  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.
