Yamabe flow on manifolds with edges

TL;DR

This paper demonstrates the evolution of metrics via Yamabe flow on manifolds with edge singularities, establishing key analytic estimates to enable local existence results in this singular geometric setting.

Contribution

It introduces Schauder estimates for heat operators on edge singularities and applies them to prove local existence of Yamabe flow on such manifolds.

Findings

Established Schauder estimates for heat operators on edge manifolds.

Proved local existence of Yamabe flow in singular edge settings.

Provided a framework for using flow techniques to study the Yamabe problem with singularities.

Abstract

Let (M,g) be a compact oriented Riemannian manifold with an incomplete edge singularity. This article shows that it is possible to evolve g by the Yamabe flow within a class of singular edge metrics. As the main analytic step we establish parabolic Schauder-type estimates for the heat operator on certain H\"older spaces adapted to the singular edge geometry. We apply these estimates to obtain local existence for a variety of quasilinear equations, including the Yamabe flow. This provides a setup for a subsequent discussion of the Yamabe problem using flow techniques in the singular setting.