Minimal surfaces from self-stresses

Minimal surfaces from self-stresses Wai Yeung Lam (Wayne) Technische Universität Berlin Brown University Edinburgh, 31 May 2016 Wai Yeung Lam (Wayne) (TU Berlin) Discrete minimal surfaces 31 May 2016 1 / 26

Smooth minimal surfaces Soap films Mean curvature H = 0 Critical points of area Christoffel dual of Gauss map Wai Yeung Lam (Wayne) (TU Berlin) Discrete minimal surfaces 31 May 2016 2 / 26

Discrete differential geometry Discrete obects, represented by a finite number of variables Approximate smooth obects AND possess similar properties Goal: a discrete theory with rich mathematical structures, in such a way that the classical smooth theory arises in the limit of refinement of the discrete one. Wai Yeung Lam (Wayne) (TU Berlin) Discrete minimal surfaces 31 May 2016 3 / 26

Holomorphic quadratic differentials Definition Given a realization z : V C of a discrete surface M = (V, E, F), a function q : E int R satisfying for every interior vertex i q i = 0, q i /(z z i ) = 0. is called a discrete holomorphic quadratic differential. Wai Yeung Lam (Wayne) (TU Berlin) Discrete minimal surfaces 31 May 2016 4 / 26

Holomorphic quadratic differentials Theorem Let Φ : C C be a Möbius transformation (i.e. a fractional linear map). Then q is a holomorphic quadratic differential on z q is a holomorphic quadratic differential on Φ z. Proof. It suffices to consider the inversion in the unit circle at the origin w := Φ(z) = 1/z. We have q i /(w w i ) = z i q i z 2 i q i /(z z i ) = 0. Wai Yeung Lam (Wayne) (TU Berlin) Discrete minimal surfaces 31 May 2016 5 / 26

q is a holomorphic quadratic differential on z if and only if for every interior vertex i 0 = q i 0 = q i z z i 2 (z z i ) N : V S 2 the inverse stereographic proection of z: 1 N := 1 + z 2 Möbius invariance implies for every interior vertex i 2 Re z 2 Im z z 2 1. 0 = q i = k i N N i 2 0 = q i N N i 2 (N N i ) = k i (N N i ) where k i := q i / N N i 2. Wai Yeung Lam (Wayne) (TU Berlin) Discrete minimal surfaces 31 May 2016 6 / 26

Lemma If N 1, then 0 = k i (N N i ) = 0 = k i N N i 2 Proof. k i N N i 2 = 2k i ( N i 2 N, N i ) = 2k i N i N, N i = 0 Theorem There is a one-to-one correspondence between discrete holomorphic quadratic differentials q on z and functions k : E int R satisfying for every interior vertex k i (N N i ) = 0. We call k a self-stress of N. Wai Yeung Lam (Wayne) (TU Berlin) Discrete minimal surfaces 31 May 2016 7 / 26

A-minimal surfaces N : V S 2 k : E int R self-stress if for i V int k i (N N i ) = 0 f : V R 3 f φl f φr = k i (N N i ) Wai Yeung Lam (Wayne) (TU Berlin) Discrete minimal surfaces 31 May 2016 8 / 26

A-minimal surfaces Definition Given a discrete surface M and its dual M, a realization f : V R 3 of M is A-minimal with Gauss map N : V S 2 if for every interior edge {i} (N N i ) (f φl f φr ) = 0 where φ l, φ r are the left and the right face of e i. 1 Planar vertex stars (edges in asymptotic line direction) 2 Generalize the integrable systems approach by Bobenko and Pinkall. Wai Yeung Lam (Wayne) (TU Berlin) Discrete minimal surfaces 31 May 2016 9 / 26

Polarity Whiteley (1987) N : V S 2 bar-and-oint framework k : E int R self-stress if for i V int k i (N N i ) = 0 ˆN : V R 3 hinged sheetwork k : E int R self-stress if for i F int k i N i N = 0 k i r i (N i N ) = 0 Wai Yeung Lam (Wayne) (TU Berlin) Discrete minimal surfaces 31 May 2016 10 / 26

C-minimal surfaces ˆN : V R 3 with face normal N k : E int R self-stress if for i F int k i N i N = 0 k i r i (N i N ) = 0 f : V R 3 fφl fφr = k i N i N l α i i tan = 2 0 Wai Yeung Lam (Wayne) (TU Berlin) Discrete minimal surfaces 31 May 2016 11 / 26

C-minimal surfaces Definition Given a discrete surface M and its dual M, a realization f : V R 3 of M is C-minimal with Gauss map N : V S 2 if f has planar faces with face normal N and the scalar mean curvature H : F int R defined by H i := l i tan α i 2 i F int = Vint vanishes identically. 1 Planar faces (edges in curvature line direction) 2 Generalize the curvature approach by Bobenko, Pottmann and Wallner. Wai Yeung Lam (Wayne) (TU Berlin) Discrete minimal surfaces 31 May 2016 12 / 26

Conugate pairs of minimal surfaces Theorem (L 2015) Suppose M = (V, E, F) simply connected. Then A-minimal surfaces f : V R 3 1-1 C-minimal surfaces f : V R 3. Wai Yeung Lam (Wayne) (TU Berlin) Discrete minimal surfaces 31 May 2016 13 / 26

Example: Orthogonal circle patterns z : V(Z 2 ) C q i = { +1 on "horizontal" edges 1 on "vertical" edges q : E(Z 2 ) R is a discrete holomorphic quadratic differential on z. Wai Yeung Lam (Wayne) (TU Berlin) Discrete minimal surfaces 31 May 2016 14 / 26

Example: Orthogonal circle patterns Wai Yeung Lam (Wayne) (TU Berlin) Discrete minimal surfaces 31 May 2016 15 / 26

Example: Orthogonal circle patterns q : E int ±1 discrete holomorphic quadratic differential Discrete integrable systems Wai Yeung Lam (Wayne) (TU Berlin) Discrete minimal surfaces 31 May 2016 16 / 26

Example: Discrete harmonic functions q is a holomorphic quadratic differential on a triangular mesh z : V C if for i V int 0 = q i = k i z z i 2 0 = q i /(z z i ) = k i (z z i ) where k i = q i / z z i 2. Wai Yeung Lam (Wayne) (TU Berlin) Discrete minimal surfaces 31 May 2016 17 / 26

Example: Discrete harmonic functions Crapo-Whiteley (1994) {k : E int R k i (z z i ) = 0 i} proection of u : V R k i (z z i ) = i(grad u ik grad u il ) How about k i z z i 2? Answer: k i z z i 2 = (cot ki + cot il)(u u i ) where {ik}, {il} are the two faces sharing {i}. Wai Yeung Lam (Wayne) (TU Berlin) Discrete minimal surfaces 31 May 2016 18 / 26

Example: Discrete harmonic functions Definition Given a non-degenerate realization z : V C of a triangulated surface, a function h : V R is a discrete harmonic function in the sense of the cotangent Laplacian if (cot ki + cot il)(h h i ) = 0 i V int where {ik} and {il} are two neighboring faces containing the edge {i}. Theorem (L-Pinkall 2016) Given a simply connected triangulated surface M and z : V C. Then holomorphic quadratic differentials 1 1 discrete harmonic functions modulo linear functions Wai Yeung Lam (Wayne) (TU Berlin) Discrete minimal surfaces 31 May 2016 19 / 26

Weierstrass representation Theorem (L. 2015) Given M simply connected and z : V C. Then for every discrete holomorphic quadratic differential q, there exists F : V C 3 such that {i} E F ik F il = q i i(z z i ) 1 z i z i(1 + z i z ) z i + z Furthermore, Re(F) is A-minimal and Im(F) is C-minimal. The converse also holds.. Re(F ik F il ) = k i (N N i ) Im(F ik F il ) = k i (N i N ) Wai Yeung Lam (Wayne) (TU Berlin) Discrete minimal surfaces 31 May 2016 20 / 26

Question: Are our discrete minimal surfaces critical points of the total area? Answer: Not all, but those possessing discrete integrable structures, e.g. from orthogonal circle patterns. Wai Yeung Lam (Wayne) (TU Berlin) Discrete minimal surfaces 31 May 2016 21 / 26

Total area of a discrete surface Given a polygon γ = (γ 0, γ 1,..., γ n = γ 0 ) in R 3, its area vector is defined by Aγ = 1 2 Area of γ:= ± Aγ n 1 γ i γ i+1. i=0 Aγ = the largest signed area over all orthogonal proections to planes. If γ is embedded on a plane, Aγ coincides with the usual notion of area. The total area of a discrete surface Area σ (f) = φ F σ φ A φ where σ : F ±1. Wai Yeung Lam (Wayne) (TU Berlin) Discrete minimal surfaces 31 May 2016 22 / 26

Total area of a discrete surface Theorem (L. 2015) Let N : V S 2 be a P-net and f θ : V R 3 the associated family of minimal surfaces. We consider Area σ (f θ ) with σ := N, A/ A. Then the gradient of Areaσ at f θ vanishes. = critical points of the total area. Wai Yeung Lam (Wayne) (TU Berlin) Discrete minimal surfaces 31 May 2016 23 / 26

Discrete minimal surfaces Christoffel dual of Gauss map Mean curvature H = 0 Critical points of area Bobenko, Pinkall (96) Schief (03), Bobenko, Pinkall, Polthier (93) Pottmann, Wallner (10) In fact, there is a unified theory in terms of discrete holomorphic quadratic differentials. Wai Yeung Lam (Wayne) (TU Berlin) Discrete minimal surfaces 31 May 2016 24 / 26

References W. Y. Lam. Discrete minimal surfaces: critical points of the area functional from integrable systems (2015). arxiv: 1510.08788. Others W. Y. Lam and U. Pinkall. Isothermic triangulated surfaces. Math. Ann. (2016). W. Y. Lam and U. Pinkall. Holomorphic vector fields and quadratic differentials on planar triangular meshes. In: Advances in Discrete Differential Geometry. Ed. by A. I. Bobenko. Springer Berlin, 2016. Wai Yeung Lam (Wayne) (TU Berlin) Discrete minimal surfaces 31 May 2016 25 / 26