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

Three-manifolds of positive Ricci curvature and convex weakly umbilic boundary