Gaussian densities and stability for some Ricci solitons

TL;DR

This paper analyzes the stability of Ricci solitons using second variations of Perelman's functionals, introduces the concept of central density, and demonstrates instability of certain Einstein manifolds under Ricci flow.

Contribution

It provides the second variation formulas for Perelman's functionals and introduces the central density concept to study Ricci solitons' stability.

Findings

Certain Einstein manifolds are unstable under Ricci flow.

The central density can predict singularity formation.

Examples in dimension 4 are explicitly computed.

Abstract

In this announcement, we exhibit the second variation of Perelman's $\lambda$ and $\nu$ functionals for the Ricci flow, and investigate the linear stability of examples. We also define the "central density" of a shrinking Ricci soliton and compute its values for certain examples in dimension 4. Using these tools, one can sometimes predict or limit the formation of singularities in the Ricci flow. In particular, we show that certain Einstein manifolds are unstable for the Ricci flow in the sense that generic perturbations acquire higher entropy and thus can never return near the original metric.