Ricci Flow on Manifolds with Boundary with Arbitrary Initial Metric

TL;DR

This paper establishes short-time existence and uniqueness of Ricci flow on manifolds with boundary, showing that certain boundary conditions and curvature properties are preserved during the flow.

Contribution

It proves the existence, uniqueness, and boundary-preserving properties of Ricci flow on manifolds with arbitrary initial metrics and boundary convexity conditions.

Findings

Flow makes boundary instantaneously umbilic

Preserves positive curvature operator and PIC conditions

Maintains boundary convexity during flow

Abstract

In this paper, we study the Ricci flow on manifolds with boundary. In the first part of the paper, we prove short-time existence and uniqueness of the solution, in which the boundary becomes instantaneously umbilic for positive time. In the second part of the paper, we prove that the flow we constructed in the first part preserves natural boundary conditions. More specifically, if the initial metric has a convex boundary, then the flow preserves positive curvature operator and the PIC1, PIC2 conditions. Moreover, if the initial metric has a two-convex boundary, then the flow preserves the PIC condition.