Convergence of the Yamabe flow on singular spaces with positive Yamabe constant

TL;DR

This paper investigates the convergence behavior of the Yamabe flow on singular spaces with positive Yamabe constant, establishing conditions for convergence and analyzing alternative scenarios including non-convergence.

Contribution

It extends the analysis of Yamabe flow convergence to stratified spaces with singularities, introducing a low energy condition and a concentration-compactness framework.

Findings

Convergence of Yamabe flow is established under a low energy condition.

A concentration-compactness dichotomy is proved for the flow.

An example of non-convergence is analyzed in detail.

Abstract

In this work, we study the convergence of the normalized Yamabe flow with positive Yamabe constant on a class of pseudo-manifolds that includes stratified spaces with iterated cone-edge metrics. We establish convergence under a low energy condition. We also prove a concentration - compactness dichotomy, and investigate what the alternatives to convergence are. We end by investigating a non-convergent example due to Viaclovsky in more detail.