1. Discrete conformal geometry of polyhedral surfaces
Feng Luo
Rutgers University, USA
Discretization in Geometry and Dynamics
Döllnsee-Schorfheide, Templin, Germany
Joint work with
D. Gu (Stony Brook), J. Sun (Tsinghua Univ.), and T. Wu (Courant)
Oct. 8, 2018

2. then T1 and T2 differ by a complex linear map.
A simple problem
topologically isomorphic
T1 and T2 are two hexagonal Delaunay triangulations of the plane. T1 T2
infinite triangulations
Problem. If (C, T1, l1) and (C, T2 , l2 ) are related by a vertex scaling,
then T1 and T2 differ by a complex linear map.
This is a Cauchy rigidity type problem.
∃ λ: {vertices} → R>0 s.t.,
l1(uv) = λ(u) λ(v) l2(uv)
i.e., the same length cross ratio at corresponding pairs of edges.

3. Recall Riemann surfaces and differential geometry
Uniformization Thm (Poincare-Koebe 1907) ∀ Riemannian metric g on a surface S, Ǝ λ: S → R>0 s.t., (S, λg) is a complete metric of curvature -1, 0, 1.
Key step: if S is simply connected, the (S, g) is conformal to S2, C, or D.
Koebe (1908): If the genus of S is zero, then (S, g) is conformal
to a domain (connected open set) in the S2.
Koebe conjecture (1908): Every domain in C is conformal to circle domain C-X s.t. connected components of X are round disks and points (i.e., circle type closed set).
Riemann mapping
circle type closed set X
circle domain
True: if it has finitely many (Koebe) or countably many (He-Schramm) boundary components.

4. Weyl Problem (1915): A metric g of positive Gaussian curvature on S2 is isometric the
boundary of a compact convex set in R3.
Levy, Alexandrov, Pogorelov and Nirenberg
Question: What about the Weyl problem in H3 for non-compact convex surfaces?
CH(Y) convex hull in H3
Known:
convex surfaces ∂CH(Y) in H3 have curvature at least -1 and genus zero.
Thurston: for any closed set Y in ∂H3, ∂CH(Y) is complete and curvature -1 (hyperbolic). Y
Conj. (L-Sun-Wu): ∀ complete hyperbolic metric on a (non-cpt) genus
zero surface is isometric to ∂CH(X) where X is a circle type closed set in ∂H3.
this conjecture, discrete uniformization, and Koebe conjecture, progresses we made.

5. Isometries of C are naturally isometries of H3.
Hyperbolic 3-space H3 and convex hull
H3=C X R>0
Geodesics:
Planes and half spaces:
Convex hull in H3 of a subset X in C:
removing maximal half-spaces missing X
Isometries of C are naturally isometries of H3.
Ideal triangles:
convex hull in H3 of {a,b,c} in ℂ . X

6. Polyhedral surface
PL metric d on (S,V) is a flat cone metric, cone points in V.
Isometric gluing of E2 triangles along edges: (S, T, l ). Curvature K=Kd: V →R, K(v)= 2π-sum of angles at v = 2π- cone angle at v
triangulation
K(v)<0
A triangulated PL metric (S, T, l)
is Delaunay: a+b ≤π at each edge e.
K(v)>0
a+b ≤ π
Delaunay triangulation always exists on (S, V, d).

7. a hyperbolic metric d* on S-V.
discrete conformality: PL surface (S,V, d) and hyperbolic metric d* on S-V
Given a PL metric d on (S,V), produce a Delaunay triangulation T of (S,V,d) t s
∀ t  T is associated an ideal hyperbolic triangle t* t
If t, s  T glued by isometry f along e, then t* and s* are glued by the same f* along e*. t* s*
a hyperbolic metric d* on S-V.
Def. (G-L-S-W). Two PL metrics d1, d2 on (S,V) are discrete conformal (d.c.)
iff d1* are d2* are isometric by an isometry homotopic to id on S-V.
Motivated by the important work of Bobenko-Pinkall-Springborn.
Equivalent to the previous def. using vertex scaling and Delaunay condition.

8. a+b≤π H3 dst* = ∂CH(V) dst =standard flat metric on (C, V)
An important example: (C, V, dst) (V discrete infinite set, infinite triangulations)
dst* H3
= ∂CH(V)
Delaunay triangulations
dst =standard flat metric on (C, V)
Boundary of hyperbolic convex hull of V =
circumcircles are maximum disks missing V
a+b≤π
Delaunay triangulation =
projection of boundary triangulation of
the hyperbolic convex hull

9. unique, up to scaling, a PL metric d* of constant curvature .
Thm 1. (Gu-L-Sun-Wu). ∀ PL metric d on a closed (S,V) is discrete conformal
unique, up to scaling, a PL metric d* of constant curvature
2𝜋𝜒 𝑆 𝑉
Question 1. Do the dis. unif. metrics d* converge to the smooth Poincare metric?
U. Bücking,
the torus case by Gu-L-Wu
Question 2. Non-compact infinite triangulated polyhedral surfaces?
Especially, simply connected ones?

10. Why dis. conf. geom. of infinite triangulated PL surfaces?
Needed to prove convergence of discrete conformal maps to conformal maps. .
Convergence of fn → Riemann mapping F?
quasi-conformality implies fn → G.
PL approximations fn
Is G conformal?
discrete uniformization thm
Riemann mapping F
Problem. A Delaunay hexagonal triangulation of C, dis.conf. to the equilateral
hexagonal triangulation, is unique up to complex linear maps.
ANS: Yes, L-Sun-Wu, Dai-Ge-Ma

11. Discrete uniformization conjecture
Discrete uniformization for non-cpt simply connected PL surfaces (S, V)
Uniformization Thm. Ɐ Riemannian metric d, (S,d) is conformal to C or D.
Discrete uniformization conjecture
Every PL surface (S,V,d) is d.c. to a unique (C, V’, dst) or (D, V’, dst).
Associated hyperbolic metric
Weyl’s problem
(S-V, d*) complete hyperbolic w/ cusp ends at V
isometric to a unique
CH(V’) in H3
CH(V’ ꓴ (S2-D))

12. a circle type closed set X.
Geometry of convex hulls in H3 and conjectures
Thurston If Y closed in S2, then CH(Y)  H3 is complete hyperbolic.
Eg.  simply connected domain in S2, Y=S2-, |Y|>1, CH(Y) isometric to H2.
Thurston’s isometry
convex hull & hyperbolic geom. 
Riemann mapping, conformal geom.
Q. Not simply connected  ?
Koebe Conj. ∀ domain  in S2 is conformal to a circle domain S2-X .
Conj (L-S-W) 1. ∀ complete hyperbolic surf (, d) of genus 0 is isometric to CH(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. CH(X) isometric CH(Y) ,
then X, Y differ by a Moebius transf.

13. for a circle type closed set X.
Conj (L-S-W) 1. ∀ complete hyperbolic surf (, d) of genus 0 is isometric to CH(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. CH(X) isometric CH(Y) ,
then X, Y differ by a Moebius transformation.
Known
1. Rivin:
2. Schlenker:
3. L-Tillmann:
Thm (L-Wu) Koebe conjecture implies Conjecture 1.
Combining the work of He-Schramm on Koebe conjecture,
Corollary (L-Wu) 3. ∀ simply connected PL surface (S,V,d) is d.c. to (C, V’, dst) or (D, V’, dst).

14. Sketch of proof of Thm 2 KC implies Conj. 1.
Goal: ∀ circle domain C-X with Poincare metric dX, find another circle domain C-Y
s.t., (C-X, dX) is isometric to CH(Y).
Step 1. Take circle type closed sets Xn converging to X in Hausdorff metric s.t. each
Xn has only finitely many connected components and
components of ∂Xn are circles.
Step 2. By Schlenker’s theorem, find a circle type closed set Yn s.t.
Ǝ isometry Fn: (C-Xn, dn) → CH(Yn). Normalize Yn.
Step 3. May assume that Yn converges in Hausdorff metric to a closed set Y.
Easy to show (C-X, dX) is isometric to CH(Y) using Alexandrov’s work.
Claim: Each connected component of Y is the Hausdorff limit of
connected components of Yn’s.

15. Step 4 (Main) which implies the Claim.
Prop. The sequence Fn is equicontinuous from (C-Xn, dE) to (CH(Yn), dE)
in the Klein model where dE is Euclidean metric.
If not, use Schramm’s transboundary extremal length and Schramm’s theorem for
some curve families whose extremal lengths tending to zero on circle domains.
But Fn is conformal, so the corresponding curve families in CH(Yn) has extremal lengths tending to zero.
However, this is not the case mainly due to:
The Euclidean area of any convex surface
in the Poincare model of H3 is at most 16π.
2. The map g(x) = 2𝑥 1+ 𝑥 sending the Poincare model
to the Klein model is 2-Lipschitz:
‖ g(x) –g(y)‖ ≤ 2 ‖x-y‖.

16. Thank you.