThusTheorem 3.1(b) says α ( w ) ≥ α (0) .The proof of Theorem 3.1 depends on the following lemma. Lemma 3.2. If w : { v , v , ..., v n } → R satisfies (a), (b), (c) in Theorem 3.1 such that there is w i < w , then there exists ˆ w ∈ R n +1 such that(a) ˆ w i ≥ w i for i = 1 , , ..., n ,(b) ˆ w i ≤ ˆ w = w for i = 1 , , ..., n ,(c) ˆ w ∗ l is a generalized Delaunay polyhedral metric on T , and(d) (8) α ( ˆ w ) > α ( w ) . Let us first prove Theorem 3.1 using Lemma 3.2.
Proof.
By replacing w by w − w ( v )(1 , , ..., , we may assume that w ( v ) = 0 . Suppose theresult does not hold, i.e., there exists w so that w = 0 , w i ≤ for i = 1 , , ..., n with one w i < , and w ∗ l is a generalized Delaunay PL metric on T so that α ( w ) ≥ α (0) . We willderive a contradiction as follows. By Lemma 3.2, we may assume, after replacing w by ˆ w , that(9) α ( w ) > α (0) . Consider the set X = { x ∈ R n +1 | w ≤ x ≤ , x = 0 , x ∗ l is a generalized Delaunay polyhedral metric on T } . Clearly w ∈ X and therefore X (cid:54) = ∅ and X is bounded. Since inner angles are continuous inedge lengths, we see that X is a closed set in R n +1 . Therefore X is compact. Let t ∈ X be amaximum point of the continuous function f ( x ) = α ( x ) on X . We claim that t = 0 . To provethis, we assume t (cid:54) = 0 and t ≤ . Then by Lemma 3.2, we can find ˆ t ≥ t such that ˆ t = 0 and ˆ t ≤ , and ˆ t ∗ l is a generalized Delaunay polyhedral metric on T with α (ˆ t ) > α ( t ) . Thiscontradicts the maximality of t . Now for t = 0 , we have α (0) = α ( t ) ≥ α ( w ) > α (0) where the last inequality follows from (9). This is a contradiction. (cid:3) Now back to the proof of Lemma 3.2.
Proof.
After replacing w by w − w (1 , , ..., , we way assume w = 0 . Let a i = a i ( w ) = a i ( w , w i , w i +1 ) , b i = b i ( w ) = b i ( w , w i − , w i ) and c i = c i ( w ) = c i ( w , w i , w i +1 ) be the innerangles ∠ v v i +1 v i , ∠ v v i − v i and ∠ v i v v i +1 in the metric w ∗ l respectively. See Figure 2. Let l i = l ( v v i ) and l i,i +1 = l ( v i v i +1 ) be the edge lengths in the metric l .Let us begin the proof for the simplest case where all triangles in w ∗ l are non-degenerate(i.e., w ∗ l is a PL metric) and w i < for all i ≥ . Let j ∈ { , , , ..., n } be the indexsuch that w ∗ l ( v v j ) = min { w ∗ l ( v v k ) | k = 1 , , ..., n } . It is well known that in a Euclideantriangle (cid:52) ABC , ∠ A < π/ if BC is not the unique largest edge. Hence, due to w ∗ l ( v v j ) ≤ w ∗ l ( v v j ± ) , in the triangles ∆ v v j v j ± , we have(10) a j ( w ) < π/ , b j ( w ) < π/ and a j ( w ) + b j ( w ) < π .Now consider ˆ w = ( w , w , ..., w j − , w j + t, w j +1 , ..., w n ) . For small t > , ˆ w ( t ) ∗ l is stilla PL metric since w ∗ l is. We claim ˆ w ∗ l is still Delaunay for small t . Indeed, by Proposition ONVERGENCE OF DISCRETE CONFORMAL MAPS 9 F IGURE
2. Star triangulation of an n -sided polygon2.1, both angles a j − and b j +1 decrease in t . On the other hand a j +1 ( w ) = a j +1 ( ˆ w ) and b j − ( w ) = b j − ( ˆ w ) . Therefore, the Delaunay conditions b j − + a j − ≤ π and b j +1 + a j +1 ≤ π hold for edges v v j ± . The Delaunay condition on the edge v v j follows from choice of j that a j + b j < π . Finally, by Proposition 2.1(b), dα ( ˆ w ) dt = cot( a j ) + cot( b j ) = sin( a j + b j )sin( a j ) sin( b j ) > .Therefore, for small t > , we have α ( ˆ w ) > α ( w ) .In the general case, the above arguments still work.Let J = { j ∈ V | w j < } . By assumption, J (cid:54) = ∅ . Claim 1. If j ∈ J , then c j ( w ) < π and c j − ( w ) < π .We prove c j − ( w ) < π by contradiction. Suppose otherwise that c j − ( w ) = π . Then thetriangle ∆ v v j v j − is degenerate in w ∗ l metric, i.e., e w j + w j − l j,j − = e w j l j + e w j − l j − . Dueto w j < and w j − ≤ , we have e w j + w j − l j,j − = e w j l j + e w j − l j − > e w j + w j − l j + e w j + w j − l j − = e w j + w j − ( l j + l j − ) . This shows l j,j − > l j + l j − which contradicts the triangle inequality for l metric. Therefore c j − ( w ) < π . By the same argument, we have c j ( w ) < π . This proves Claim 1.Let I = { i > | w i = 0 } and β ( w ) = (cid:88) i ∈ I ( b i ( w ) + a i ( w )) and γ ( w ) = (cid:88) j ∈ J ( b j ( w ) + a j ( w )) . Note that the cone angle at v is α ( w ) = n (cid:88) i =1 ( π − a i ( w ) − b i +1 ( w )) = πn − β ( w ) − γ ( w ) . By the assumption that α ( w ) ≥ α (0) , we have(11) β ( w ) + γ ( w ) ≤ β (0) + γ (0) . Claim 2. If I (cid:54) = ∅ , then β ( w ) > β (0) . Indeed, if i ∈ I , i.e., w i = 0 , then in the triangle ∆ v v i v i ± , we have w = 0 , w i = 0 and w i ± ≤ . Since ∆ v v i v i − are generalized triangles in both l and w ∗ l metrics, by proposition ∆ v v i v i − is a generalized triangle in ( w , ..., w i − , tw i − , w i , ..., w n ) ∗ l for t ∈ [0 , . By Proposition 2.1 and w i − ≤ , b i ( w , ..., w i − , tw i − , w i , ..., w n ) is increasing in t ≥ and is strictly increasing in t ≥ if w i − < . Therefore, b i ( w ) = b i ( w , w i − , w i ) ≥ b i ( w , , w i ) = b i ( w , w , ..., w i − , , w i , ..., w n ) = b i (0) , and b i ( w ) > b i (0) if w i − < . Apply the same argument to ∆ v v i v i +1 and a i , we have a i ( w ) ≥ a i (0) and a i ( w ) > a i (0) if w i +1 < . Therefore β ( w ) ≥ β (0) . On the other hand,since J (cid:54) = ∅ , there exists an i ∈ I so that either i − or i + 1 is in J . Say i − ∈ J , i.e., w i − < . Then we have b i ( w ) > b i (0) and β ( w ) > β (0) .By claim 2 and (11), if I (cid:54) = ∅ , we conclude that(12) γ ( w ) = (cid:88) j ∈ J ( a j ( w ) + b j ( w )) < γ (0) . Since w ∗ l and l are Delaunay, we have a i ( w ) + b i ( w ) ≤ π and a i (0) + b i (0) ≤ π for all i = 1 , , ..., n . This implies, by (12), that there exists j ∈ J so that(13) a j ( w ) + b j ( w ) < π. If I = ∅ , let j ∈ J = { , , , ..., n } be the index such that w ∗ l ( v v j ) = min { w ∗ l ( v v k ) | k =1 , , ..., n } . Then the same argument used in showing (10) and Claim 1 imply (13) still holds.(Here Claim 1 is used to show that ( w , ..., w j − , w j + t, w j +1 , ..., w n ) ∗ l is a generalized PLmetric for small t > .)Fix this j ∈ J as above. To finish the proof, we will show that there exists a small t > sothat for ˆ w = ( w , w , ... , w j − , w j + t, w j +1 , ..., w n ) ∈ R n +1 ≤ the following hold:(i) ˆ w ∗ l is a generalized polyhedral metric on T ;(ii) ˆ w ∗ l satisfies the Delaunay condition;(iii) α ( ˆ w ) > α ( w ) .Since w j < , any t ∈ (0 , − w j ) will make ˆ w ∈ R n +1 ≤ .To see part (i), by Claim 1 and (13) which imply a j ( w ) , b j ( w ) , c j ( w ) , c j − ( w ) < π , thetriangle (∆ v v j v j +1 , w ∗ l ) (or ( ∆ v v j v j − , w ∗ l ) ) is either non-degenerate or is degeneratewith π -angle at v j , i.e., b j +1 ( w ) = π (or a j − ( w ) = π respectively). Therefore by Proposition2.1(c), for small t > , ˆ w ∗ l is still a generalized PL metric.To see part (ii), we check the sum of opposite angles at the following edges: v v j − , v v j +1 and v v j . At the edge v v j , due to (13) and continuity, we see a j ( ˆ w ) + b j ( ˆ w ) < π for small t > . At the edge v v j − (or similarly v v j +1 ), by Proposition 2.1(c) that a j − ( ˆ w ∗ l ) and b j +1 ( ˆ w ∗ l ) are strictly decreasing functions in t > and b j − ( ˆ w ) = b j − ( w ) , we have a j − ( ˆ w ) + b j − ( ˆ w ) < a j − ( w ) + b j − ( w ) ≤ π. Similarly, we have the Delaunay condition for ˆ w ∗ l at the edge v v j +1 .Finally, to see (iii), by Proposition 2.1 and (13), we have ddt | t =0 α ( ˆ w ) = ddt | t =0 ( c j ( ˆ w )) + ddt | t =0 ( c j − ( ˆ w ))= cot( b j ( ˆ w )) + cot( a j ( ˆ w )) > . Therefore, for small t > , α ( ˆ w ) > α ( w ) . (cid:3) Lemma 3.3.
Let ( P N , T ) be a star triangulation of an N -gon with boundary vertices v , ..., v N labelled cyclically and one interior vertex v and l : E ( T ) → R > be a flat generalizedPL metric on T . There is a constant λ ( l ) depending on l such that if ( P N , T , w ∗ l ) with ONVERGENCE OF DISCRETE CONFORMAL MAPS 11 w : { v , ..., v N } → R is a generalized PL metric with zero curvature at v , then the ratio ofedge lengths satisfies (14) w ∗ l ( v i v ) w ∗ l ( v i v i +1 ) ≤ λ ( l ) for all indices. F IGURE
3. Triangulated hexagon and length ratio
Proof.
Let x i ( w ) = w ∗ l ( v v i ) and y i ( w ) = w ∗ l ( v i v i +1 ) be the edge lengths in the metric w ∗ l where v N +1 = v . By definition,(15) x i +2 y i +1 = λ i x i y i , where λ i > depends on l . Then(16) x i +1 y i +1 ≥ x i +2 − y i +1 y i +1 = x i +2 y i +1 − λ i x i y i − . We prove by contradiction. If the result of lemma 3.3 is not true, than there exists a sequenceof conformal factors w ( n ) such that x i ( w ( n ) ) y i ( w ( n ) ) → ∞ , for some i . Without loss of generality, assume i = 1 and then by (16) inductively we have x ( w ( n ) ) y ( w ( n ) ) → ∞ , x ( w ( n ) ) y ( w ( n ) ) → ∞ , ..., x N ( w ( n ) ) y N ( w ( n ) ) → ∞ . Then the angle a i ( w ( n ) ) at v in the triangle ∆ v v i v i +1 (in w ( n ) ∗ l metric) converge to , forany i . But that contradicts the fact that the curvature π − (cid:80) Ni =1 a i ( w ( n ) ) at v is zero. (cid:3) The next result concerns linear discrete conformal factor and spiral hexagonal triangulations.It is a counterpart of Doyle spiral circle packing in the discrete conformal setting. Unlike Doylespiral circle packing, not all choices of linear functions produce generalized PL metrics.We begin by recalling the developing maps. If ( S, T , l ) is a flat generalized PL metric ona simply connected surface S (i.e., K v = 0 for all interior vertices v ), then a developing map φ : ( S, T , l ) → C for ( T , l ) is an isometric immersion determined by | φ ( v ) − φ ( v (cid:48) ) | = l ( vv (cid:48) ) for v ∼ v (cid:48) . It is constructed as follows. Fix a generalized triangle t ∈ T and isometricallyembeds t to C . This defines φ | t . If s is a generalized triangle sharing a common edge e with t , we can extend φ | t to φ | t ∪ s by isometrically embedding s to φ ( s ) ⊂ C sharing the edge φ ( e ) with φ ( t ) such that φ ( s ) and φ ( t ) are on different sides of φ ( e ) . Since the surface is simply connected, by the monodromy theorem, we can keep extending φ to all triangles in T andproduce a well defined isometric immersion. As a convention, if τ is triangle in T and l is ageneralized PL metric on T , we use ( τ, l ) to denote the induced generalized PL metric on τ .Given a lattice L in C , there exists a Delaunay triangulation T st = T st ( L ) of C with vertexset L such that the T st is invariant under the translation action of L . In particular T st descendsto a 1-vertex triangulation of the torus C /L . Therefore, the degree of each vertex v ∈ T st is six,i.e., this triangulation is topologically the same as the standard hexagonal triangulation of C .Let l : E ( T st ) → R > be the edge length function of ( C , T st ( L ) , d st ) where d st is the standardflat metric on C . Let τ be a triangle in T st with vertices , u , u . Then L = u Z + u Z and { u , u } is called a geometric basis of L . Note that two vertices v, v (cid:48) ∈ L are joint by an edge e ∈ T st if and only if v − v (cid:48) ∈ {± u , ± u , ± ( u − u ) } . Proposition 3.4.
Suppose ( C , T st , l ) is a hexagonal Delaunay triangulation of the plane withvertex set a lattice V = u Z + u Z where { u , u } is a geometric basis. Let w : V → R be anon-constant linear function w ( nu + mu ) = n ln( λ ) + m ln( µ ) , m, n, ∈ Z , such that w ∗ l is a generalized Delaunay PL metric on T st . Then the following hold.(a) The generalized PL metric ( T st , w ∗ l ) is flat.Let φ be the developing map for the flat metric ( T st , w ∗ l ) .(b) If there exists a non-degenerate triangle in the generalized PL metric w ∗ l , then thereare two distinct non-degenerate triangles σ and σ in ( T st , w ∗ l ) such that φ ( int ( σ )) ∩ φ ( int ( σ )) (cid:54) = ∅ .(c) Suppose all triangles in w ∗ l are degenerate. Then there exists an automorphism ψ ofthe triangulation T st such that w ( ψ ( nu + mu )) = n ln( γ ( V )) + m ln( γ ( V )) where γ i ( V ) are two explicit numbers depending only on V . We remark that parts (a) and (b) for the lattice Z + e πi/ Z were proved in [43]. Proof.
Consider two automorphisms A and B of the topological triangulation T st defined by A ( v ) = v + u and B ( v ) = v + u for v ∈ V . By definition, we have AB = BA and A, B generate the group < A, B > ∼ = Z acting on T st . Any triangle in T st is equivalent, under theaction of < A, B > , to exactly one of the two triangles T or T where the vertices of T are , u , u and the vertices of T are , − u , − u . In the generalized PL metric w ∗ l , the maps A and B satisfy w ∗ l ( A ( e )) = λ w ∗ l ( e ) and w ∗ l ( B ( e )) = µ w ∗ l ( e ) for each edge e ∈ T . It follows that for any triangle τ ∈ T st , the generalized triangle ( A ( τ ) , w ∗ l ) (resp. ( B ( τ ) , w ∗ l ) ) is the scalar multiplication of ( τ, w ∗ l ) by λ (resp. by µ ). Hence there areonly two similarity types of triangles in ( C , T st , w ∗ l ) . For each v ∈ V , the six angles at v arecongruent to the six inner angles in T and T in w ∗ l metric. Therefore, ( T , w ∗ l ) is a flatmetric. See Figure 4(b).By the assumption that w is not a constant, we have ( λ, µ ) (cid:54) = (1 , . Say λ (cid:54) = 1 . Using thedeveloping map φ , there exist two complex affine maps α and β of the complex plane C suchthat φA = αφ and φB = βφ . Since A is a scaling by the factor λ (cid:54) = 1 and φ is a local isometry,the affine map α is of the form α ( z ) = λ ∗ z + c where | λ ∗ | = λ (cid:54) = 1 and α has a unique fixedpoint p ∈ C . By AB = BA , it follows αβ = βα . Therefore, from β ( p ) = βα ( p ) = αβ ( p ) ,we conclude β ( p ) = p . After replacing the developing map φ by ρ ◦ φ for an isometry ρ of C , we may assume that α and β both fix , i.e., α ( z ) = λ ∗ z and β ( z ) = µ ∗ z are both scalarmultiplications. Let G = < α, β > be the abelian group generated by α, β which acts on C byscalar multiplications. ONVERGENCE OF DISCRETE CONFORMAL MAPS 13 F IGURE
4. Flatness of spiral hexagonal triangulationsTo see part (b), let Ω be the image φ ( C ) of the developing map which is invariant underthe action of G . By the assumption that there are non-degenerate triangles in ( T st , w ∗ l ) ,the image Ω has non-empty interior. There are two cases we have to consider. In the firstcase, there exists a pair of integers ( n, m ) (cid:54) = (0 , so that α n β m is the identity element in thegroup G . In this case, we take σ to be any non-degenerate triangle and σ = A n B m ( σ ) . Bydefinition, we have φ ( σ ) = φ ( σ ) . Therefore, the result holds. In the second case that forall ( n, m ) (cid:54) = (0 , , α n β m (cid:54) = id , i.e., the group G is isomorphic to Z . Since both α ( z ) and β ( z ) are scalar multiplications, this implies that the action of the group G on int (Ω) is notdiscontinuous. In particular, for any non-empty open set U ⊂ Ω , there is α n β m ∈ G − { id } so that α n β m ( U ) ∩ U (cid:54) = ∅ . Take σ to be a non-degenerate triangle, U = φ ( int ( σ )) and σ = A n B m ( σ ) . Then we have φ ( int ( σ )) ∩ φ ( int ( σ )) (cid:54) = ∅ .To see part (c), since each triangle is degenerate, the inner angles a, b, c and x, y, z of twotriangles T and T are or π as shown in Figure 4(b). Composing with an automorphism of T st , we may assume that a = π , and then by the Delaunay condition, y (cid:54) = π .There are two cases depending on ( x, y, z ) = ( π, , or (0 , , π ) . The two cases differ bythe automorphism ρ of the lattice u Z + u Z and of T st such that ρ ( u ) = u , ρ ( u ) = u − u and ρ (0) = 0 . Thus it suffices to consider the case: z = π . Let the lengths of u , u and u − u in l -metric be b , b and b respectively. The lengths of the corresponding edges in w ∗ l metric are λb , µb and λµb . By the same computation, one works out the edge lengthsof the triangle with vertices , u and u − u in w ∗ l metric to be µ λ b , µb and µλ b . SeeFigure 4(c).We obtain two equations for edge lengths of degenerate triangles: λb + µb = λµb (dueto a = π ) and µ λ b = µb + µλ b (due to z = π ) . See Figure 4(c). These are same as λb + µb = λµb and µb = λb + b . Solving µ in terms of λ , we obtain a quadratic equationin λ :(17) b b λ + ( b − b − b ) λ − b b = 0 . Since b i > , this equation has a unique positive solution which we call γ ( V ) . The solutionin µ is called γ ( V ) . (cid:3)
4. R
IGIDITY OF HEXAGONAL TRIANGULATIONS OF THE PLANE
We begin with, F IGURE
5. Spiral hexagonal triangulations
Definition 4.1.
A flat generalized PL metric on a simply connected surface ( X, T , l ) withdeveloping map φ is said to be embeddable into C if for every simply connected finite sub-complex P of T , there exists a sequence of flat PL metrics on P whose developing maps φ n converge uniformly to φ | P and φ n : P → C is an embedding.For instance, all geometric triangulations of open sets in C are embeddable. However, thespiral flat triangulations produced in Proposition 3.4 are not embeddable. The main result inthis section works for embeddable flat PL metrics only.The following lemma is a consequence of definition. Lemma 4.2.
Suppose ( X, T , l ) is a flat generalized PL metric on a simply connected surfacewith a developing map φ .(a) Suppose φ is embeddable. If t , t are two distinct non-degenerate triangles or twodistinct edges in T , then φ ( int ( t )) ∩ φ ( int ( t )) = ∅ .(b) If φ is the pointwise convergent limit lim n →∞ ψ n of the developing maps ψ n of embed-dable flat generalized PL metrics ( X, T , l n ) , then ( X, T , l ) is embeddable.Proof. To see (a), if otherwise that φ ( int ( t )) ∩ φ ( int ( t )) (cid:54) = ∅ , then φ is not embeddable.Indeed, take P to be a finite simply connected subcomplex containing t and t , then thedeveloping maps φ n defined on P which converge uniformly to φ | P must satisfy φ n ( int ( t )) ∩ φ n ( int ( t )) (cid:54) = ∅ for n large. This contradicts that φ n are embedding.Part (b) follows from the fact that ψ n converges to φ uniformly on compact subsets andthe fact that if lim n →∞ a n = a and lim m →∞ b n,m = a n , then a = lim j →∞ b j,n j for somesubsequence. (cid:3) Let T st be a hexagonal Delaunay triangulation of the plane S = C with vertex set the lattice V = { u n + u m | n, m ∈ Z } and l : E ( T st ) → R > be the edge length function associatedto ( S, T st , d st ) . Given a flat generalized PL metric ( S, T st , l ) , its normalized developing map φ = φ l : S → C is a developing map such that φ (0) = 0 and φ ( u ) is in the positive x-axis.Suppose { u , u } is a geometric basis of the lattice u Z + u Z . Two vertices v, v (cid:48) are adjacentin T st , i.e., v ∼ v (cid:48) , if and only if v = v (cid:48) + δ for some δ ∈ {± u , ± u , ± ( u − u ) } . Giventwo vertices v, v (cid:48) ∈ V , the combinatorial distance d c ( v, v (cid:48) ) between v, v (cid:48) is the length of theshortest edge path joining them.The goal of this section is to prove the following stronger version of theorem 1.4. ONVERGENCE OF DISCRETE CONFORMAL MAPS 15
Theorem 4.3.
Suppose ( S, T st , l ) is a hexagonal Delaunay triangulation whose vertex set is alattice in C and ( S, T st , w ∗ l ) is a flat generalized Delaunay PL metric on T st . If ( S, T st , w ∗ l ) is embeddable into C , then w is a constant function. We will deduce Theorem 1.4 from Theorem 4.3 in §
7. Theorem 4.3 will be proved usingseveral lemmas.4.1.
Limits of discrete conformal factors.
The following lemma is a corollary of Theorem3.1.
Lemma 4.4.
Suppose ( S, T st , w ∗ l ) is a flat generalized Delaunay PL metric surface. Thenfor any δ ∈ V and u : V → R defined by u ( v ) = w ( v + δ ) − w ( v ) , u ∗ ( w ∗ l ) = ( u + w ) ∗ l is a flat generalized Delaunay PL metric on T st . In particular, if u ( v ) = max { u ( v ) | v ∈ V } ,then u is constant. The next lemma shows how to produce discrete conformal factors w such that w ( v + δ ) − w ( v ) are constants. Lemma 4.5.
Suppose w ∗ l is a flat generalized Delaunay PL metric on T st . Then for any δ ∈ {± u , ± u , ± ( u − u ) } , there exist v n ∈ V such that w n ∈ R V defined by w n ( v ) = w ( v + v n ) − w ( v n ) satisfies(a) for all v ∈ V , the following limit exists w ∞ ( v ) = lim n →∞ w n ( v ) ∈ R , (b) w n ∗ l and w ∞ ∗ l are flat generalized Delaunay PL metric on T st ,(c) w ∞ ( v + δ ) − w ∞ ( v ) = a for all v ∈ V where a = sup { w ( v + δ ) − w ( v ) | v ∈ V } ,(d) the normalized developing maps φ w n ∗ l of w n ∗ l converges uniformly on compact sets in S to the normalized developing map φ ∞ of w ∞ ∗ l . In particular, if ( S, T st , w ∗ l ) is embeddable,then ( S, T st , w ∞ ∗ l ) is embeddable.Proof. By Lemma 3.3, there is a constant M = M ( V ) depending only on the lattice V = u Z + u Z such that a = sup { w ( v + δ ) − w ( v ) | v ∈ V } ≤ M ( V ) . Take v n ∈ V so that w ( v n + δ ) − w ( v n ) ≥ a − n . By definition,(18) w n (0) = 0 , w n ( δ ) ≥ a − n , w n ( v + δ ) − w n ( v ) ≤ a, and(19) sup {| w n ( v ) − w n ( v (cid:48) ) || v ∼ v (cid:48) } < ∞ . By Lemma 3.3, if v ∈ V is of combinatorial distance m to , then, using w n (0) = 0 , we have(20) | w n ( v ) | ≤ mM ( V ) . By (20) and the diagonal argument, we see that there exists a subsequence of { v n } , still denotedby { v n } for simplicity, so that w n converges to w ∞ ∈ R V in the pointwise convergent topology.By lemma 4.4, each w n ∗ l is a flat generalized Delaunay PL metric. By lim n →∞ w n = w ∞ and continuity, we conclude that w ∞ ∗ l is again a flat generalized Delaunay PL metric on T st .By (18), w ∞ ( δ ) − w ∞ (0) = max { w ∞ ( v + δ ) − w ∞ ( v ) | v ∈ V } . By Lemma 4.4, we see that conclusion (c) holds. Since the developing map φ w ∗ l dependscontinuously on w ∈ R V , lim n →∞ φ w n ∗ l ( v ) = φ ∞ ( v ) for each vertex v ∈ V . On the otherhand, a developing map φ is determined by its restriction to V . We see that φ w n ∗ l converges to φ ∞ uniformly on compact subsets of the plane. The last statement follows from Lemma 4.2(b)since each φ w n ∗ l is embeddable by definition. (cid:3) Proof of Theorem 4.3.
Suppose w ∗ l is a flat generalized Delaunay PL metric on T st with an embeddable developing map φ . Our goal is to show that w : V → R is a constant.Suppose otherwise, we will derive a contradiction by showing that the developing map φ is notembeddable.Since w is not a constant, we can choose δ ∈ {± u , ± u , ± ( u − u ) } such that a =sup { w ( v + δ ) − w ( v ) | v ∈ V } > . By Lemma 4.5 applied to w ∗ l and δ = δ , we producea function w ∞ : V → R so that w ∞ ∗ l is a flat generalized Delaunay PL metric on T st and w ∞ ( v + δ ) = w ∞ ( v ) + a for all v ∈ V . Now applying Lemma 4.5 to w ∞ ∗ l with δ ∈ {± u , ± u , ± ( u − u ) } − {± δ } , we obtain a second function ˆ w = ( w ∞ ) ∞ : V → R and b ∈ R such that ˆ w ∗ l is a flat generalized Delaunay PL metric on T st and ˆ w ( v + δ ) = ˆ w ( v ) + a , ˆ w ( v + δ ) = ˆ w ( v ) + b , for all v ∈ V . This shows that ˆ w : V → R is a non-constant affine function, i.e., ˆ w ( n + me πi/ ) = a n + b m + c for some a , b , c ∈ R .Let ˆ φ , φ ∞ and φ be the normalized developing maps for ˆ w ∗ l , w ∞ ∗ l and w ∗ l respectively.Since φ is embeddable, by Lemmas 4.5, ˆ φ and φ ∞ are embeddable.If ˆ w ∗ l contains a non-degenerate triangle, then by Proposition 3.4, there exist two non-degenerate triangles t and t in ( T st , ˆ w ∗ l ) so that ˆ φ ( int ( t )) ∩ ˆ φ ( int ( t )) (cid:54) = ∅ . By Lemma4.2(a), this contradicts that ˆ w ∗ l is embeddable.F IGURE
6. Angles a and z are zero in ˆ w ∗ l . Part (b) is the developing imageof corresponding set in w ∗ l ONVERGENCE OF DISCRETE CONFORMAL MAPS 17
Therefore all triangles in the generalized PL metric ˆ w ∗ l are degenerate, i.e., all angles intriangles are either or π . We will use the same notations used in the proof of Proposition 3.4.By Proposition 3.4(c) and Figure 6, we may assume, after composing with an automorphismof T st and subtracting by a constant, that ˆ w ( nu + mu ) = n ln( γ ( V )) + m ln( γ ( V )) where ( γ ( V ) , γ ( V )) are given by the solutions of (17) and the angles a, b, c, x, y, z of T and T are ( a, b, c, x, y, z ) = ( π, , , , , π ) .Let P = u − u , P = u − u , P = 0 and P = u in V . See Figure 6(c). In the caseof a = z = π , we claim that the length µλ b of the edge P P is strictly less than the sum of thelengths λb of the edge P P and µ λ b of the edge P P , i.e.,(21) µλ b < λb + µ λ b . Indeed, by the equations λb + µb = λµb and µb = λb + b derived in the proof ofProposition 3.4, we obtain b b = λ + µ (1 + λ ) µ . Equation (21) says b b < λ + µ λ µ . Thus it suffices to show that λ + µ (1+ λ ) µ < λ + µ λ µ . This is the same as, λ ( λ + µ ) < (1 + λ )( λ + µ ) , i.e., λ + λ µ < λ + λ µ + λ + µ . The last inequality clearly holds since both λ and µ are positive.Now consider the oriented edge path P P P P (oriented from P to P ) in T st and its imageunder the developing map ˆ φ of ˆ w ∗ l in C . By the assumption that a = z = π , the angles ofthe polygonal path ˆ φ ( P P P P ) at ˆ φ ( P ) and ˆ φ ( P ) are π . See Figure 6(c). Also the sum ofthe lengths of ˆ φ ( P P ) and ˆ φ ( P P ) is larger than the length of ˆ φ ( P P ) by the claim above.On the other hand, since ˆ φ is embeddable, there exists a sequence of flat PL metrics on T st whose developing maps φ n are embedding and φ n converges uniformly on compact sets to ˆ φ .This implies, that for n large the two line segments φ n ( P P ) and φ n ( P P ) intersect in theirinteriors. This contradicts the assumption that φ n is an embedding.This ends the proof of Theorem 4.3 Remark 4.6.
The above argument also gives a new proof of Rodin-Sullivan’s hexagonal circlepacking theorem.
The following will be used to show that the limit of discrete uniformization maps is con-formal. Let B n ( v ) = { i ∈ V ( T st ) | d c ( i, v ) ≤ n } and B n ( v ) be the subcomplex of T st whosesimplices have vertices in B n ( v ) . Lemma 4.7.
Take the standard hexagonal lattice V = Z + e π/ Z and its associated standardhexagonal triangulation whose edge length function is l : V → { } . There is a sequence s n of positive numbers decreasing to zero with the following property. For any integer n and avertex v , there exists N = N ( n, v ) such that if m ≥ N and ( B m ( v ) , w ∗ l ) is a flat Delaunaytriangulated PL surface with embeddable developing map, then the ratio of the lengths of anytwo edges sharing a vertex in B m ( v ) is at most s n . The proof of the lemma is exactly the same as that of Rodin-Sullivan [34, pages 353-354]since we have Lemma 3.3 and Theorem 4.3 which play the roles of Rodin-Sullivan’s RingLemma and rigidity of hexagonal circle packing in [34, pages 352-353]).5. E
XISTENCE OF DISCRETE UNIFORMIZATION METRICS ON POLYHEDRAL DISKS WITHSPECIAL EQUILATERAL TRIANGULATIONS
By a polygonal disk we mean a flat PL surface ( P , V, d ) which is isometrically embeddedin the complex plane C and P is homeomorphic to the closed disk. The goal of this sec-tion is to prove the existence of a discrete conformal metric by regular subdividing the giventriangulations.An equilateral triangulation T of a polyhedral surface is a geometric triangulation whosetriangles are equilateral. The edge length function of an equilaterally triangulated connectedpolyhedral surface will be denoted by the constant function l st : E ( T ) → R . Given an equi-lateral Euclidean triangle ∆ ⊂ C and n ∈ Z ≥ , the n -th standard subdivision of ∆ is theequilateral triangulation of ∆ by n equilateral triangles. See Figure 7. If T is an equilateraltriangulation of a polyhedral surface, its n -th standard subdivision , denoted by T ( n ) , is the equi-lateral triangulation obtained by replacing each triangle in T by its n -th standard subdivision.We use V ( n ) to denote V ( T ( n ) ) .1st 2nd 3rd 6thF IGURE
7. The standard subdivisionsThe main result of this section is the following theorem.
Theorem 5.1.
Suppose ( P , T , l st ) is a flat polygonal disk with an equilateral triangulation T such that exactly three boundary vertices p, q, r have curvature π . Then for sufficientlylarge n , there is discrete conformal factor w n : V ( n ) → R for the n -th standard subdivision ( P , T ( n ) , l st ) such that the discrete curvature K of w n ∗ l st satisfies(a) K i = 0 for all i ∈ V ( n ) − { p, q, r } ,(b) K i = π for all i ∈ { p, q, r } , and(c) there is a constant (cid:15) > independent of n such that all inner angles of triangles in ( T ( n ) , w n ∗ l st ) are in the interval [ (cid:15) , π/ (cid:15) ] , the sum of two angles facing each interior edgeis at most π − (cid:15) and each angle facing a boundary edge is at most π/ − (cid:15) . Conditions (a) and (b) imply that the underlying metric space of ( P , T ( n ) , w n ∗ l st ) is an equi-lateral triangle. Condition (c) says that the metric doubles of ( P , T ( n ) , l st ) and ( P , T ( n ) , w n ∗ l st ) are two Delaunay triangulated polyhedral 2-spheres differing by a vertex scaling.There are two steps involved in the construction of the discrete conformal factor w n in The-orem 5.1. In the first step, we produce a discrete conformal factor w (1) : V ( n ) → R suchthat w (1) vanishes outside the union of combinatorial balls of radius [ n/ (the integral part of n/ ) centered at non-flat vertices v (cid:54) = p, q, r and the discrete curvature K i ( w (1) ∗ l st ) = 0 if ONVERGENCE OF DISCRETE CONFORMAL MAPS 19 d c ( i, v ) < [ n/ and K i ( w (1) ∗ l st ) = O (1 / (cid:112) ln( n )) if d c ( i, v ) = [ n/ . This step diffuses thenon-zero discrete curvatures π/ , − π/ and − π/ (at non-flat vertices v ) to small curvaturesat vertices defined by d c ( i, v ) = [ n/ . In the second step, by choosing n large such that allcurvatures are very small, we use a perturbation argument to show that there is w (2) : V ( n ) → R such that w (2) ∗ ( w (1) ∗ l st ) satisfies the conditions in Theorem 5.1. The required discrete con-formal factor w n is w (1) + w (2) since ( w (2) + w (1) ) ∗ l st = w (2) ∗ ( w (1) ∗ l st ) . The basic tools to be used for proving Theorem 5.1 are discrete harmonic functions, theirgradient estimates and ordinary differential equations (ODE). We begin by recalling the relatedmaterial.5.1.
Laplace operator on a finite graph.
Given a graph ( V, E ) , the set of all oriented edgesin ( V, E ) is denoted by ¯ E . If i ∼ j in V , we use [ ij ] ∈ ¯ E to denote the oriented edge from i to j . If x ∈ R V and y ∈ R ¯ E , we use x i and y ij to denote x ( i ) and y ([ ij ]) respectively. A conductance on G is a function η : ¯ E → R ≥ so that η ij = η ji . Definition 5.2.
Given a finite graph ( V, E ) with a conductance η , the gradient (cid:79) : R V → R ¯ E is the linear map ( (cid:79) f ) ij = η ij ( f i − f j ) , the Laplace operator associated to η is the linear map (cid:52) : R V → R V defined by ( (cid:52) f ) i = (cid:88) j ∼ i η ij ( f i − f j ) , and the Dirichlet energy of f ∈ R V on ( V, E, η ) is E ( f ) = 12 (cid:88) i ∼ j η ij ( f i − f j ) . The following is well known (see [7]).
Proposition 5.3 (Green’s identity) . Given a finite graph ( V, E ) with a conductance η ,(a) for any subset V ⊂ V , (cid:88) i ∈ V f i ( (cid:52) g ) i − g i ( (cid:52) f ) i = (cid:88) i ∈ V ,j ∼ i,j / ∈ V η ij ( g i f j − f i g j ) . (b) (cid:80) i ∈ V ( (cid:52) f ) i = 0 . Given a set V ⊂ V and g : V → R , the Dirichlet problem asks for a function f : V → R so that(22) ( (cid:52) f ) i = 0 , ∀ i ∈ V − V , and f | V = g. The
Dirichlet principle states that solutions f to the Dirichlet problem (22) are the same as min-imum points of the Dirichlet energy function restricted to the affine subspace { h ∈ R V | h | V = g } , i.e.,(23) E ( f ) = min {E ( h ) | h ∈ R V and h | V = g } . In particular, the Dirichlet problem (22) is always solvable.A subset U ⊂ V in a graph ( V, E ) is called connected if any two vertices i, j ∈ U can bejoint by an edge path whose vertices are in U . For instance, a connected graph ( V, E ) means V is a connected. The following is well known (see [7]). Proposition 5.4.
Suppose ( V, E ) is a finite connected graph with a conductance η ij > forall edges [ ij ] and V ⊂ V . Let f be a solution to the Dirichlet problem (22). Then,(a) (Maximum principle) for V (cid:54) = ∅ , max i ∈ V f i = max i ∈ V f i . (b) (Strong maximum principle) If V − V is connected and max i ∈ V − V f i = max i ∈ V f i , then f | V − V is a constant function. A system of ODE associated to discrete conformal change.
Let ( S, T , l ) be a com-pact connected polyhedral surface with discrete curvature K . Given a subset V ⊂ V and afunction K ∗ : V − V → ( −∞ , π ) , we try to find a function w : V → R such that w ∗ l is a PL metric whose curvature K ( w ) is equal to K ∗ on V − V and w | V = 0 . In the PLmetric w ∗ l , let θ ijk = θ ijk ( w ) be the angle at vertex i in the triangle ∆ ijk and η ij = η ij ( w ) be cot( θ kij ) + cot( θ lij ) if [ ij ] is an interior edge and η ij = cot( θ kij ) if [ ij ] is a boundary edge.The associated Laplacian ∆ : R V → R V is (∆ f ) i = (cid:80) j ∼ i η ij ( f i − f j ) . We will construct w by choosing a smooth 1-parameter family w ( t ) ∈ R V such that w (0) = 0 and w ( t ) ∗ l is a PLmetric whose curvature K i ( t ) = K i ( w ( t ) ∗ l st ) satisfies(24) ∀ i ∈ V − V , K i ( t ) = (1 − t ) K i + tK ∗ i ; and ∀ i ∈ V , w i ( t ) = 0 . The required vector w is defined to be w (1) . Note that by definition K (0) = K . Due to thecurvature evolution equation (7) that dK i ( t ) dt = (cid:80) j ∼ i η ij ( w ( t ))( w (cid:48) i − w (cid:48) j ) where w (cid:48) i ( t ) = dw i ( t ) dt ,we obtain the following system of ODE in w ( t ) equivalent to (24):(25) ∀ i ∈ V − V , (cid:88) j ∼ i η ij ( w (cid:48) i − w (cid:48) j ) = K ∗ i − K i ; ∀ i ∈ V , w (cid:48) i ( t ) = 0; and w (0) = 0 . Using (cid:52) f , we can write Equation (25) as(26) ∀ i ∈ V − V , (∆ w (cid:48) ) i = K ∗ i − K i ; ∀ i ∈ V , w (cid:48) i ( t ) = 0; and w (0) = 0 .We will show, under some assumptions on ( T , l ) , that the solution to (25) exists for all t ∈ [0 , .Let W ⊂ R V be the open set(27) W = { w ∈ R V | w ∗ l is a PL metric on T and η ij ( w ) > for all edges [ ij ] } . Lemma 5.5.
Suppose V (cid:54) = ∅ and ∈ W . The initial valued problem (25) defined on W hasa unique solution in a maximum interval [0 , t ) with t > such that if t < ∞ , then either lim inf t → t − θ ijk ( w ( t )) = 0 for some angle θ ijk or lim inf t → t − η ij ( w ( t )) = 0 for some edge [ ij ] .Proof. Indeed Equation (25) can be written as Y ( w ) · w (cid:48) ( t ) = β and w (0) = 0 where Y ( w ) isa square matrix valued smooth function of w ∈ W and w (cid:48) ( t ) is considered as a column vector.We claim that Y ( w ) is an invertible matrix for w ∈ W . If Y ( w ) is invertible, then (25) can bewritten as w (cid:48) ( t ) = Y ( w ) − β and by the Picard’s existence theorem, there exists an interval onwhich the ODE (25) has a solution. Now Y ( w ) is invertible if and only if the following systemof linear equations has only trivial solution x = 0 ,(28) Y ( w ) · x = 0 . By (25), Equation (28) is the same as (∆ x ) i = 0 for i ∈ V − V and x i = 0 for i ∈ V . Further-more w ∈ W implies η ij ( w ) > for all edges [ ij ] . By the maximum principle (Proposition5.4), we see that x = 0 . ONVERGENCE OF DISCRETE CONFORMAL MAPS 21 If t < ∞ and t ↑ t , then w ( t ) leaves every compact set in W . For each δ > , we claimthat W δ = { w ∈ W | θ ijk ≥ δ, | w i | ≤ δ , η ij ≥ δ } is compact. Clearly W δ is bounded bydefinition. To see that W δ is closed in R V , take a sequence x n ∈ W δ such that lim n →∞ x n = y ∈ R V . Then y ∗ l is a generalized PL metric with all angles θ ijk ≥ δ . Since each degeneratetriangle has an angle which is zero, therefore y ∗ l is a PL metric. Also by continuity, wehave θ ijk ( y ) ≥ δ , η ij ( y ) ≥ δ and | y i | ≤ δ , i.e., y ∈ W δ . Since w ( t ) leaves every W δ foreach δ > , one of the following three occurs: lim inf t → t − θ ijk ( w ( t )) = 0 for some θ ijk , or lim inf t → t − η ij ( w ( t )) = 0 for some edge [ ij ] , or lim sup t → t − | w i ( t ) | = ∞ for some i ∈ V .However lim sup t → t − | w i ( t ) | = ∞ for one vertex i implies that lim inf t → t − θ ljk ( w ( t )) = 0 forsome θ ljk . Indeed, if otherwise, lim inf t → t − θ ljk ( w ( t )) ≥ δ > for all θ ljk for some δ . It is wellknown that in a Euclidean triangle whose angles are at least δ , the ratio of two edge lengthsis at most δ ) . Therefore, in each triangle ∆ v i v j v k in T , we have e w i ( t ) ≤ e w j ( t ) l ( v j v k ) l ( v i v k ) sin( δ ) .Since w j ( t ) = 0 for j ∈ V and the surface S is connected, we conclude that all w k ( t ) , k ∈ V ,are bounded for all t . This contradicts lim sup t → t − | w i ( t ) | = ∞ . (cid:3) Standard subdivision of an equilateral triangle.Theorem 5.6.
Let S = ∆ ABC be an equilateral triangle, T be the n -th standard subdivisionof S with the associated PL metric l st : V = V ( T ) → { n } and V = { v ∈ V | v is in the edge BC of the triangle ∆ ABC } . Given any α ∈ [ π , π ] , there exists a smooth family of vectors w ( t ) ∈ R V for t ∈ [0 , such that w (0) = 0 and w ( t ) ∗ l st is a PL metric on T with curvature K ( t ) = K ( w ( t ) ∗ l st ) satisfying,(a) K A ( t ) = − tα + (2 + t ) π (angle at A is tα + (1 − t ) π ),(b) K i ( t ) =0 for all i ∈ V − { A } ∪ V ,(c) w i ( t ) = 0 for all i ∈ V ,(d) all inner angles θ ijk ( t ) in metric w ( t ) ∗ l st are in the interval [ π − | α − π | , π + | α − π | ] ⊂ [ π , π ] ,(e) θ ijk ( t ) ≤ π for i (cid:54) = A ,(f) | K i ( t ) − K i (0) | ≤ √ ln( n ) for i (cid:54) = A and (29) (cid:88) i ∈ V | K i ( t ) − K i (0) | ≤ π . Remark 5.7.
The discrete conformal map from (∆ ABC, T , l st ) to (∆ ABC, T , w (1) ∗ l st ) isa discrete counterpart of the analytic function f ( z ) = z α/π . F IGURE
8. Discrete conformal maps of equilateral triangles and their unions
Our proof of Theorem 5.6 relies on the following two lemmas about estimates on discreteharmonic functions on T . Lemma 5.8.
Assume ∆ ABC, n, T , V are as given in Theorem 5.6. Let τ : T → T bethe involution induced by the reflection of ∆ ABC about the angle bisector of ∠ BAC and η : E → R ≥ be a conductance so that ητ = η and η ij = η ji . Let ∆ : R V → R V be theLaplace operator defined by (∆ f ) i = (cid:80) j ∼ i η ij ( f i − f j ) . If f ∈ R V satisfies (∆ f ) i = 0 for i ∈ V − { A } ∪ V and f | V = 0 , then for all edges [ ij ] , the gradient ( (cid:79) f ) ij = η ij ( f i − f j ) satisfies (30) | η ij ( f i − f j ) | ≤ | ∆( f ) A | . Lemma 5.9.
Assume ∆ ABC, n, T , V are as given in Theorem 5.6. Let η : E ( T ) → [ M , M ] be a conductance function for some M > and ∆ be the Laplace operator on R V associatedto η . If f : V → R solves the Dirichlet problem (∆ f ) i = 0 , ∀ i ∈ V − { A } ∪ V , f | V = 0 and (∆ f ) A = 1 , then for all u ∈ V , | (∆ f ) u | ≤ M √ ln n . We will prove Lemma 5.8, Lemma 5.9, and Theorem 5.6 in order.The simplest way to see Lemma 5.8 is to use the theory of electric network. We put aresistance of η ij Ohms at the edge [ ij ] (if η ij = 0 , the resistance is ∞ , or remove edge [ ij ] from the network). Now place a one-volt battery at vertex A and ground every vertex in V .Then Kirchhoff’s laws show that the voltage f i at the vertex i solves the Dirichlet problem (∆ f ) i = 0 for all i ∈ V − { A } ∪ V , f A = 1 and f | V = 0 . Ohm’s law says η ij ( f i − f j ) isthe electric current through the edge [ ij ] . Since the resistance is symmetric with respect to thesymmetry τ , the currents in the network are the same as the currents in the quotient network T /τ . In the quotient network T /τ , there is only one edge e A from the vertex A. Therefore,the current through any edge is at most the current | (∆ f ) A | through e A (in the network T /τ ).This shows | η ij ( f i − f j ) | ≤ | (∆ f ) A | . Proof of lemma 5.8.
Removing all edges [ ij ] for which η ij = 0 from the graph ( V, E ) , we ob-tain a finite collection of disjoint connected subgraphs Γ , ..., Γ N from ( V, E ) . By construction,the associated Laplace operators on Γ i with conductance η | E (Γ i ) is the restriction of the Laplaceoperator ∆ to V (Γ i ) . By the maximum principle (Proposition 5.4), the function f | V (Γ m ) is aconstant and (30) holds unless Γ m contains the vertex A and some vertex in V . Therefore, itsuffices to prove the lemma for those edges [ ij ] in the connected graph Γ m = ( V (cid:48) , E (cid:48) ) such that A ∈ V (cid:48) and V (cid:48) ∩ V (cid:54) = ∅ . Let A , A = τ ( A ) be the vertices adjacent to A . Since τ ( A ) = A , ητ = η and V (cid:48) ∩ V (cid:54) = ∅ , we have τ (Γ m ) = Γ m and A , A ∈ V (cid:48) .We will work on the graph Γ m = ( V (cid:48) , E (cid:48) ) from now on. Using the maximum principle for f − f τ , we see that f = f τ . By replacing f by − f if necessary, we may assume that f A > .By the maximum principle, we have that ≤ f i < f A for all i ∈ V (cid:48) − { A } .Take an edge [ ij ] in the graph Γ m . If τ { i, j } = { i, j } , then τ i = j and τ j = i . This implies f i = f τ i = f j and (30) holds. If τ { i, j } = { i (cid:48) , j (cid:48) } (cid:54) = { i, j } , say τ i = i (cid:48) , τ j = j (cid:48) , then f i = f i (cid:48) , f j = f j (cid:48) . We may assume that f i ≤ f j . If f i = f j , then (30) holds. Hence we mayassume f i < f j . If j = A , then i = A or A . Due to f A = f A , then (30) holds. If j (cid:54) = A ,then by the maximum principle applied to f on the subgraph ( V (cid:48) − { A } , E (cid:48) − { AA , AA } ) ,we conclude that f A ≥ f j > f i . Let U = { k ∈ V (cid:48) − { A }| f k > f i } . By definition, j, j (cid:48) , A , A ∈ U , i, i (cid:48) , A / ∈ U , and V ∩ U = ∅ . This shows (∆ f ) k = 0 for all k ∈ U and hence ONVERGENCE OF DISCRETE CONFORMAL MAPS 23 (cid:80) k ∈ U (∆ f ) k = 0 . By Green’s formula (5.3), (cid:88) k ∈ U (∆ f ) k = (cid:88) k ∈ U,l / ∈ U,k ∼ l η kl ( f k − f l ) = 0 . If l / ∈ U ∪ { A } , then by definition f i ≥ f l . Therefore, if k ∈ U , k ∼ l , and l / ∈ U ∪ { A } , then f k > f i ≥ f l . This shows, (cid:88) k ∈ U,l / ∈ U,l ∼ k η kl ( f k − f l )= (cid:88) k ∈ U,l / ∈ U ∪{ A } ,l ∼ k η kl ( f k − f l ) + (cid:88) k ∼ A η kA ( f k − f A ) ≥ ( (cid:79) f ) ji + ( (cid:79) f ) j (cid:48) i (cid:48) − (∆ f ) A . Therefore, | (∆ f ) A | ≥ | ( (cid:79) f ) ij | since ( (cid:79) f ) ij = ( (cid:79) f ) i (cid:48) j (cid:48) . (cid:3) Proof of Lemma 5.9.
For the given u ∈ V , construct a function g : V → R by solving theDirichlet problem: (∆ g ) i = 0 , ∀ i ∈ V − V , g u = 1 and g | V −{ u } = 0 . By the maximumprinciple (Proposition 5.4), ≤ g i ≤ for all i . Using Green’s identity that (cid:80) i ∈ V [ f i (∆ g ) i − g i (∆ f ) i ] = 0 , we obtain g A (∆ f ) A + g u (∆ f ) u = 0 . Since (∆ f ) A = 1 and g u = 1 , we see (∆ f ) u = − g A . F IGURE
9. Layers in triangle ABCTherefore, it suffices to show that | g A | ≤ M √ ln n . For this purpose, take k ≤ [ n ] and define U k = { i ∈ V | d c ( i, u ) = k } where d c ( i, j ) is the combinatorial distance in the graph T (1) . Let G k be the subgraph of T (1) whose edges are [ ij ] where i, j ∈ U k . Due to k ≤ [ n ] , U k ∩ V (cid:54) = ∅ ,and G k is topologically an arc. By the maximum principle applied to g on the subgraph whoseedges consist of [ ij ] with i, j ∈ { v ∈ V | d c ( v, u ) ≥ k } , we obtain g A ≤ max i ∈ U k g i . Let v k ∈ U k such that g v k = max i ∈ U k g i and edge path E k be the shortest edge path in G k joining v k to a point u k in V − { u } . By construction g u k = 0 . Since U k contains at most k + 1 vertices, the length of E k is at most k . The Dirichlet energy E ( g ) of g on T (1) is given by E ( g ) = 12 (cid:88) i ∼ j η ij ( g i − g j ) ≥ [ n ] (cid:88) k =1 E k , where(31) E k = 12 (cid:88) [ ij ] ∈ ¯ E k η ij ( g i − g j ) , and ¯ E k be the set of oriented edges in E k . Suppose w = v k ∼ w ∼ w ∼ ... ∼ w l k = u k are the vertices in the edge path E k where l k ≤ k . Using the Cauchy-Schwartz inequality, weobtain(32) E k = l k (cid:88) i =1 η w i w i − ( g w i − g w i − ) ≥ M l k (cid:88) i =1 ( g w i − g w i − ) ≥ M l k [ l k (cid:88) i =1 ( g w i − g w i − )] ≥ kM ( g v k − g u k ) = g v k kM ≥ g A kM . By (5.3) and (32), we obtain(33) E ( g ) ≥ g A M [ n ] (cid:88) k =1 k ≥ g A ln( n )100 M .
On the other hand, the Dirichlet principle says E ( g ) = min h ∈ R V { (cid:80) i ∼ j η ij ( h i − h j ) | h u =1 , h | V −{ u } = 0 } . Take h ∈ R V to be h u = 1 and h i = 0 for all i ∈ V − { u } . We obtain E ( g ) ≤ (cid:88) i ∼ j η ij ( h i − h j ) ≤ M. Combining this with (33), we obtain g A ln( n )100 M ≤ M, i.e., g A ≤ M (cid:112) ln( n ) . (cid:3) Proof of Theorem 5.6.
We construct the smooth family w ( t ) ∈ R V by solving the system ofordinary differential equations (25) where ( S, T , l ) = (∆ ABC, T , l st ) , K ∗ | V − V ∪{ A } = 0 , K ∗ A = π − α and w i ( t ) = 0 for i ∈ V . By the assumption that θ ijk (0) = π (i.e., T is anequilateral triangulation), ∈ W where the space W is defined by (27). By Lemma 5.5, thereexists a maximum s > such that a solution w ( t ) to (25) exists and condition (d) holds forall t ∈ [0 , s ) . We claim that s ≥ , w (1) exists and w (1) ∗ l st is a PL metric. In particular, w (1) ∗ l st satisfies condition (d) and w (1) ∈ W . Without loss of generality, let us assume that s < ∞ . By lemma 5.5 and condition (d), we obtain the following two conclusions:(34) lim inf t → s − η ij ( w ( t )) = 0 for some [ ij ] , or lim sup t → s − | θ ijk ( w ( t )) − π | = | α − π | for some θ ijk . ONVERGENCE OF DISCRETE CONFORMAL MAPS 25
The conclusion lim inf t → s − θ ijk ( w ( t )) = 0 is ruled out by condition (d) which implies θ ijk ( w ( t )) ≥ π . We prove the claim that s ≥ as follows. Since α ∈ [ π , π ] , we have π + | α − π | ≤ π and π − | α − π | ≥ π . This shows, by ( d ) ,(35) θ ijk ( t ) ∈ [ π , π for all t ∈ [0 , s ) . In particular, cot( θ kij ) ≥ and η ij ≥ cot( θ kij ) ≥ . Hence by definition we have | ( (cid:79) w (cid:48) ) ij | = η ij | w (cid:48) i − w (cid:48) j | ≥ cot( θ kij ) | w (cid:48) i − w (cid:48) j | . By Lemma 5.8 and the variation formula (7) that dK i dt = (∆ w (cid:48) ) i , we obtain | ( (cid:79) w (cid:48) ) ij | ≤ | (∆ w (cid:48) ) A | = | dK A dt | = | α − π | . This implies, by (6), the following,(36) | dθ kij dt | ≤ cot( θ ijk ) | w (cid:48) j − w (cid:48) k | + cot( θ jik ) | w (cid:48) i − w (cid:48) k | ≤ | ( (cid:79) w (cid:48) ) jk | + | ( (cid:79) w (cid:48) ) ik | ≤ | α − π | . Therefore, for all t ∈ [0 , s ) ,(37) | θ kij ( t ) − π | = | θ kij ( t ) − θ kij (0) | ≤ (cid:90) t | dθ kij ( t ) dt | dt ≤ t | α − π | ≤ s | α − π | . The above inequality shows that s ≥ . Indeed, if otherwise that s < , using (37), we concludethat θ ijk ( t ) ∈ [ π − s | α − π | , π + s | α − π | ] . In particular, lim inf t → s − η ij ( t ) ≥ cot( π + s | α − π | ) > and lim sup t → s − | θ kij ( t ) − π/ | < | α − π/ | . This contradicts (34).To see part (e), by (37), if t ∈ [0 , ] , we have | θ ijk ( t ) − π | ≤ | α − π | ≤ π , i.e., θ ijk ( t ) ∈ [ π , π
12 ] . Now if [ ij ] is an interior edge, then for t ∈ [0 , ] (38) | ( (cid:79) w (cid:48) ) ij | = (cot( θ kij ) + cot( θ lij )) | w (cid:48) i − w (cid:48) j |≥ (1 + cot( θ lij )cot( θ kij ) ) cot( θ kij ) | w (cid:48) i − w (cid:48) j |≥ (1 + cot( 5 π
12 )) cot( θ kij ) | w (cid:48) i − w (cid:48) j |≥
54 cot( θ kij ) | w (cid:48) i − w (cid:48) j | . If θ ijk is an angle with i (cid:54) = A , then either one of the two edges [ ij ] , [ ik ] is an interior edge,or i ∈ { B, C } . In the first case, say [ ij ] is an interior edge, using (38) and Lemma 5.8, for t ∈ [0 , / , we have(39) | dθ ijk dt | ≤ cot( θ kij ) | w (cid:48) i − w (cid:48) j | + cot( θ jik ) | w (cid:48) i − w (cid:48) k |≤ | ( (cid:79) w (cid:48) ) ij | + | ( (cid:79) w (cid:48) ) ik |≤ ( 45 + 1) | (∆ w (cid:48) ) A | | α − π | ≤ · π π . In the second case that i ∈ { B, C } , one of the edges [ ij ] or [ ik ] , say [ ij ] is in the edge BC of ∆ ABC , i.e., w (cid:48) i = w (cid:48) j = 0 . Therefore by Lemma 5.8, for t ∈ [0 , / , we have | dθ ijk dt | ≤ cot( θ kij ) | w (cid:48) i − w (cid:48) j | + cot( θ jik ) | w (cid:48) i − w (cid:48) k | ≤ | ( (cid:79) w (cid:48) ) ik | (40) ≤ | (∆ w (cid:48) ) A | | α − π | ≤ π . Therefore if θ ijk is not the angle at A and t ∈ [0 , , by (39) and (5.3), we have | θ ijk ( t ) − π | = | θ ijk ( t ) − θ ijk (0) | ≤ (cid:90) t | dθ ijk dt | dt ≤ (cid:90) | dθ ijk dt | dt = (cid:90) / | dθ ijk dt | dt + (cid:90) / | dθ ijk dt | dt ≤ · π
20 + 12 | α − π | ≤ π
40 + 12 · π π . Therefore, θ ijk ( t ) ∈ [ π , π ] ⊂ ( π , π ) for all t ∈ [0 , . Since conditions (d) and (e) hold forall t ∈ [0 , , by definition of η ij , we see lim inf t → η ij ( w ( t )) > . Now we prove that w (1) isdefined and w (1) ∗ l st is a PL metric. By the estimates above, there exists δ > such that forall t ∈ [0 , , w ( t ) ∈ W δ = { w ∈ W | θ iij ≥ δ, η ij ≥ δ } . By Lemma 5.5, the maximum time t for which w ( t ) exists on [0 , t ) must be greater than . Therefore, w (1) exists and w (1) ∈ W .Since (d) and (e) are closed conditions, it follows that w (1) ∗ l st satisfies (d) and (e).Now we prove part (f). By parts (d) and (e), we have θ ijk ( t ) ∈ [ π , π ] for i (cid:54) = A and θ Ajk ∈ [ π , π ] . Since the conductance η ij is either cot( θ kij ) or a sum cot( θ kij ) + cot( θ lij ) , weobtain for all edges [ ij ] in T , η ij ( t ) ∈ [cot( π ) , π )] ⊂ [ , . Let K i ( t ) be thecurvature of the metric w ( t ) ∗ l st at the vertex i . By Lemma 5.9 for f = | α − π/ | dw ( t ) dt and M = 100 , we conclude that for all i ∈ V , | dK i ( t ) dt | = | (∆ w (cid:48) ) i | ≤ | α − π/ | (cid:112) ln( n ) ≤ (cid:112) ln( n ) . Therefore, | K i ( t ) − K i (0) | ≤ (cid:82) t | dK i ( t ) dt | dt ≤ (cid:82) | dK i ( t ) dt | dt ≤ √ ln( n ) . Finally to prove (29), if α = π/ , then all w ( t ) = 0 and K ( t ) = K (0) and the resultfollows. If α (cid:54) = π/ , we first claim that w (cid:48) A ( t ) (cid:54) = 0 for each t . Indeed, if otherwise that w (cid:48) A ( t ) = 0 for some t , then by the maximum principle applied to the Dirichlet problem: (∆ w (cid:48) ( t )) i = 0 for i ∈ V − { A } ∪ V and w (cid:48) i ( t ) = 0 for i ∈ V ∪ { A } , we conclude w (cid:48) i ( t ) = 0 for all i ∈ V . In particular, α − π/ w (cid:48) ) A = 0 at t = t which is acontradiction. Therefore w (cid:48) A ( t ) (cid:54) = 0 and by the maximum principle again w (cid:48) A ( t ) w (cid:48) i ( t ) ≥ .Now if i ∈ V , then K (cid:48) i ( t ) = (cid:80) j ∼ i η ji ( w (cid:48) i − w (cid:48) j ) = − (cid:80) j ∼ i η ji w (cid:48) j . Since η ij ≥ , therefore w (cid:48) A ( t ) K (cid:48) i ( t ) ≤ for i ∈ V . It follows that for all i ∈ V , ( K i ( t ) − K i (0)) w (cid:48) A ( t ) ≤ . At thevertex A , | K A ( t ) − K A (0) | = | t ( α − π ) | ≤ π . Therefore by the Gauss-Bonnet theorem that K A ( t ) + (cid:80) i ∈ V K i ( t ) = K A ( t ) + (cid:80) i ∈ V K i ( t ) = 2 π and that K i ( t ) − K i (0) have the same signsfor i ∈ V , we obtain (cid:80) i ∈ V | K i ( t ) − K i (0) | = | (cid:80) i ∈ V ( K i ( t ) − K i (0)) | = | K A ( t ) − K A (0) | ≤ π . (cid:3) ONVERGENCE OF DISCRETE CONFORMAL MAPS 27
A gradient estimate of discrete harmonic functions.
The proof Theorem 5.1 is basedon the following estimate. Given a triangulated surface ( S, T ) , v ∈ V ( T ) and r > , we use B r ( v ) = { j ∈ V ( T ) | d c ( j, v ) ≤ r } to denote combinatorial ball of radius r centered at thevertex i where d c is the combinatorial distance on T (1) . Proposition 5.10.
Suppose ( P , T (cid:48) , l ) is polygonal disk with an equilateral triangulation and T is the n -th standard subdivision of the triangulation T (cid:48) with n ≥ e . Let η : E = E ( T ) → [ M , M ] be a conductance function and ∆ : R V → R V be the associated Laplace operator. Let V ⊂ V ( T ) be a thin subset such that for all v ∈ V and m ≤ n/ , | B m ( v ) ∩ V | ≤ m .If f : V → R satisfies (∆ f ) i = 0 for i ∈ V − V , | (∆ f ) i | ≤ M √ ln( n ) for i ∈ V and (cid:80) i ∈ V | (∆ f ) i | ≤ M , then for all edges [ uv ] in T , | f u − f v | ≤ M (cid:112) ln(ln( n )) . Proof.
Fix an edge [ uv ] in the triangulation T . Construct a function g : V = V ( T ) → R bysolving the Dirichlet problem (∆ g ) i = 0 for i (cid:54) = u, v , and g u = 1 , g v = 0 . By the maximumprinciple, we have ≤ g i ≤ . By the identity (cid:80) i ∈ V (∆ g ) i = 0 and that g is not a constant, weobtain (∆ g ) u = − (∆ g ) v (cid:54) = 0 . Using the Green’s identity that (cid:80) i ∈ V ( f i (∆ g ) i − g i (∆ f ) i ) = 0 and the assumptions of f, g , we obtain f u (∆ g ) u + f v (∆ g ) v − (cid:88) i ∈ V g i (∆ f ) i = 0 . Since (∆ g ) v = − (∆ g ) u , this shows f u − f v = 1(∆ g ) u (cid:88) i ∈ V g i (∆ f ) i . On the other hand, by the maximum principle g u − g j ≥ , we have | (∆ g ) u | = | (cid:80) j ∼ u η ju ( g j − g u ) | = (cid:80) j ∼ u η ju ( g u − g j ) ≥ M ( g u − g v ) = M . Therefore,(41) | f u − f v | ≤ M | (cid:88) i ∈ V g i (∆ f ) i | . To estimate the right-hand side of (41), take r = [ (cid:112) ln( n )] and select a / ∈ B r ( u ) . Then using (cid:80) i ∈ V (∆ f ) i = (cid:80) i ∈ V (∆ f ) i , | g i | ≤ , (41) and the Lemma 5.11 below, we obtain | f u − f v | ≤ M | (cid:88) i ∈ V g i (∆ f ) i | = M | (cid:88) i ∈ V ( g i − g a )(∆ f ) i | ≤ M (cid:88) i ∈ V | ( g i − g a ) || (∆ f ) i |≤ M ( (cid:88) i ∈ V ∩ B r ( u ) | g i − g a || (∆ f ) i | + (cid:88) i ∈ V − B r ( u ) | g i − g a || (∆ f ) i | ) ≤ M ( 2 M (cid:112) ln( n ) | V ∩ B r ( u ) | + 100 M (cid:112) ln( r ) (cid:88) i ∈ V | (∆ f ) i | ) ≤ M [ 20 M (cid:112) ln( n ) (cid:112) ln( n ) + 100 M (cid:113) ln( (cid:112) ln( n )) ] ≤ M (cid:112) ln(ln( n )) . In the last two steps, we have used | V ∩ B r ( u ) | ≤ r = 10 √ ln n and n ≥ e to ensure (cid:113) ln( √ ln( n )) ≥ √ ln( n ) √ ln( n ) . (cid:3) Lemma 5.11.
Assume ( P , T (cid:48) , l ) , T , E, M, η and ∆ are as given in Proposition 5.10, and g isas given in the proof of Proposition 5.10, i.e., (∆ g ) i = 0 for i (cid:54) = u, v , and g u = 1 , g v = 0 . If ≤ r ≤ n and { a, b } ∩ B r ( u ) = ∅ , then | g a − g b | ≤ M (cid:112) ln( r ) . The strategy of the proof to Lemma 5.11 is similar to that of Lemma 5.9.
Proof.
For k ≤ r/ , let U k = { i ∈ V | d c ( i, u ) = k } . Since T is an equilateral triangulationof a flat surface, we have | U k | ≤ k . Recall that a subset U of V = V ( T ) is called connectedif any two points in U can be joint by an edge path in T (1) whose vertices are in U . Eachsubset U ⊂ V is a disjoint of connected subsets which are called connected components of U .We claim that there exists a connected component G k of U k such that { a, b } lie in a connectedcomponents of V − G k . To see this, note that since T is the n -th standard subdivision of T (cid:48) ,for all k ≤ r/ ≤ n/ , the set B k ( u ) = { i ∈ V | d c ( i, u ) ≤ k } is connected and B k ( u ) c = { i ∈ V | d c ( i, u ) > k } has at most two connected components which are also connected componentsof V − U k . If B k ( u ) c is connected, then U k is connected and we take G k = U k . If B k ( u ) c hastwo connected components R and R , then there exists a non-flat boundary vertex v (cid:48) ∈ R such that d c ( u, v (cid:48) ) ≤ k ≤ r . This shows that v (cid:48) ∈ B r ( u ) . See Figure 10. The component R is contained in B r ( u ) due to d c ( v (cid:48) , u ) ≤ r . Since a, b / ∈ B r ( u ) , it follows that a, b are in R .We take G k to be the connected component of U k such that R is a connected component of V − G k . Therefore, the claim follows.F IGURE
10. Triangulated polygonal disksLet us assume without loss of generality that g a ≤ g b . By the maximum principle applied to g on the connected graph whose vertex set is the connected component of V − G k containing { a, b } , there exist two vertices u k , u (cid:48) k ∈ G k so that g u k ≥ g b , and g u (cid:48) k ≤ g a . ONVERGENCE OF DISCRETE CONFORMAL MAPS 29
Let E k be the shortest edge path with vertices in G k connecting u k to u (cid:48) k and ¯ E k be the setof all oriented edges in E k . The length of E k is at most | G k | ≤ k . The Dirichlet energy of g on the graph T (1) is(42) E ( g ) = 12 (cid:88) i ∼ j η ij ( g i − g j ) ≥ M (cid:88) i ∼ j ( g i − g j ) ≥ M [ r ] (cid:88) k =1 (cid:88) [ ij ] ∈ ¯ E k ( g i − g j ) . Suppose w = u k ∼ w ∼ w ∼ ... ∼ w l k = u (cid:48) k is the edge path E k where l k ≤ k . Then bythe Cauchy-Schwartz inequality, we have(43) (cid:88) [ ij ] ∈ ¯ E k ( g i − g j ) = l k (cid:88) i =1 ( g w i − g w i − ) ≥ l k ( l k (cid:88) i =1 ( g w i − g w i − )) ≥ l k ( g u k − g u (cid:48) k ) ≥ ( g a − g b ) k . Combining (42) and (43), we obtain(44) E ( g ) ≥ M ( g a − g b ) [ r ] (cid:88) i =1 k ≥ ( g a − g b ) ln( r )100 M .
On the other hand by the Dirichlet principle we have E ( g ) ≤ (cid:80) i ∼ j η ij ( h i − h j ) for any h ∈ R V such that h u = 1 , h v = 0 . Take h to be h u = 1 and h i = 0 , ∀ i ∈ V − { u } . We obtain E ( g ) ≤ (cid:80) i ∼ j η ij ( h i − h j ) ≤ M . Therefore, ( g b − g a ) ln( r )100 M ≤ M which implies | g b − g a | ≤ M (cid:112) ln( r ) . (cid:3) A proof of Theorem 5.1.
For simplicity, a boundary vertex v ∈ P − { p, q, r } with non-zero curvature will be called a corner . Note that corners in T and its n -th standard subdivision T ( n ) are the same. In particular, the total number of corners is independent of n . Let V c bethe set of all corner vertices. Since P is embedded in C , given a corner v ∈ V c , the degree m of v has to be 3,5 or 6. Consider the combinatorial ball B [ n/ ( v ) of radius [ n/ centered at acorner v ∈ V c in T ( n ) . By construction B [ n/ ( v ) ∩ B [ n/ ( v (cid:48) ) = ∅ for distinct corners v, v (cid:48) . Each B [ n/ ( v ) is a union of m − n/ -th standard subdivided equilateral triangles ∆ , ..., ∆ m − in T . Applying Theorem 5.6 with α = πm − to the triangulated equilateral triangle (∆ i , v ) for each i = 1 , , ..., m − , we produce a discrete conformal factor w (∆ i ) ∈ R V (∆ i ) foreach ∆ i such that if a vertex u ∈ V (∆ i ) ∩ V (∆ j ) , then w u (∆ i ) = w u (∆ j ) . In particularthere is a well defined discrete conformal factor w ( B [ n/ ( v )) on B [ n/ ( v ) obtained by gluingthese w (∆ i ) . See Figure 8. Define w (1) : V ( T ( n ) ) → R as follows: if u ∈ ∪ v ∈ V c B [ n/ ( v ) ,then w (1) u = w u ( B [ n/ ( v )) for u ∈ B [ n/ ( v ) and w (1) ( u ) = 0 for u / ∈ ∪ v ∈ V c B [ n/ ( v ) . Let ˆ l = w (1) ∗ l st be the PL metric on T ( n ) and ˆ K be its the discrete curvature. Let K ∗ : V ( n ) → R be defined by K ∗ i = 0 if i / ∈ { p, q, r } , and K ∗ i = π if i ∈ { p, q, r } . By Theorem 5.6, the PLmetric ˆ l and ˆ K satisfy the following:(a) the curvature ˆ K i = K ∗ i at all vertices i such that d c ( i, v ) (cid:54) = [ n/ for some corner v ∈ V c ;(b) w (1) i = 0 for i / ∈ ∪ v ∈ V c B [ n/ ( v ) ;(c) all inner angles at a corner v ∈ V c are in [ π , π ] ; (d) all inner angles at a non-corner vertex are in [ π , π ] ;(e) | ˆ K i − K ∗ i | ≤ √ ln( n ) and (cid:80) i ∈ V | ˆ K i − K ∗ i | ≤ πN where N is the number of corners in P .We will find a discrete conformal factor w (2) : V ( n ) → R such that w (2) ∗ ˆ l and its curvaturesatisfy Theorem 5.1 by solving the following system of ordinary differential equations in w ( t ) :(45) dK i ( w ( t ) ∗ ˆ l ) dt = K ∗ i − ˆ K i , ∀ i ∈ V ( T ( n ) ) −{ p, q, r } ; w s ( t ) = 0 , s ∈ { p, q, r } ; and w (0) = 0 . Let K ( t ) = K ( w ( t ) ∗ ˆ l ) . Note that (45) and the Gauss-Bonnet formula imply that K (cid:48) p ( t ) = K ∗ p − ˆ K p . By Lemma 5.5, the solution w ( t ) exists on some interval [0 , (cid:15) ) . Our goal is to showthat for n large, the solution w ( t ) exists on [0 , . In this case, the conformal factor w (2) istaken to be w (1) . The required discrete conformal factor w n in Theorem 5.1 is taken to be w (1) + w (2) .Consider the maximum time t such that the solution w ( t ) to (45) exists for t ∈ [0 , t ) andthe PL metrics w ( t ) ∗ ˆ l satisfy:( c (cid:48) ) all inner angles at a corner v ∈ V c are in [ π − π , π + π ] ;( d (cid:48) ) all inner angles at a non-corner vertex are in [ π − π , π + π ] .Let V = ∪ v ∈ V c { i ∈ V ( n ) | d c ( i, v ) = [ n/ } . By construction, | B r ( i ) ∩ V | ≤ r for all r ≤ n/ . Then (cid:80) i ∈ V | (∆ w (cid:48) ( t )) i | = (cid:80) i ∈ V | K (cid:48) i ( t ) | ≤ (cid:80) i ∈ V | ˆ K i − K ∗ i | ≤ πN and | (∆ w (cid:48) ) i | = | K (cid:48) i ( t ) | ≤ | ˆ K i − K ∗ i | ≤ √ ln( n ) . Choose M = max { , πN } . Then by ( c (cid:48) ) , ( d (cid:48) ) , ( e ) andthe formula cot( a ) + cot( b ) = sin( a + b )sin( a ) sin( b ) , for all t ∈ [0 , t ) , we have η ij ( t ) = η ij ( w ( t ) ∗ ˆ l ) ∈ [ , ⊂ [ M , M ] , (∆ w (cid:48) ) i = 0 for i ∈ V ( T ( n ) ) − V , | (∆ w (cid:48) ) i | ≤ M √ ln( n ) and (cid:80) i ∈ V | (∆ w (cid:48) ) i | ≤ M . In summary, f = w (cid:48) satisfies conditions in Proposition 5.10 for all t ∈ [0 , t ) . By Proposition 5.10, if i ∼ j , then | w (cid:48) i ( t ) − w (cid:48) j ( t ) | ≤ M (cid:112) ln(ln( n )) . On the other hand, by the variation of angle formula (6) and M ≥ | cot( θ kij ) | , we have | dθ kij dt | ≤ | cot( θ ijk )( w (cid:48) j − w (cid:48) k ) | + | cot( θ jik )( w (cid:48) i − w (cid:48) k ) | ≤ M ( | w (cid:48) j − w (cid:48) k | + | w (cid:48) i − w (cid:48) k | ) ≤ M (cid:112) ln(ln( n )) . Therefore, for t ∈ [0 , t ) and sufficiently large n ,(46) | θ kij ( w ( t )) − θ kij (0) | ≤ (cid:90) t | dθ kij ( w ( t )) dt | dt ≤ M t (cid:112) ln(ln( n )) ≤ πt . It follows that t > (or t = ∞ ) since otherwise, by (46), the choices of angles in (c),(d), ( c (cid:48) ),( d (cid:48) ) and Lemma 5.5, we can extend the solution w ( t ) to [0 , t + (cid:15) ) for some (cid:15) > such that ( c (cid:48) ) and ( d (cid:48) ) still hold. To be more precise, by Lemma 5.5 on the maximality of t , we have either lim sup t → t | θ ijk ( t ) − π | = π for an inner angle θ ijk at a corner i ∈ V c , or lim sup t → t θ ijk ( t ) = π − π or lim sup t → t θ ijk ( t ) = π + π for an angle θ ijk at a non-corner vertex i . But, dueto (46), none of these conditions holds if t ≤ . Therefore the solution w (1) exists. By ONVERGENCE OF DISCRETE CONFORMAL MAPS 31 construction, the curvature K (1) of w (1) ∗ ˆ l is K (0) + (cid:82) K (cid:48) ( t ) dt = ˆ K + K ∗ − ˆ K = K ∗ .Furthermore, condition (c) in Theorem 5.1 follows from ( c (cid:48) ) and ( d (cid:48) ) .6. A PROOF OF THE CONVERGENCE THEOREM
We will prove the following theorem.
Theorem 6.1.
Let Ω be a Jordan domain in the complex plane and { p, q, r } ⊂ ∂ Ω . There existsa sequence of triangulated polygonal disks (Ω n , T n , d st , ( p n , q n , r n )) where T n is an equilateraltriangulation and p n , q n , r n are three boundary vertices such that(a) Ω = ∪ ∞ n =1 Ω n with Ω n ⊂ Ω n +1 , and lim n p n = p , lim n q n = q and lim n r n = r ,(b) discrete uniformization maps f n associated to (Ω n , T n , d st , ( p n , q n , r n )) exist and con-verge uniformly to the Riemann mapping associated to (Ω , ( p, q, r )) . Before giving the proof, let us recall Rado’s theorem and its generalization to quasiconfor-mal maps. If φ : D → Ω is a K-quasiconformal map onto a Jordan domain Ω , then φ extendscontinuously to a homeomorphism φ : D → Ω between their closures (see [1, corollary onpage 30]). If K = 1 , φ is the Caratheodory extension of the Riemann mapping. A sequenceof Jordan curves J n in C is said to converge uniformly to a Jordan J curve in C if there existhomeomorphisms φ n : S → J n and φ : S → J such that φ n converges uniformly to φ .Rado’s theorem [32] and its extension by Palka [30, corollary 1] states that, Theorem 6.2 (Rado, Palka) . Suppose Ω n is a sequence of Jordan domains such that ∂ Ω n converges uniformly to ∂ Ω . If f n : D → Ω n is a K-quasiconformal map for each n such thatthe sequence { f n } converges to a K-quasiconformal map f : D → Ω uniformly on compactsets of D , then f n converges to f uniformly on D . The following compactness result is a consequence of Palka’s theorem ([30, corollary 1])and Lehto-Virtanen’s work [25, Theorems 5.1, 5.5].
Theorem 6.3.
Suppose Ω n is a sequence of Jordan domains such that ∂ Ω n converges uniformlyto ∂ Ω and K > is a constant. Let p n , q n , r n ∈ ∂ Ω n and p, q, r ∈ ∂ Ω be distinct points suchthat lim n p n = p, lim n q n = q, lim n r n = r and h n : D → Ω n be K-quasiconformal maps suchthat h n sends (1 , √− , − to ( p n , q n , r n ) . Then there exists a subsequence { h n i } of { h n } converging uniformly on D to a K-quasiconformal map h : D → Ω sending (1 , √− , − to ( p, q, r ) . Now we prove Theorem 6.1.
Proof.
Given a Jordan domain Ω with three distinct points p, q, r in ∂ Ω , construct a sequence ofapproximating polygonal disks Ω n such that (1) each Ω n is triangulated by equilateral trianglesof side lengths tending to , (2) ∂ Ω n converges uniformly to the Jordan curve ∂ Ω such that Ω n ⊂ Ω n +1 , (3) there are three boundary vertices p n , q n , r n ⊂ ∂ Ω n such that lim n p n = p , lim n q n = q and lim n r n = r , and (4) the curvatures of Ω n at p n , q n , r n are π and curvatures of Ω n at all other boundary vertices are not π .By Theorem 5.1, we produce a standard subdivision T n of Ω n and w n ∈ R V ( T n ) such that (Ω n , T n , w n ∗ l st ) is isometric to the equilateral triangle (∆ ABC, T (cid:48) n , l (cid:48) n ) with a Delaunay tri-angulation T (cid:48) n and A, B, C correspond to p n , q n , r n . Let f n : (∆ ABC, T (cid:48) n , ( A, B, C )) → (Ω n , T n , ( p n , q n , r n )) be the associated discrete conformal map and ¯ f : (∆ ABC, ( A, B, C )) → (Ω , ( p, q, r )) be the Riemann mapping. We claim that f n converges uniformly to ¯ f on ∆ ABC . To establish the claim, first by Theorem 5.1, we know all angles of triangles in the trian-gulated PL surface (∆ ABC, T (cid:48) n , l (cid:48) n ) are at least (cid:15) > . Therefore the discrete conformalmaps f n are K-quasiconformal for a constant K independent of n . By Theorem 6.3, it fol-lows that every limit function g of a convergence subsequence { f n i } is a K-quasiconformalmap from int (∆ ABC ) to Ω which extends continuously to ∆ ABC sending
A, B, C to p, q, r respectively. We claim that the limit map g is conformal. Indeed, by Lemma 4.7, the discreteconformal map f − n , when restricted to a fixed compact set R of Ω , maps equilateral trianglesin T n which are inside R to triangles of T (cid:48) n that become arbitrarily close to equilateral triangleas n → ∞ . Therefore the limit map g of the subsequence f n i is 1-conformal and thereforeconformal in int (∆ ABC ) . The continuous extension of g sends A, B, C to p, q, r respectivelyby Theorem 6.3. On the other hand, there is only one Riemann mapping f : int (∆) → Ω whose continuous extension sends A, B, C to p, q, r respectively. Therefore, g = f . Thisshows all limits of convergence subsequences of { f n } are equal f . Therefore { f n } convergesto f uniformly on compact sets in int (∆ ABC ) . By Theorem 6.2, f n converges uniformly to ¯ f . (cid:3)
7. A
CONVERGENCE CONJECTURE ON DISCRETE UNIFORMIZATION MAPS
We discuss a general approximation conjecture and the related topics of discrete conformalequivalence of polyhedral metrics.7.1.
A strong version of convergence of discrete conformal maps.
As discussed before,the main drawback of the vertex scaling operation on polyhedral metrics is the lacking of anexistence theorem. For instance, given a PL metric on a closed triangulated surface ( S, T , l ) ,there is in general no discrete conformal factor w : V → R such that the new PL metric ( S, T , w ∗ l ) has constant discrete curvature.The recent work of [13] established an existence and a uniqueness theorem for polyhedralmetrics by allowing the triangulations to be changed. Definition 7.1. (Discrete conformality of PL metrics [13] ) Two PL metrics d, d (cid:48) on ( S, V ) arediscrete conformal if there exist sequences of PL metrics d = d, ..., d m = d (cid:48) on ( S, V ) andtriangulations T , ..., T m of ( S, V ) satisfying(a) (Delaunay) each T i is Delaunay in d i ,(b)(Vertex scaling) if T i = T i +1 , there exists a function w : V → R so that if e is an edge in T i with end points v and v (cid:48) , then the lengths l d i +1 ( e ) and l d i ( e ) of e in d i and d i +1 are relatedby (47) l d i +1 ( e ) = e w ( v )+ w ( v (cid:48) ) l d i ( e ) , (c) if T i (cid:54) = T i +1 , then ( S, d i ) is isometric to ( S, d i +1 ) by an isometry homotopic to the identityin ( S, V ) . The main theorem proved in [13] is the following.
Theorem 7.2.
Suppose ( S, V ) is a closed connected marked surface and d is a PL metric on ( S, V ) . Then for any K ∗ : V → ( −∞ , π ) with (cid:80) v ∈ V K ∗ ( v ) = 2 πχ ( S ) , there exists a PLmetric d ∗ , unique up to scaling and isometry homotopic to the identity, on ( S, V ) such that d ∗ is discrete conformal to d and the discrete curvature of d ∗ is K ∗ . Furthermore, the metric d ∗ can be found using a finite dimensional (convex) variational principle. ONVERGENCE OF DISCRETE CONFORMAL MAPS 33
There is a close relation between the discrete conformal equivalence defined in Definition7.1 and convex geometry in hyperbolic 3-space. The first work relating vertex scaling operationand hyperbolic geometry is in the paper by Bobenko-Pinkall-Springborn [4]. They associatedeach polyhedral metric on ( S, T , l ) a hyperbolic metric with cusp end on the punctured surface S − V ( T ) . However, the Delaunay condition on the triangulation T was missing in theirdefinition. The discrete conformal equivalence defined in Definition 7.1 is equivalent to thefollowing hyperbolic geometry construction. Let ( S, V, d ) be a PL surface. Take a Delaunaytriangulation T of ( S, V, d ) and consider the PL metric d as isometric gluing of Euclideantriangles τ ∈ T . Consider each triangle τ in T as the Euclidean convex hull of three points v , v , v in the complex plane C . Let τ ∗ be the convex hull of { v , v , v } in the upper-halfspace model of the hyperbolic 3-space H . Thus τ ∗ is an ideal hyperbolic triangle having thesame vertices as that of τ . If σ and τ are two Euclidean triangles in T glued isometrically alongtwo edges by an isometry f considered as an isometry of the Euclidean plane, we glue τ ∗ and σ ∗ along the corresponding edges using the same map f considered as an isometry of H . Herewe have used the fact each isometry of the complex plane extends naturally to an isometry ofthe hyperbolic 3-space H . The result of the gluing of these τ ∗ produces a hyperbolic metric d ∗ on the punctured surface S − V . It is easy to see that d ∗ is independent of the choicesof Delaunay triangulations. It is shown in [13] (see also [14]) that two PL metrics d and d on ( S, V ) are discrete conformal in the sense of Definition 7.1 if and only if the associatedhyperbolic metrics d ∗ and d ∗ are isometric by an isometry homotopic to the identity on S − V .Using this hyperbolic geometry interpretation, one defines the discrete conformal map be-tween two discrete conformally equivalent PL metrics d and d as follows (see [4] and [13]).The vertical projection of the ideal triangle τ ∗ to τ induces a homeomorphism φ d : ( S − V, d ∗ ) → ( S − V, d ) . Suppose d and d are two discrete conformally equivalent PL metrics on ( S, V ) . Then the discrete conformal map from ( S, V, d ) to ( S, V, d ) is given by φ d ◦ ψ ◦ φ − d where ψ : ( S, V, d ∗ ) → ( S, V, d ∗ ) is the hyperbolic isometry. Note that in this new setting,discrete conformal maps are piecewise projective instead of piecewise linear.Theorem 7.2 can be used for approximating Riemann mappings for Jordan domains. Givena simply connected polygonal disk with a PL metric ( D, V, d ) and three boundary vertices p, q, r ∈ V , let the metric double of ( D, V, d ) along the boundary be the polyhedral 2-sphere ( S , V (cid:48) , d (cid:48) ) . Using Theorem 7.2, one produces a new polyhedral surface ( S , V (cid:48) , d ∗ ) suchthat: 11) ( S , V (cid:48) , d ∗ ) is discrete conformal to ( S , V (cid:48) , d (cid:48) ) ; ( 2) the discrete curvatures of d ∗ at p, q, r are π/ ; (3) the discrete curvatures of d ∗ at all other vertices are zero; and (4)the area of ( S , V (cid:48) , d ∗ ) is √ / . Therefore ( S , V (cid:48) , d ∗ ) is isometric to the metric double( D (∆ ABC ) , V (cid:48)(cid:48) , d (cid:48)(cid:48) ) of an equilateral triangle ∆ ABC of edge length 1. Let F be the dis-crete conformal map from ( D (∆ ABC ) , V (cid:48)(cid:48) , d (cid:48)(cid:48) ) to ( S , V (cid:48) , d (cid:48) ) such that F sends A, B, C to p, q, r respectively. Due to the uniqueness part of Theorem 7.2, we may assume that f = F | : ∆ ABC → D and f sends A, B, C to p, q, r respectively. We call f the discreteuniformization map associated ( D, V, d, ( p, q, r )) .A strong form of the convergence is the following, Conjecture 7.3.
Let (Ω , ( p, q, r ) ) be a Jordan domain in the complex plane with three markedboundary points and (Ω n , T n , d st , ( p n , q n , r n )) be any sequence of triangulated flat polygonaldisks with three marked boundary vertices such that(a) T n is an equilateral triangulation,(b) ∂ Ω n converges uniformly to ∂ Ω ,(c) the edge length of T n goes to zero, (d) lim n p n = p , lim n q n = q and lim n r n = r .Then discrete uniformization maps f n associated to (Ω n , T n , d st , ( p n , q n , r n )) converge uni-formly to the Riemann mapping associated to (Ω , ( p, q, r )) . Discrete conformal equivalence and convex sets in the hyperbolic 3-space.
We nowdiscuss the relationship between discrete conformal equivalence defined in Definition 7.1, idealconvex sets in the hyperbolic 3-space H and the motivation for Conjectures 1.5 and 1.6.The classical uniformization theorem for Riemann surfaces follows from the special casethat every simply connected Riemann surface is biholomorphic to C , D or S . The discreteanalogous should be the statement that each non-compact simply connected polyhedral sur-face is discrete conformal to either ( C , V, d st ) or ( D , V, d st ) where V is a discrete set and d st is the standard Euclidean metric. Furthermore, the set V is unique up to M¨obius transforma-tions. For a non-compact polyhedral surface ( S, V, d ) with an infinite set V , the hyperbolicgeometric view point of discrete conformality is a better approach. Namely discrete conformalequivalence between two PL metrics is the same as the Teichm¨uller equivalence between theirassociated hyperbolic metrics. For instance, if we take a Delaunay triangulation T of the com-plex plane ( C , d st ) with vertex set V , then the associated hyperbolic metric d ∗ st on C − V isisometric to the boundary of the convex hull ∂C H ( V ) in H . Therefore, a PL surface ( S, V (cid:48) , d ) is discrete conformal to ( C , V, d st ) for some discrete subset V ⊂ C if and only if the associ-ated hyperbolic metric d ∗ is isometric to the boundary of the convex hull ∂C H ( V ) . It showsdiscrete uniformization is the same as realizing hyperbolic metrics as the boundaries of convexhulls (in H ) of closed sets in ∂ H . One can formulate the conjectural discrete uniformizationtheorem as follows. Given a discrete set V (cid:48) in C or D , let ˆ d be the unique conformal completehyperbolic metric on C − V (cid:48) or D − V (cid:48) . Then ˆ d is isometric to the boundary of the convex hullof a discrete set V ⊂ C or ( C ∪ {∞} − D ) ∪ V where V is discrete and unique up to M¨obiustransformations. This is the original motivation for proposing Conjectures 1.5 and 1.6.These two conjectures bring discrete uniformization close to the classical Weyl problem onrealizing surfaces of non-negative Gaussian curvature as the boundaries of convex bodies in the3-space. In the hyperbolic 3-space H , convex surfaces have curvature at least − . The workof Alexandrov [2] and Pogorelov [31] show that for each path metric d on the 2-sphere SS of curvature ≥ − , there exists a compact convex body, unique up to isometry, in H whoseboundary is isometric to ( SS , d ) . The interesting remaining cases are non-compact surfaces ofgenus zero in the hyperbolic 3-space H . A theorem of Alexandrov [2] states that any completesurface of genus zero whose curvature is at least − is isometric to the boundary of a closedconvex set in H . On the other hand, given a closed set X ⊂ C , W. Thurston proved that theintrinsic metric on ∂C H ( X ) is complete hyperbolic (see [10] for a proof). Putting these twotheorems together, one sees that each complete hyperbolic metric on a surface of genus zero isisometric to the boundary of the convex hull of a closed set in the Riemann sphere. However,in this generality, the uniqueness of the convex surface is false. Conjectures 1.5 and 1.6 saythat one has both the existence and uniqueness if one imposes restricts to the boundaries of theconvex hulls of closed sets.There are some evidences supporting Conjectures 1.5 and 1.6. The work of Rivin [33] andSchlenker [35] show that Conjectures 1.5 and 1.6 hold if Ω has finite area (i.e., X is a finiteset) or if Ω is conformal to the 2-sphere with a finite number of disjoint disks removed (i.e., X is a finite disjoint union of round disks). Our recent work [28] shows that Conjectures 1.5holds for Ω having countably many topological ends using the work of He-Schramm on K¨obeconjecture. ONVERGENCE OF DISCRETE CONFORMAL MAPS 35
One should compare Conjectures 1.5 and 1.6 with the K¨obe circle domain conjecture whichstates that each genus zero Riemann surface is biholomorphic to the complement of a circletype closed set in the Riemann sphere. The work of He-Schramm [17] shows that K¨obe con-jecture holds for surfaces with countably many ends and the circle type set is unique up toM¨obius transformations. Uniqueness is known to be false for the K¨obe conjecture in general.Our recent work [28] shows that the K¨obe conjecture is equivalent to Conjecture 1.5. Otherrelated works are [5], [11], [23], [24], [33], [35], [36], and [38].We end this paper by proposing the following the conjecture. The work of Rodin-Sullivan[34] and Theorem 1.4 show the rigidity phenomena for the two most regular patterns (regularhexagonal circle packing and regular hexagonal triangulation) in the plane. These rigidityresults can be used to approximate the Riemann mappings and the uniformization metrics. Thethird regular pattern in the plane is the hexagonal square tiling in which each square of sidelength one interests exactly six others. See figure 11.F
IGURE
11. Regular hexagonal square tiling
Conjecture 7.4.
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