The Yamabe flow on incomplete manifolds

TL;DR

This paper develops an analytic framework for second order nonlinear parabolic equations on singular manifolds and applies it to prove the existence of unique local solutions to the Yamabe flow on incomplete manifolds.

Contribution

It introduces maximal regularity techniques for nonlinear parabolic equations on singular manifolds and demonstrates their application to the Yamabe flow with incomplete initial metrics.

Findings

Existence and uniqueness of solutions in an Lp-framework.

Local solutions to the Yamabe flow on incomplete manifolds.

Application of maximal regularity tools to geometric flows.

Abstract

This article is concerned with developing an analytic theory for second order nonlinear parabolic equations on singular manifolds. Existence and uniqueness of solutions in an Lp-framework is established by maximal regularity tools. These techniques are applied to the Yamabe flow. It is proven that the Yamabe flow admits a unique local solution within a class of incomplete initial metrics.