Ricci curvature on polyhedral surfaces via optimal transportation Benoît Loisel and Pascal Romon Abstract The problem of defining correctly geometric objects such as the curvature is a hard one in discrete geometry. In 2009, Ollivier [Oll09] defined a notion of curvature applicable to a wide category of measured metric spaces, in particular to graphs. He named it coarse Ricci curvature because it coincides, up to some given factor, with the classical Ricci curvature, when the space is a smooth manifold. Lin, Lu & Yau [LLY11], Jost & Liu [JL11] have used and extended this notion for graphs giving estimates for the curvature and hence the diameter, in terms of the combinatorics. In this paper, we describe a method for computing the coarse Ricci curvature and give sharper results, in the specific but crucial case of polyhedral surfaces. Keywords: discrete curvature; optimal transportation; graph theory; discrete lapla- cian; tiling. MSC: 05C10, 68U05, 90B06 1 The coarse Ricci curvature of Ollivier Let us first recall the definition of the coarse Ricci curvature as given originally by Ollivier in [Oll09]. Since our focus is on polyhedral objects, we will use, for greater legibility, the language of graphs and matrices, rather than more general measure theoretic formulations. In this section, we will assume that our base space X (V,E) is a simple graph (no loops, no Æ multiple edges between vertices), unoriented and locally finite (vertices at finite distance are in finite number) for the standard distance (the length of the shortest chain between x and x0): © ª x,x0 V, d(x,x0) inf n, n N¤, x1,...,xn 1, x x0 x1 xn x0 8 2 Æ 2 9 ¡ Æ » » ¢¢¢ » Æ arXiv:1402.0644v1 [math.DG] 4 Feb 2014 where x y stands for the adjacency relation and adjacent vertices are at distance 1. We shall » call this distance the uniform distance; the more general case will be considered in section4. Polyhedral surfaces (a.k.a. two-dimensional cell complexes) studied afterwards will be seen as a special case of graphs. We do not require our graphs to be finite, but we will nevertheless use vector and matrix notations (with possibly infinitely many indices). P For any x X , a probability measure ¹ on X is a map V R such that y V ¹(y) 1. 2 ! Å 2 Æ Assume that for any vertex x we are given a probability measure ¹x . Intuitively ¹x (y) is the 1 probability of jumping from x to y in a random walk. For instance we can take ¹x to be the 1 1 1 uniform measure on the sphere (or 1-ring) at x, Sx {y V, y x}, namely ¹ (y) if y x, Æ 2 » x Æ dx » where d S denotes the degree of x, and 0 elsewhere. A coupling or transference plan » x Æ j x j between ¹ and ¹0 is a measure on X X whose marginals are ¹,¹0 respectively: £ X X »(y, y0) ¹(y) and »(y, y0) ¹0(y0). Æ Æ y0 y Intuitively a coupling is a plan for transporting a mass 1 distributed according to ¹ to the same mass distributed according to ¹0. Therefore »(y, y0) indicates which quantity is taken from y and sent to y0. Because mass is nonnegative, one can only take from x the quantity ¹(x), no more no less, and the same holds at the destination point, ruled by ¹0. We may view measures as vectors indexed by V and couplings as matrices, and we will use that point of view later. The cost of a given coupling » is X c(») »(y, y0)d(y, y0) Æ y,y V 02 where cost is induced by the distance traveled. The Wasserstein distance W1 between proba- bility measures ¹,¹0 is X W1(¹,¹0) inf »(y, y0)d(y, y0) Æ » y,y V 02 where the infimum is taken over all couplings » between ¹ and ¹0 (such a set will never be empty and we will show that its infimum is attained later on). Let us focus on two simple but important examples: 1. Let ¹ be the Dirac measure ± , i.e. ± (y) equals 1 when y x and zero elsewhere. x x x Æ Then there is only one coupling » between ± and ± and it satisfies »(y,z) 1 if y x x x0 Æ Æ and z x0, and vanishes elsewhere. Obviously W (x,x0) d(x,x0). Æ 1 Æ 1 1 2. Consider now ¹ ,¹ the uniform measures on unit spheres around x and x0 respec- x x0 tively; then a coupling » vanishes on (y, y0) whenever y lies outside the sphere Sx or y0 outside Sx0 . So the (a priori infinite) matrix » has at most dx dx0 nonzero terms, and 1 we can focus on the dx dx submatrix »(y, y0)y Sx ,y S whose lines sum to and 0 0 x0 dx 1 £ 2 2 columns to d . For instance, » could be the uniform coupling x0 0 1 1 1 1 ¢¢¢ ¢¢¢ B . C » @ . A Æ dx dx 0 1 1 ¢¢¢ ¢¢¢ where we have written only the submatrix. 3. A variant from the above measure is the measure uniform on the ball B {x} S . x Æ [ x 1 Ollivier’s coarse Ricci curvature between x and x0 (which by the way need not be neighbors) measures the ratio between the Wasserstein distance and the distance. Precisely, we set W ( 1 , 1 ) 1 1 ¹x ¹x · (x,x0) 1 0 Æ ¡ d(x,x0) ¢ 1Also called Wasserstein curvature. 2 1 1 Since ¹x is the uniform measure on the sphere, then · compares the average distance between the spheres Sx and Sx0 with the distance between their centers, which indeed depends on the Ricci curvature in the smooth case (see [Oll09, Oll10] for the analogy with riemannian manifolds which prompted this definition). We will rather use the definition of Lin, Lu & Yau [LLY11] (see also Ollivier [Oll10]) for a t smooth time variable t: let ¹x be the lazy random walk 8 1 t if y x t < ¡t Æ ¹ (y) if y Sx x Æ dx 2 : 0 otherwise t 1 so that ¹x (1 t)±x t¹x interpolates linearly between the Dirac measure and the uniform Æ ¡ Å W (¹t ,¹t ) 2 t 1 x x t measure on the sphere . We let · (x,x0) 1 0 , and · (x,x0) 0 as t 0. We then Æ ¡ d(x,x0) ! ! set t · (x,x0) r ic(x,x0) liminf Æ t 0 t ! and we will call r ic the (asymptotic) Ollivier–Ricci curvature. The curvature r ic is attached to a continuous Markov process whereas ·1 corresponds to a time-discrete process3. Lin, Lu & Yau [LLY11] prove the existence of the limit r ic(x,x0) using concavity properties. In the next section, we give a different proof by linking the existence to a linear programming problem with convexity properties. The relevance of such a definition comes from the analogy with riemannian manifolds but can also be seen through its applications, e.g. the existence of an upper bound for the diameter of X depending on r ic (see Myers’ theorem below). 2 A linear programming problem In the case of graphs, the computation of W1 is surprisingly simple to understand and imple- t t t ment numerically. Recall that a coupling » between ¹x and ¹x is completely determined by a 0 t (dx 1) (dx0 1) submatrix, and henceforward we will identify » with this submatrix. A cou- Å £ Å (dx 1)(d 1) pling is actually any matrix in R Å x0 Å with nonnegative coefficients, subject to the follow- t t t ing t-dependent linear constraints: y B , » ,L ¹ (y) and y0 B , » ,C ¹ (y0), 8 2 x h y i Æ x 8 2 x0 h y0 i Æ x0 for all y B and y0 B , where L and C are the following matrices 2 x 2 x0 y y0 0 . 1 0 1 . 1 . 0 B C ¢¢¢ ¢¢¢ ¢¢¢ B . C B 1 1 C B 0 . 0 0 C Ly B ¢¢¢ ¢¢¢ C, Cy B C Æ B 0 C 0 Æ B . C @ ¢¢¢ ¢¢¢ ¢¢¢ A B . C 0 @ . A ¢¢¢ ¢¢¢ ¢¢¢ . 1 . M,N t MN is the standard inner product between matrices. We will write the nonneg- h i Æ ativity constraint E ,»t 0, where E is the basis matrix whose coefficients all vanish h y y0 i ¸ y y0 2In [LLY11], a different notation is used: the lazy random walk is parametrized by ® 1 t and the limit point Æ ¡ corresponds to ® 1. Æ 3However both approaches are equivalent, by considering weighted graphs and allowing loops (i.e. weights wxx ). See [BJL11] and also §4 for weighted graphs. 3 t except at (y, y0). The set of possible couplings is therefore a bounded convex polyhedron K (dx 1)(d 1) contained in the unit cube [0,1] Å x0 Å . In the following, we will also need the limit set K 0 {E } which contains a unique coupling (see case 1 above). Æ xx0 In order to compute ·t , we want to minimize the cost function c, which is actually linear: X X c(»t ) »t (x, y)d(x, y) »t (x, y)d(x, y) »t ,D Æ Æ Æ h i y,y V y Bx ,y B 02 2 02 x0 where D stands for the distance matrix restricted to B B , so that D is the (constant) L2 x £ x0 gradient of c. Clearly the infimum is reached, and minimizers lie on the boundary of K t . Then either the gradient D is perpendicular to some facet of K t , and the minimizer can be freely chosen on that facet, or not, and the minimizer is unique and lies on a vertex of K t .