On the Ricci flow on the rotationally symmetric two-ball
  In this paper we study a boundary value problem for the Ricci flow in the two dimensional ball endowed with a rotationally symmetric metric. We show 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 positive geodesic curvature then the flow uniformizes de curvature along a sequence of times. 
Jean Cortissoz