Variational Principles and Rigidity on Triangulated Surfaces

Feng Luo Rutgers University

Geometry & Topology Down Under The Hyamfest Melbourne, Australia July 17-22, 2011 arXiv:1010.3284

Polyhedral surfaces (S, T, L)

Isometric gluing of E2 (Euclidean) triangles along edges

We also use S2 or H2 triangles.

Eg. Boundary of generic convex polytopes in E3, S3, H3.

(S, T) =triangulated surface E = all edges in T V= all vertices in T

A polyhedral metric L on (S,T) = edge length function L : E → R s.t., L(ei)+L(ej) > L(ek) In S2 case, we add that the sum of three lengths < 2π.

Curvature

Def. The curvature of (S,T, L) is K: V→ R

Polyhedral metrics ↔ Riemannian metrics Z ↔ R

Problem: Relationship among metric L, curvature K, topology et al.

Eg. 1. Gauss-Bonnet:

Eg. 2. Under what condition does K determine the metric L?

Eg. 3. Given (S,T), is T a geometric triangulation? i.e., find H2 or E2 or S2 metrics on (S, T) with K=0. (discrete uniformization)

Eg. 4. What is the meaning of conformality of (S,T,L) and (S, T, L’) ? (discrete Riemann surface)

Eg. 5. Given K*:V ->R, find L: E→ R>0 with K* as its curvature. (prescribing curvature problem, shape design in graphics).

Eg. 6. What does the Laplace operator tell us about (S,T,L)? (discrete spectral geometry)

Example: Thurston’s circle packing (CP)

A (tangential) circle packing (CP) on (S,T) is r: V → R>0.

The edge length L: E → R is given by L(uv) = r(u) + r(v)

Thm(Thurston,1978). A E2 or H2 CP metric on (S,T) is determined determined up to scaling by its curvature K.

Use of CP: calculate Riemann map.

Images supplied by D. Gu working with S.T. Yau.

Bowers-Stephenson first used CP for brain imaging.

Inversive distance

• inversive distance I(C,C’) between circles C, C’ is I(C,C’)=(l2-r2-R2)/(2rR)

I(C,C’) is invariant under Mobius transformation.

• I(C,C’) in (-1,1)

• I(C,C’) =1

• I(C,C’) > 1

Bowers-Stephenson suggested using disjoint circle packing for applications.

Bowers-Stephenson Conjecture (2003)

Given (S,T), CP’s on (S,T) with given inversive distance I:E→[1,∞) are determined by their curvature K up to scaling.

Thurston, Andreev: CP’s with given inversive distance I: E →[0,1] are determined by K.

Thm 1. Given (S,T) and I: E -> [0, ∞), then CP’s on (S,T) with given inversive distance I are determined by curvature K up to scaling.

Variational Principles (VP) on triangulated spaces

Basic example of finite dim VP:

F: n-sided polygons in R2 → R

F(P)= area(P) / length2(∂P)

maximum of F are the regular n-gons.

This is 1.5-dim.

We are interested in the 2-dimensional analogy of above.

Schlaefli (1858): for a tetrahedron, w = ∑ai dli is closed and

S(l) = ∫l w satisfies

∂S/∂li=ai

Variational Principles (VP) on triangulated 3-mfds

Regge calculus, discrete general relativity (1962)

(M3, T) triangulated 3-manifold a polyhedral metric L: E → R>0 Einstein action W(L) = ∑

t S(t) - 2π ∑

e L(e)

sum over all tetra t and edges e.

due to Schlafli: ∂S/∂L1=ai

W(L) = ∑t S(t) - 2π ∑

e L(e)

∂W/∂L1 = a1+a2+…+ak – 2π = -K(e1)

a1, a2,…., ak are dihederal angles at e1

K: E → R is the curvature.

grad(W) = -K Thm (Regge): Critical points of W(L) are flat metrics.

A 2-D Schlaefli: Colin de Verdiere (1991):

w=∑ ai dui is closed,

F(u)=∫u w concave in u and

∂F/∂ui=ai

ui=ln(ri)

Colin de Verdiere’s variational proof of Thurston’s thm

Given (S, T), for u: V → R, define r: V→R by r(v) = eu(v) .

W(u) = ∑tF(t, u)-2π∑i ui, sum over all triangles t and all ui’s.

W: RV → R is concave s.t., ∂W/∂u1=a1+…+ak-2π

grad(W) = -K

Injectivity Lemma If U open convex in Rn, W: U → R is C1 strictly convex, then grad(W): U → Rn is 1-1.

W restricted to P= u | ∑ ui =0 is strictly concave

so r to K is 1-1.

Cohen-Kenyon-Propp (2001). For E2 triangles

w= ∑ ai dui is closed and F(u) = ∫u w is locally convex.

the domain of F(u) = u | eui+euj>euk is NOT convex in R3. The injectivity lemma applies locally only.

If the injectivity lemma applies, then Cohne-Kenyon-Propp formula implies:

Thm(Rivin) (1994). A E2 polyhedral surface (S,T,L) is determined up to scaling by its φ0 :E →R sending e to a+b.

Eg. ai+bi determine tetra

φ0 is a new kind of curvature.

Curvatures in PL = quantities depending on inner angles.

Q: Can you find all 2D Schlaefli formulas?

Thm 2 . For E2 triangles, all 2D schlaefli are (up to scaling) integrations of the closed 1-forms for some λ ϵ R,

(1) ∫ wλ, wλ = ∑i (∫ai sinλ(t) dt /li

λ+1 ) dli

(2) ∫ uλ, uλ = ∑i (∫ai cotλ(t/2)dt/ri

λ+1 )dri

Furthermore, these functions are locally convex/concave.RM. λ=0 corresponds to Colin de Verdiere and Cohne-Kenyon-Propp.RM. There are similar theorems for S2, H2 triangles.

New curvatures Let λ ϵ R. For E2, or S2, or H2 polyhedral metric (S, T, L), define discrete curvatures kλ, ψλ, φλ as follows:

φλ(e) = ∫aπ/2 sinλ(t) dt + ∫b

π/2 sinλ(t) dt

ψλ(e)= ∫0

(a-x-y)/2 cosλ(t) dt + ∫0

(b-z-w)/2 cosλ(t) dt

kλ (v) = (4-m)π/2 -Σa ∫aπ/2 tanλ(t/2) dt

where a’s are angles at the vertex v of degree m.

Examples

kλ (v) = (4-m)π/2 -Σa ∫aπ/2 tanλ(t/2) dt

• K0= classical K = 2π –angle sum at v

φλ(e) = ∫aπ/2 sinλ(t) dt + ∫b

π/2 sinλ(t) dt

• φ0 ( e ) = a+b-π : E → R (Rivin)

• φ-2(e) = cot(a) + cot(b): E →R discrete cotangent Laplacian operator

• φ1 (e) = cos(a) + cos(b), φ-1 (e) = tg(a/2) + tg(b/2)

ψ0 –curvature was introduced by G. Leibon 2002

Thm 3. For any λ ϵ R and any (S, T),

(i) a E2 or H2 (tangential) CP metric on (S, T) is determined up to scaling by its kλ.

(ii) a E2 or S2 polyhedral metric on (S, T) is determined up to scaling by its φλ curvature.

(iii) a H2 polyhedral metric on (S, T) is determined by its ψλ curvature.

RM 1. (iii) for λ=0 was a theorem of G. Leibon (2002). RM 2. We proved thm 3 (ii) and (iii) in 2007 under some assumptions on λ.

Corollary 4. (a) (Guo-Gu-L-Zeng) Discrete Laplacian determines E2 polyhedral metric (S,T,L) up to scaling.

(b) Discrete Laplacian determines S2 polyhedral metric (S,T,L).

RM. We don’t know the answer for H2 polyhedral metrics.

Eg. A E2 tetrahedron is determined up to scaling by any of the following six tuples: i=1,…,6. (…, ai+bi ,.. ) (Rivin) 0

(.., cos(ai)+cos(bi) ,…) 1

(.., tg(ai/2)+tg(bi/2) ,…) -1

(.., cot(ai)+cot(bi) ,..) -2

Proofs of thm 1, 3 use variational principles (VP)

Thm 3 uses VP from them 2Thm 1 uses VP discovered by R. Guo (2009).

Guo proved a local rigidity version of thm 1 using his VP.

The main problem: domain of the action function is not convex for some λ so the injectivity Lemma does not apply.

Key observationAll those locally convex/concave functions in thm 2 and Guo’s action function defined on non-convex open sets can be naturally extended to be convex/concave functions defined in open convex sets.

Thus injectivity lemma still applies.

Cohen-Keynon-Propp’s VP and its convex extension

w =∑ ai dxi closed

w defined on Ω = x | which is not convex in R3

Lemma. ∆3 =l ϵ R3>0

| li+lj >lk = the space of all E2 triangles. Then

a1: ∆3→ R can be extended to a C0-smooth

a1*: R3>0

→R

s.t., a1 * is constant on each component of R3>0

-∆3.

Pf.

The extension

Extending w from Ω to R3 by w* = ∑ a*

i dxi , C0-smooth 1-form

w* is closed: ∫ ᵟ w*=0

F*(x) = ∫x w* is well defined.

(a) F* is C1 -smooth (b) F* is locally convex on Ω (Cohen-Kenyon-Propp)

(c) F* is linear on each component W of R3-Ω since grad(F) on W is a constant by the construction

(a)+(b)+(c) imply F* is convex in R3.

Happy Birthday Hyam!

Q. Find all non-constant functions W(z1,z2,z3), f(t), g(t) so that for all E2 triangles,

We have also proved that Schlaefli in 3D is unique up to scaling in the above sense.