Quasilinear Parabolic Equations and the Ricci Flow on Manifolds with Boundary

TL;DR

This paper establishes short-time existence results for quasilinear parabolic equations and the Ricci flow on manifolds with boundary, introducing a new boundary condition for the flow.

Contribution

It introduces a novel boundary condition for the Ricci flow and proves short-time existence results, extending the analysis to manifolds with boundary.

Findings

Short-time existence for quasilinear parabolic equations on manifolds with boundary.

New boundary condition for Ricci flow on manifolds with boundary.

Existence results for Ricci flow with the proposed boundary condition.

Abstract

The first part of the paper discusses a second-order quasilinear parabolic equation in a vector bundle over a compact manifold $M$ with boundary $\partial M$. We establish a short-time existence theorem for this equation. The second part of the paper is devoted to the investigation of the Ricci flow on $M$. We propose a new boundary condition for the flow and prove two short-time existence results.