Three-manifolds of positive curvature and convex weakly umbilic boundary

Abstract

In this paper we study a boundary value problem in manifolds with weakly umbilic boundary (the Second Fundamental form of the boundary is a constant multiple of the metric). We show that if we start with a metric of positive curvature and convex boundary (positive Second Fundamental form), the Ricci flow uniformizes the curvature. In the case of a metric with rotational symmetry, we indicate how to weaken the hypothesis to positive Ricci curvature and positive Second Fundamental form.