Singular Ricci solitons and their stability under the Ricci flow

TL;DR

This paper introduces spherically symmetric singular Ricci solitons, analyzes their evolution under Ricci flow, and proves local stability of their singularity opening behavior under symmetric perturbations.

Contribution

It presents the first study of singular Ricci solitons' stability under Ricci flow, including well-posedness results near singular initial data.

Findings

Singular Ricci solitons exist in all dimensions ≥3 with curvature blow-up at a point.

Under Ricci flow, these solitons immediately open up and become incomplete but non-singular.

The stability analysis shows the singularity opening persists under symmetric perturbations.

Abstract

We introduce certain spherically symmetric singular Ricci solitons and study their stability under the Ricci flow from a dynamical PDE point of view. The solitons in question exist for all dimensions $n+1\ge 3$, and all have a point singularity where the curvature blows up; their evolution under the Ricci flow is in sharp contrast to the evolution of their smooth counterparts. In particular, the family of diffeomorphisms associated with the Ricci flow "pushes away" from the singularity causing the evolving soliton to open up immediately becoming an incomplete (but non-singular) metric. In the second part of this paper we study the local-in time stability of this dynamical evolution, under spherically symmetric perturbations of the initial soliton metric. We prove a local well-posedness result for the Ricci flow near the singular initial data, which in particular implies that the "opening up" of the singularity persists for the perturbations also.