ISSN 1066-369X, Russian Mathematics (Iz. VUZ), 2007, Vol. 51, No. 12, pp. 30–51. c  Allerton Press, Inc., 2007. * The Ricci Flow on the Two Ball with a Rotationally Symmetric Metric J. C. Cortissoz 1 1 University Los Andes, Carrera 1 N 18A 10 Bogot ´ a, Colombia 1 Received April 13, 2007 Abstract—In this paper we study a boundary-value problem for the Ricci flow in the two- dimensional ball endowed with a rotationally symmetric metric of positive Gaussian curvature and prove short and long time existence results. We construct families of metrics for which the flow uniformizes the curvature along a sequence of times. Finally, we show that if the initial metric has positive Gaussian curvature and the boundary has positive geodesic curvature then the flow uniformizes the curvature along a sequence of times. DOI: 10.3103/S1066369X07120031 1. INTRODUCTION Let M be a compact two-dimensional manifold with boundary, and g(t) be a one-parameter family of metrics on M (the parameter t will be called “time”). Let us denote by R(t) the scalar curvature of g(t), and by k g (t) the geodesic curvature of the boundary ∂M with respect to the outward unit normal. Then one can consider the following boundary-value problem for the Ricci flow on M : ⎧ ⎪ ⎨ ⎪ ⎩ ∂g ∂t = −Rg in M × (0,T ), k g = ψ on ∂M × (0,T ), g(·, 0) = g 0 . (1) In [1] S. Brendle proved the following result. Theorem 1. Let M be a compact surface with boundary ∂M . Then for every initial metric with vanishing geodesic curvature at the boundary, the initial boundary-value problem (1) has a unique solution. The solution to the normalized flow is defined for all t ≥ 0. For t →∞ the solution converges exponentially to a metric with constant Gauss curvature and vanishing geodesic curvature. In the present paper we consider the following boundary-value problem for a one-parameter family of metrics g(t) on the two-dimensional ball B 2 : ⎧ ⎪ ⎨ ⎪ ⎩ ∂g ∂t = −Rg in B 2 × (0,T ), k g = k 0 on ∂B 2 × (0,T ), g(·, 0) = g 0 , (2) where g 0 is a rotationally symmetric metric on B 2 : ds 2 = dr 2 + f (r) 2 dω 2 , ∗ The text was submitted by the author in English. 1 E-mail: jcortiss@uniandes.edu.co. 30