Discrete uniformiza0on theorem for polyhedral surfaces and hyperbolic convex polytopes

Transcription

1 Discrete uniformiza0on theorem for polyhedral surfaces and hyperbolic convex polytopes Feng Luo Quantum invariants and low- dimensional topology Matrix Ins0tute, Australia Joint work with D. Gu (Stony Brook), J. Sun (Tsinghua Univ.), S. Tillmann (Sydney), Tianqi Wu (Courant) Dec. 14, 2016

2 S = connected surface Thm(Poincare- Koebe 1907) Riemannian metric g on S, Ǝ λ: S R >0 s.t., (S, λg) is a complete metric of curvature - 1, 0, 1. Q1: Can one compute the uniformiza0on metrics and maps? Q2: Is there discrete unif. thm. for polyhedral surfaces? Does it converge (to smooth case)? ANS (Gu- L- Sun- Wu): yes

3 Corollary. (Riemann mapping ) Any s.c. domain Ω C is conformal to D. Riemann mapping can be computed. B. Beeker, B. Loustau, based on C. Collins & K. Stephenson Thurston- Stephenson s circle packing

4 The Scharwz- Christoffel method by L. Trefethen, L., and T. Driscoll Thm (Gu- L- Sun- Wu). Uniformiza0on metrics and maps are computable.

5 PL metrics d on marked surface (S, V) are flat cone metrics on S, cone points V Isometric gluing of E 2 triangles along edges: (S, T, l ). d is determined by edge lengths l: E={all edges in T } R triangulation K(v)<0 Curvature K=K d : V={all ver0ces} R, K(v)= 2π- sum of angles at v K(v)>0 Gauss- Bonnet A triangulated PL metric (S, T, l) is Delaunay: a+b π at each edge e.

6 Polyhedral metrics d on cpt (S,V) and hyperbolic metric d* on S- V Given d on (S,V), produce a Delaunay triangula0on T of (S,V,d) t T is associated an ideal hyperbolic triangle t* If t, s T glued by isometry f along e, then t* and s* are glued by the same f along e*. a hyperbolic metric d* on S- V. Bobenko- Pinkall- Springborn Def. (G- L- S- W). Two PL metrisc d 1, d 2 on (S,V) are discrete conformal iff d 1 * are d 2 * are isometric by an isometry homotopic to id on S- V.

7 Delaunay trianguladon = hyperbolic convex hull V C discrete set Delaunay triangulation T of C E (V) (Euclidean convex hull): t triangle in T iff its circumdisk B contains no V in its interior. The hyperbolic convex hull C H (V) hull of V in H 3. C H (V)= H 3 - int(c H (B)) B are max balls in S 2 missing V. a+a π triangle t=c E (a,b,c) in T corresponds to triangle t* =C H (a,b,c) in C H (V).

8 Does no change d* d st * H 3 Eg. d st Boundary of hyperbolic convex hull of V

9 Thm 1. (Gu- L- Sun- Wu) PL metric d on a closed (S,V) and K # : V (-, 2π), s.t., K # (v) =2πχ(S), Ǝ a PL metric d #, unique up to scaling, on (S,V) s.t., (a) d # is discrete conformal to d, (b) the discrete curvature of d # is K #. For K # = 2πχ(S)/ V, d # is a discrete uniformiza0on metric. Eg.1. Any PL metric on (S 1 XS 1,V) is d.c. to a unique flat (S 1 XS 1, V, d # ) where K # =0 (Fillastre).

10 . K # =4π/3 at a,b,c Eg 2. A polygonal disk (D, V; a,b,c) in C is d.c. to the equilateral triangle (ΔABC, V, {A,B,C}) Thm 2 (L- Sun- Wu). Given a Jordan domain Ω and p,q,r Ω, Ǝ polygonal disks (Ω n, V n ; p n,q n,r n ) approxima0ng it, s.t., (a) (Ω n, V n ; p n,q n,r n ) triangula0on T n by equilateral triangles of length 0, (b) the associated discrete uniformiza0on maps f n Riemann mapping for (Ω;p,q,r). Counterpart of Thurston s circle packing conjecture: F n converges to the Riemann mapping. A Riemann mapping sending the triangle to (Ω;p,q,r). B C

11 Discrete uniformiza0on for simply connected non- cpt polyhedral surfaces S=non- cpt simply connected topological surface Unif. Thm. Every complex structure on S is conformal to C or D. Discrete uniformiza0on conjecture. Every PL surface (S,V,d) is d.c. to a unique (C, V, d st ) or (D, V, d st ). Associated hyperbolic metric Weyl s problem on convex embedding, Alexandrov, Nirenberg, Pogorelov (S- V, d) complete hyperbolic C(V ) in H 3 isometric

12 Geometry of convex hulls in H 3 and conjectures Thurston. If X closed in S 2, then C H (X) H 3 is complete hyperbolic. Eg. Ω simply connected domain in C, X=S 2 - Ω. Then C H (X) isometric to H 2. Thurston s isometry convex hull geom. Ω Riemann mapping, conformal geom. QuesDon: not simply connected Ω? X is of circle type Koebe Conjecture. Every domain Ω in S 2 is conformal to S 2 - X s.t., connected components of X are points or round disks. Conj (L- S- W) 1. complete hyperbolic surf (Σ, d) of genus 0 is isometric to C H (X) for a circle type closed set X. Conj.(L- S- W) 2. If X and Y are two circle type closed sets s.t. C H (X) isometric C H (Y), then X, Y differ by a Moebius transf. Thm (Rivin). Conj. 1&2 hold for X = finite set. Thm (Schlenker). Conj. 1&2 hold for X = finite union of disks. Thm (L- Tillmann). Conj. 1&2 hold for X = a union of one disk and a finite set. Thm (L- Wu). Conjecture 1 holds if has countably many top ends.

13 Conjecture (Koebe). For any closed set Y in S 2 with connected complement, S 2 - Y is conformal to S 2 - X for a circle type closed set X. Conjecture (L- S- W). For any closed set Y in S 2 with connected complement, C H (Y) is isometric to a unique C H (X) for a circle type closed set X. Thm 5. (L- Sun- Wu) If Y C is discrete s.t. isometry C H (Y) C H (Z+ τ Z) preserving cell structures, then Y and Z+τ Z differ by a linear map. It implies limit of approxima0ng F n is conformal.

14 Thm 1. (Gu- L- Sun- Wu) PL metric d on a closed (S,V) and K # : V (-, 2π), s.t., K # (v) =2πχ(S), Sketch of proof of Theorem 1. Ǝ a PL metric d #, unique up to scaling, on (S,V) s.t., (a) d # is discrete conformal to d, (b) the discrete curvature of d # is K #. Vertex scaling: given l: E R and u: V R, define u * l(vv ) = e u(v)+u(v ) l(vv ).

15 Sketch of proof thm 1 Step 1. There exists a c 1 - smooth map A: {PL metrics d on (S,V)}/ ~ Teich(S- V) s.t., A(d)=A(d ) iff d and d are discrete conformal. ~ = isometry homotopic to iden0ty Step 2. for any PL metric d on (S,V) P= {[d ] d disc. conf. to d} / ~ R V. Step 3. The discrete curvature map K: P/R >0 > (-, 2π) V {Gauss- Bonnet equa0on} is 1-1, onto. (GB: x Є R V, v ЄV x(v) = 2π χ (S).) We prove: K is smooth, locally 1-1 (a varia0onal principle), image of K is closed (degenera0on analysis+ Akiyoshi).

16 A varia0onal principle associated to vertex scaling Vertex scaling: given l: E R and u: V R, u * l(vv ) = e u(v)+u(v ) l(vv ). Prop (L, 2004) Fix a triangle Δ of lengths l 1, l 2, l 3, let a 1, a 2, a 3 be the angles of the vertex scaled triangle of lengths l i e u j +u k where a i =a i (u 1,u 2,u 3 ). Then there is a locally concave func0on F(u) s.t. F=(a 1, a 2, a 3 ). Prof. The matrix [ - a i / u j ] is symmetric and semi- posi0ve definite.

17 Thank you.

18 Thank you.

19 For (S,V), define PL Teichmuller space T pl (S,V)={ (S,V,d) PL metric d on (S,V)}/ ~ (S,V,d) ~ (S,V, d ) iff an isometry homotopic to id. Known (Troyanov) T pl (S,V) is homeomorphic to R - 3χ (S- V). For a triangula0on T of (S,V), let D pl (T)={ [S,V,d] T is Delaunay in d} Rivin s thm: D pl (T) s form a cell decomposi0on of T pl (S,V). T pl (S,V) = U T D pl (T)

20 Penner s decorated Teichmuller space T d (S,V) Decorated ideal triangle: It has angles a i and length l i For any l 1, l 2, l 3, a unique decorated triangle of lengths l 1,l 2,l 3.

21 Decorated Teichmuller space Let d=complete hyperbolic metric of finite area on S- V. Construct at each cusp v a horoball H(v). One has the decorated metric (S- V, d, w) where w=(w 1,, w n ) in R V, e w i=length of H(v i ) T d (S- V) ={(S- V, d, w) decorated metrics}/ isometry id T d (S- V) = T(S- V) X R n preserving marking

22 Penner s coordinate For Ɐ triangula0on T of (S,V), Ɐ x: E(T) R >0, a decorate metric d x on (S,V) having ln(x) edge length. For any l 1, l 2, l 3 >0, a unique decorated triangle of lengths ln(l i ) This produces the decorated metric on (S,V)

23 For a triangula0on T, let D(T) be the set of all [(S- V,d, w)] s, s.t., T is Delaunay in d. Thm(Penner) D(T) s form a cell decomposi0on of T d (S- V), i.e. T d (S- V)= U T D(T). Define a map F T : D pl (T) T d (S- V): One shows: 1. F T (D pl (T)) D(T) (Euclidean Delaunay implies hyperbolic Delaunay) 2. F T (D pl (T)) =D(T) (Delaunay implies triangular ineq.) 3. F T Dpl (T) D pl (T) =F T Dpl (T) D pl (T )

24 F T Dpl (T) Dpl(T ) = F T Dpl(T) Dpl (T ) This is Penner s Ptolemy iden0ty: Thm: The gluing of F T s produces a C 1 diffeomorphism F: T pl (S,V) T d (S- V) preserving cell decomposi0ons and d, d discrete conformal iff Proj(F(d)) =Proj(F(d )) where Proj: T d (S- V) = T(S- V) X R n T(S- V) is the natural projec0on.

25 Final proof of discrete uniformiza0on theorem Take a p T(S- V), consider the composi0on map h K =discrete curvature F - 1 R n p X R n T d (S- V) T pl (S,V) (-, 2π) n { x i =2πχ(S)} Discrete Unif thm: the map h: P={ x i =0} to Q is 1-1 onto. We will show that h is a homeomorphism. K Q Step 1. h is 1-1: due to a varia0onal principle developed by Luo in Namely, It is shown that h is the gradient of a strictly convex func0on. Step 2. h(p) is closed in Q. This implies h is onto using (a) dim(p)=dim(q) (b) h is 1-1 implies h(p) open in Q (c) Q connected and h(p) open and closed in Q implies h(p)=q.

26 Sketch of Proof of convergence thm Thm (L- Sun- Wu). Given a Jordan domain (Ω;p,q,r), p,q,r Ω, Ǝ polygonal disks (Ω n, V n ; p n,q n,r n ) approxima0ng it, s.t., (a) (Ω n, V n ; p n,q n,r n ) triangulated T n by equilateral triangles of length 0, (b) the associated discrete uniformiza0on maps f n Riemann mapping for (Ω;A,B,C).. f n Two steps: 1. There exists L>0 s.t. all f n are L- quasi- conformal 2. Every limit of convergent subsequences of f n is 1- quasiconformal

27 To see h(q) closed Take seq {x k } in P= { x i =1} s.t., x k leaves each cpt set in P. Want: h(x k ) leaves each cpt set in Q, i.e., some curvature >2π 1. Akiyoshi Thm: Each px R n >0 intersects only finitely many D(T) s in T d (S- V). 2. May assume that x k are dis. conf. factors of PL metrics Delaunay in one T. 5. x k leaves each cpt set in P means some coord of x k goes to. 6. Take v Є V s.t., x k (v) and v adjacent to v s.t., x k (v ) t Є [0, ). Claim: the curvature K at v 2π.

28 Limi0ng map lim f ni is conformal Circle packing case, f not conformal non- regular hexagonal circle packing of an open set in C. Thm (Rodin- Sullivan). Hexagonal circle packings of an open set in C are regular. Discrete conformal case, f not conformal non- regular Delaunay hexagonal triangula0on T of an open set in C which is a vertex scaling of T st. T st

29 Thm(L- Sun- Wu). If T is a geometric hexagonal triangula0on of an open set in C s.t. 1. it is Delaunay, 2. g: V R >0 sa0sfying length(vv )=g(v)g(v ), then g = constant. edges vv, Thm. If isometry C H (V) C H (Z+ e πi/3 Z) preserving cell structures, then V and Z+ e πi/3 Z differ by a linear map. Thm (Rodin- Sullivan). If T is a geometric hexagonal triangula0on of an open set in C s.t. r: V R >0 sa0sfying length(vv )=r(v)+r(v ), edges vv, then r=constant.

30 discrete harmonic func0ons on la ce, a new proof of Rodin- Sullivan thm CP: Using Thurston s max principle and taking limits of circle packings, if the result is false, a hexagonal circle packing of an open set in C whose radius func0on r: V=Z+e πi/3 Z R >0 sa0sfies ln(r): V R is non- const. linear. Doyle s theorem: spiral circle packing cannot be embedded in C. DC: if the theorem is false, using a max principle and taking limits, a Delaunay hexagonal triangula0on of an open set in C whose length func0on l(vv )=w(v)w(v ) sa0sfies that ln(w): V R is non- constant linear. Prop. Spiral hexagonal triangula0on of a simply connected surface cannot be embedded into C.

31 Hyperbolic surface (S- V,d*). f n discrete uniformiza0on thm Thm (L- Sun- Wu). Given any polygonal disk (D, V; p,q,r) with a regular triangula0on T s. t, all boundary ver0ces except p,q,r have angles other than π/3, we can sufficiently subdivide T to a new regular triangula0on T s.t., (1) no flips are used in the discrete uniformiza0on process for (D, V, T ), (2) all angles are within [1/1000, π/2+1/1000]. Corollary. The discrete unif maps f n and piecewise linear maps g n are L- quasi- conformal.

32 Classical theorems on convex surfaces Cauchy. If P,Q are cpt convex polytopes in R 3 s.t. Ǝ f: P à Q an isometry preserving cell structures, then P and Q differ by a rigid mo0on. Alexandrov. Any polyhedral metric on S 2 with K 0 is isometric to P for a compact convex polytope P in R 3. Pogorelov. If P,Q compact convex bodies in R 3 with isometric boundaries, then P and Q differ by a rigid mo0on.

33 Koebe- Andreev- Thurston theorem A simplicial triangula0on of a disk can be realized by a circle packing of the unit disk. Circle packing map Thurston s discrete Riemann mapping conjecture, Rodin- Sullivan s theorem Regular hexagonal circle packing K. Stephenson s picture