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

TL;DR

This paper proves that three-manifolds with positive Ricci curvature and convex weakly umbilic boundary can be deformed through Ricci flow to achieve constant curvature and totally geodesic boundary, extending geometric classification results.

Contribution

It establishes a new deformation result for three-manifolds with specific boundary conditions under Ricci flow, linking boundary geometry to curvature uniformization.

Findings

Ricci flow deforms the metric to constant curvature

Boundary becomes totally geodesic after deformation

Initial conditions ensure convergence to a canonical form

Abstract

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.