The Ricci flow on manifolds with boundary

TL;DR

This paper establishes short-time existence, uniqueness, and conditions for the continuation of Ricci flow on manifolds with boundary, focusing on boundary data control and geometric regularity.

Contribution

It proves the short-time existence and uniqueness of Ricci flow with prescribed boundary conditions on manifolds with boundary, extending previous results to include boundary data control.

Findings

Short-time existence and uniqueness of Ricci flow with boundary conditions.

Conditions under which the flow persists based on boundary and curvature bounds.

Analysis of boundary regularity for Ricci-DeTurck equations.

Abstract

We study the short-time existence and regularity of solutions to a boundary value problem for the Ricci-DeTurck equation on a manifold with boundary. Using this, we prove the short-time existence and uniqueness of the Ricci flow prescribing the mean curvature and conformal class of the boundary, with arbitrary initial data. Finally, we establish that under suitable control of the boundary data the flow exists as long as the ambient curvature and the second fundamental form of the boundary remain bounded.