Long-time existence of the edge Yamabe flow

TL;DR

This paper proves the long-time existence, uniqueness, and convergence of the normalized Yamabe flow on compact manifolds with incomplete edge singularities and negative Yamabe invariant, using novel maximum principle techniques.

Contribution

It introduces new maximum principle methods and bounds for the edge Yamabe flow on singular spaces, enabling analysis of long-term behavior and solving the Yamabe problem in this setting.

Findings

Proved long-time existence and convergence of the edge Yamabe flow.

Established uniform bounds without Krylov-Safonov estimates.

Solved the Yamabe problem for incomplete edge metrics with negative invariant.

Abstract

This article presents an analysis of the normalized Yamabe flow starting at and preserving a class of compact Riemannian manifolds with incomplete edge singularities and negative Yamabe invariant. Our main results include uniqueness, long-time existence and convergence of the edge Yamabe flow starting at a metric with everywhere negative scalar curvature. Our methods include novel maximum principle results on the singular edge space without using barrier functions. Moreover, our uniform bounds on solutions are established by a new ansatz without in any way using or redeveloping Krylov-Safonov estimates in the singular setting. As an application we obtain a solution to the Yamabe problem for incomplete edge metrics with negative Yamabe invariant using flow techniques. Our methods lay groundwork for studying other flows like the mean curvature flow as well as the porous medium equation in the singular setting.