  In this paper we consider three-manifolds with weakly umbilic boundary (the
Second Fundamental form of the boundary is a constant multiple of the metric).
We show that if the initial manifold has positive Ricci curvature and the
boundary is convex (nonnegative Second Fundamental form), its metric can be
deformed via the Ricci flow to a metric of constant curvature and totally
geodesic boundary.
