1. 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

2. Polyhedral surfaces (S, T, L ) Isometric gluing of E 2 (Euclidean) triangles along edges We also use S 2 or H 2 triangles.

3. Eg. Boundary of generic convex polytopes in E 3, S 3, H 3. (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(e i )+L(e j ) > L(e k ) In S 2 case, we add that the sum of three lengths < 2π.

4. Curvature Def. The curvature of (S,T, L) is K: V→ R Polyhedral metrics ↔ Riemannian metrics Z ↔ R

5. 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 H 2 or E 2 or S 2 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)

6. 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)

7. Thm (Thurston,1978 ). A E 2 or H 2 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.

8. Inversive distance inversive distance I(C,C’) between circles C, C’ is I(C,C’)=(l 2 -r 2 -R 2 )/(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.

9. 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.

10. Variational Principles (VP) on triangulated spaces Basic example of finite dim VP: F: {n-sided polygons in R 2 } → R F(P)= area(P) / length 2 (∂P) maximum of F are the regular n-gons. This is 1.5-dim. We are interested in the 2-dimensional analogy of above.

11. Schlaefli (1858): for a tetrahedron, w = ∑a i dl i is closed and S(l) = ∫ l w satisfies ∂S/∂l i =a i Variational Principles (VP) on triangulated 3-mfds Regge calculus, discrete general relativity (1962) (M 3, 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.

12. due to Schlafli: ∂S/∂L 1 =a i W(L) = ∑ t S(t) - 2π ∑ e L(e) ∂W/∂L 1 = a 1 +a 2 +…+a k – 2π = -K(e 1 ) a 1, a 2,…., a k are dihederal angles at e 1 K: E → R is the curvature. grad(W) = -K Thm (Regge): Critical points of W(L) are flat metrics.

13. A 2-D Schlaefli: Colin de Verdiere (1991): w=∑ a i du i is closed, F(u)=∫ u w concave in u and ∂F/∂u i =a i u i =ln(r i )

14. Colin de Verdiere’s variational proof of Thurston’s thm Given (S, T), for u: V → R, define r: V→R by r(v) = e u(v). W(u) = ∑ t F(t, u)-2π∑ i u i, sum over all triangles t and all u i ’s. W: R V → R is concave s.t., ∂W/∂u 1 =a 1 +…+a k -2π grad(W) = -K Injectivity Lemma If U open convex in R n, W: U → R is C 1 strictly convex, then grad(W): U → R n is 1-1. W restricted to P={ u | ∑ u i =0} is strictly concave so r to K is 1-1.

15. Cohen-Kenyon-Propp (2001). For E 2 triangles w= ∑ a i du i is closed and F(u) = ∫ u w is locally convex. the domain of F(u) ={ u | e u i +e u j >e u k } is NOT convex in R 3. The injectivity lemma applies locally only.

16. If the injectivity lemma applies, then Cohne-Kenyon-Propp formula implies: Thm(Rivin) (1994). A E 2 polyhedral surface (S,T,L) is determined up to scaling by its φ 0 :E →R sending e to a+b. Eg. a i +b i determine tetra φ 0 is a new kind of curvature. Curvatures in PL = quantities depending on inner angles.

17. Q: Can you find all 2D Schlaefli formulas? Thm 2. For E 2 triangles, all 2D schlaefli are (up to scaling) integrations of the closed 1-forms for some λ ϵ R, (1)∫ w λ, w λ = ∑ i (∫ a i sin λ (t) dt /l i λ+1 ) dl i (2) ∫ u λ, u λ = ∑ i (∫ a i cot λ (t/2)dt/r i λ+1 )dr i 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 S 2, H 2 triangles.

18. New curvatures Let λ ϵ R. For E 2, or S 2, or H 2 polyhedral metric (S, T, L), define discrete curvatures k λ, ψ λ, φ λ as follows: φ λ (e) = ∫ a π/2 sin λ (t) dt + ∫ b π/2 sin λ (t) dt ψ λ (e)= ∫ 0 (a-x-y)/2 cos λ (t) dt + ∫ 0 (b-z-w)/2 cos λ (t) dt k λ (v) = (4-m)π/2 - Σ a ∫ a π/2 tan λ (t/2) dt where a’s are angles at the vertex v of degree m.

19. Examples k λ (v) = (4-m)π/2 -Σ a ∫ a π/2 tan λ (t/2) dt K 0 = classical K = 2π –angle sum at v φ λ (e) = ∫ a π/2 sin λ (t) dt + ∫ b π/2 sin λ (t) dt φ 0 ( e ) = a+b-π : E → R (Rivin) φ -2 (e) = cot(a) + cot(b): E →R discrete cotangent Laplacian operator φ 1 (e) = cos(a) + cos(b), φ -1 (e) = tg(a/2) + tg(b/2) ψ 0 –curvature was introduced by G. Leibon 2002

20. Thm 3. For any λ ϵ R and any (S, T), (i)a E 2 or H 2 (tangential) CP metric on (S, T) is determined up to scaling by its k λ. (ii) a E 2 or S 2 polyhedral metric on (S, T) is determined up to scaling by its φ λ curvature. (iii) a H 2 polyhedral metric on (S, T) is determined by its ψ λ 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.

21. Corollary 4. (a) (Guo-Gu-L-Zeng ) Discrete Laplacian determines E 2 polyhedral metric (S,T,L) up to scaling. (b)Discrete Laplacian determines S 2 polyhedral metric (S,T,L). RM. We don’t know the answer for H 2 polyhedral metrics. Eg. A E 2 tetrahedron is determined up to scaling by any of the following six tuples: i=1,…,6. (…, a i +b i,.. ) (Rivin) 0 (.., cos(a i )+cos(b i ),…) 1 (.., tg(a i /2)+tg(b i /2),…) -1 (.., cot(a i )+cot(b i ),..) -2

22. Proofs of thm 1, 3 use variational principles (VP) Thm 3 uses VP from them 2 Thm 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.

23. Key observation All 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.

24. Cohen-Keynon-Propp’s VP and its convex extension w =∑ a i dx i closed w defined on Ω ={ x | } which is not convex in R 3 Lemma. ∆ 3 ={l ϵ R 3 >0 | l i +l j >l k } = the space of all E 2 triangles. Then a 1 : ∆ 3 → R can be extended to a C 0 -smooth a 1 *: R 3 >0 →R s.t., a 1 * is constant on each component of R 3 >0 -∆ 3. Pf.

25. The extension Extending w from Ω to R 3 by w* = ∑ a* i dx i, C 0 -smooth 1-form w* is closed: ∫ ᵟ w*=0 F*(x) = ∫ x w* is well defined. (a) F* is C 1 -smooth (b) F* is locally convex on Ω (Cohen-Kenyon-Propp) (c) F* is linear on each component W of R 3 -Ω since grad(F) on W is a constant by the construction (a)+(b)+(c) imply F* is convex in R 3.

26. Happy Birthday Hyam!

27. Q. Find all non-constant functions W(z 1,z 2,z 3 ), f(t), g(t) so that for all E 2 triangles, We have also proved that Schlaefli in 3D is unique up to scaling in the above sense.