N ov 2 00 2 Combinatorial Ricci Flows on Surfaces

We show that the analog of Hamilton's Ricci flow in the combinatorial setting produces solutions which converge exponentially fast to Thurston's circle packing on surfaces. As a consequence, a new proof of Thurston's existence of circle packing theorem is obtained. As another consequence, Ricci flow suggests a new algorithm to find circle packings. §1. Introduction 1.1. For a compact surface with a Riemannian metric (X, g ij), R. Hamilton in [Ha] introduced the 2-dimensional Ricci flow defined by the equation dg ij dt = −2Kg ij where K is the Gaussian curvature of the surface. It is proved in [Ha] and [Cho] that for any closed surface with any initial Riemannian metric, the solution of the Ricci flow exists for all time, and after normalizing the solution to have a fixed area, the solution converges to the constant curvature metric conformal to the initial metric as time goes to infinity. The purpose of the paper is to study the analogous flow in the combinatorial setting. The simplest case of our setup is as follows. Given a triangulated surface, we assign to each vertex a positive radius and realize the surface by a piecewise flat cone metric with singularities at the vertices where the length of an edge is the sum of the two radii associated to the end points. We call these the (tangential type) circle-packing metrics based on the triangulation. We consider a natural flow (the combinatorial Ricci flow) on the space of all tangential type circle-packing metrics. Our results give a complete description to the asymptotic behavior of the solution of the Ricci flow. If we use hyperbolic geometry as the background metric, then for compact surfaces of negative Euler characteristic, we show that the combinatorial Ricci flow has solutions for all time for any initial metric and converges exponentially fast to the circle packing metric constructed by Thurston. 1.2. For simplicity, we shall state our results for triangulations on compact surfaces without boundary. For the most general version involving cellular-decompositions, see §2, §5 and §6. Suppose X is a closed surface and T is a triangulation on X. Let V = {v 1 , v 2 , ..., v N } be the set of vertices in T. The set of all edges and triangles in T are denoted by E and F. Unless mentioned otherwise a weight on the triangulation is a function Φ : E …