A Survey on the Ricci flow on Singular Spaces

TL;DR

This survey reviews recent advances in Ricci flow on spaces with isolated conical singularities, highlighting stability, preservation of scalar curvature, and connections to Perelman's entropies.

Contribution

It summarizes new results on Ricci de Turck flow on singular spaces, emphasizing the preservation of conical singularities and stability of Ricci flat metrics.

Findings

Ricci flow preserves conical singularities under certain conditions

Ricci flat metrics with isolated conical singularities are stable

Positive scalar curvature is maintained during the flow

Abstract

In this survey we provide an overview of our recent results concerning Ricci de Turck flow on spaces with isolated conical singularities. The crucial characteristic of the flow is that it preserves the conical singularity. Under certain conditions, Ricci flat metrics with isolated conical singularities are stable and positive scalar curvature is preserved under the flow. We also discuss the relation to Perelman's entropies in the singular setting, and outline open questions and future reseach directions.