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Discrete conformal geometry of polyhedral surfaces Discrete conformal geometry of polyhedral surfaces

Discrete conformal geometry of polyhedral surfaces - PowerPoint Presentation

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Discrete conformal geometry of polyhedral surfaces - PPT Presentation

Feng Luo Rutgers University D Gu Stony Brook J Sun Tsinghua Univ and T Wu Courant Oct 12 2017 Geometric Analysis Roscoff France ID: 756265

discrete metric thm hyperbolic metric discrete hyperbolic thm closed conformal isometric conj curvature set convex complete uniformization connected surface

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Slide1

Discrete conformal geometry of polyhedral surfaces

Feng Luo Rutgers University

D. Gu (Stony Brook), J. Sun (Tsinghua Univ.), and T. Wu (Courant)

Oct. 12, 2017

Geometric Analysis,

Roscoff

, France

Joint work with Slide2

Koebe

conjecture (1908): Every open connected set U in C is conformal toan open connected set in

C whose boundary components are circles and points.

True: if U has finitely many (

Koebe

) or countably many (He-Schramm) boundary components.

Weyl problem:

A metric of positive Gaussian curvature on

S2 is isometric the boundary of a convex body in R3.

Q1 (L-Sun-Wu): A complete path metric on U of curvature ≥ -1 is isometric to a complete convex surface S in H3 s.t. components of ∂S are circles and points in ∂H3.

Conj. (L-Sun-Wu): Every complete hyperbolic metric on U is isometric to ∂CH(X) where X is a closed set in ∂H3 whose components are round disks and points.

Brock-Dumas

Ω

=

S

2-X

∂CH(X) where X is a closed set in ∂H3

is hyperbolic.

X

Alexandrov

ThurstonSlide3

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.

Q2: Is there discrete unif. thm. for polyhedral surfaces? Does it converge (to smooth case)?

Q1: Can one compute the uniformization maps? ANS (Gu-L-Sun-Wu): yes

λg is conformal to gQ3: What is the discrete conformal equivalence for PL metrics?Slide4

Isometric gluing of E

2 triangles along edges: (S, T, l

).

Curvature

K=

K

d

: V →R, K(v)= 2π

-sum of angles at v = 2π- cone angle at v

A triangulated PL metric (S,

T, l

) is Delaunay:

a+b ≤π at each edge e.

Gauss-Bonnet

K(v)>0

K(v)<0

triangulation

(Closed surfaces)

Polyhedral surface

Delaunay triangulation always exists on (S, V, d).

PL metric d on (S,V) is a flat cone metric, cone points in V.

a+b

πSlide5

The hyperbolic 3-space H3

=C X

R>0Geodesics

:Planes and half spaces:

Ideal triangles:

convex hull in

H

3 of {a,b,c} in

C. Convex hull in H3 of a subset X in C

:removing maximal half-spacesmissing X Isometries

of H3 = Mobius transformations of C. Isometries of

C are naturally isometries of H

3. Hyperbolic 3-space and convex hull

XSlide6

PL surface

(S,V, d) and hyperbolic metric d* on S-V

Def. (G-L-S-W

). Two PL metrics d1, d

2

on (S,V) are

discrete conformal iff

d1* are d2* are isometric by an isometry homotopic to id on S-V. Given a PL metric d

on (S,V), produce a Delaunay triangulation 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.

First Definition: geometrical

t

t

s

The work of

Bobenko-Pinkall-Springborn

relates Euclidean triangle to decorated ideal triangle. Slide7

An important example

Boundary of hyperbolic convex hull of Vdst

=standard flat metric on (C , V

)

d

st

* = ∂CH

(V)H3

Delaunay triangulations

circumcircles are maximum disks

a+b

π

Slide8

vertex scaling

Same triangulation, scale edge lengths from vertex weights

Def. (

Vertex scaling, L 2004

) Given

λ

: V →

R>0 and l : {edges} →

R, λ*l(uv) = λ(u) λ(v)

l(uv).

 

u

v

.

(Gu-L-Wu, 2015)

 

prototype of discrete conformality

Second definition: analyticalSlide9

A variational principle

Prop (L, 2004).

 

Hence f(u) =

a

i

du

i

is locally concave such that

 

Then

Thm

(

Bobenko-Pinkall-Springborn

, 2010).

f can be extended to a concave function on

R

3 and is explicit.

Corollary (BPS, 2010). If l

and u*l are two PL metrics on (S,

T

, V) with the same discrete curvature, then u ≡ c.

However, given

l

on

T

, there are in general no constant curvature metrics of the form u

*

l

.

3x3

is

negative semi-definite.

 

Pf. The map sending dis. conf. factor u to the curvature is the gradient

 Slide10

Answer is no in general.

Triangulation independence ?

Def.(G-L-S-W) .

Two PL metrics on a closed surface (S,V) are discrete conformal iff they

are related by a sequence of vertex scaling and flip operations on

Delaunay triangulations

.

Q: is (T1

, l3)=(T1, w*l1)?

isometries

But yes if all triangulations are Delaunay.

Thm (GLSW). These two definitions of discrete conformality are equivalent.Slide11

Thm (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.

,

d

# is discrete conformal to d,the

discrete curvature of d# is K#.For K#= 2πχ(S)/|V|, d#

is a discrete uniformization metric. Eg.2. Any PL metric on (S1XS1,V) is d.c. to a unique flat (S

1XS1, V, d#) where K#=0 (Fillastre

2008).

Thm

(

Fillastre

2008) Every cusped hyperbolic puncture torus is isometric the boundary of the convex hull of a finite set of points in a Fuschian hyperbolic 3-manifolds. Slide12

Convergence

Let (

S,d) be a closed Riemannian surface of genus g, λd is the uniformization metric.

Goal: compute λd.

A sequence of triangulations (S,

T

n

, ln) is regular if there exist δ>0, C

>0, qn →0 s.t. (1) all angles in Tn are in (δ,

- δ), (2) all lengths of edges in Tn are in (

qn).

Thm (Gu-L-Wu). If (S, Tn, ln

) is a regular sequence of triangulated polyhedral tori approximating a Riemannian torus (S,d) and (S, Tn’, ln’) is the flat polyhedral torus discrete conformal to (S,

Tn, ln) of area 1, then (S, Tn’, l

n’) converges uniformly to the uniformization metric λd associated to (S, d). Conj. It should be true for all closed Riemannian surfaces.

 Slide13

Discrete uniformization for non-cpt

simply connected PL surfaces (S, V) Discrete uniformization conjecture

Every PL surface (S,V,d) is d.c.

to a unique (C, V’,

d

st

) or (

D, V’, dst).

Uniformization Thm. Ɐ Riemannian metric d, (S,d) is conformal to C or D.

Associated hyperbolic metric

(S-V, d) complete hyperbolic w/ cusp ends at VCH(V’) in H3

isometric

Weyl’s problem

CH(V’

ꓴ (S2-D)) Slide14

Conj (L-S-W) 1. ∀ complete hyperbolic surf (, 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. CH(X) isometric CH(Y) ,

then X, Y differ by a Moebius transf. X is of circle type

Thurston. If Y closed in S2, then CH(Y)  H3

is complete hyperbolic. Eg.  simply connected domain in C, Y=S2-

. Then C

H(Y) isometric to H2.

Riemann mapping,

conformal geom.

Thurston’s isometry

Geometry of convex hulls in

H

3 and conjectures

Koebe Conjecture. Every domain 

in S2 is conformal to S2

-

X

s.t.

, connected components of

X

are points or round disks.

convex hull geom.

Question

: not simply connected

?

Alexandrov

.

complete hyperbolic surf (

,d) of genus 0 is isometric to

C

H

(Y).Slide15

Conj (L-S-W) 1. ∀ complete hyperbolic surf (, 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. CH(X) isometric CH

(Y) , then X, Y differ by a Moebius transformation.Thm (Rivin). Conj. 1&2 hold for (∑, d) of finite area. (X = finite set).

Thm (Schlenker). Conj. 1 & 2 hold for (∑, d) of finite topology, no cusp ends. (X = finite union of disks).Thm (L-Tillmann

). Conj. 1 & 2 hold for (∑,d): (X = a union of one disk and a finite set).Thm (L-Wu) . Conj. 1 holds for surfaces ∑ with countably many topological ends.

Thm (L-Wu) . Koebe conjecture implies Conjecture 1.

This implies the existence part of discrete uniformization conjecture.Slide16

Thm (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.

,

d

# is discrete conformal to d,the

discrete curvature of d# is K#.Slide17

Sketch of proof thm

Step 1. There exists a c1

-smooth map φ: {PL metrics d on (S,V)}/

~ → Teich(S-V) s.t.

,

φ(d)=

φ

(d’) iff d and d’ are discrete conformal. ~ = isometry homotopic to identity

Step 2. for any PL metric d on (S,V) P = {[d’ ]| d’ disc. conf. to d} /~ ≈ RV.Step 3. The discrete curvature map

K: P/R>0 __> (-∞, 2π)V ∩ {Gauss-Bonnet equation} is 1-1, onto.

(GB: x Є RV, ∑v ЄV x(v) = 2π χ (S).)We prove: K is smooth, locally 1-1 (a

variational principle), image of K is closed (degeneration analysis+ Akiyoshi).Slide18

Thank you.