The Ricci flow on surfaces with boundary

TL;DR

This paper studies the Ricci flow on surfaces with boundary, showing convergence to constant curvature metrics under certain conditions and establishing long-term existence for symmetric cases.

Contribution

It provides new results on the behavior of Ricci flow on surfaces with boundary, including convergence criteria and long-term existence in symmetric settings.

Findings

Convergence to constant curvature metrics with convex boundary and positive initial curvature.

Existence of the normalized flow for all time when boundary geodesic curvature is nonpositive and symmetry holds.

Identification of conditions leading to totally geodesic boundary in the limit.

Abstract

We show for a non homogeneous boundary value problem for the Ricci flow on the disk that when the initial metric has positive curvature and the boundary is convex then the initial metric is deformed, via the normalized flow and along sequences of times, to a metric of constant curvature and totally geodesic boundary. We also show that when the geodesic curvature of the boundary is nonpositive and the metric is rotationally symmetric, the normalized version of the flow exists for all time.