Stability and instability of Ricci solitons

Abstract

We consider the volume-normalized Ricci flow close to compact shrinking Ricci solitons. We show that if a compact Ricci soliton $(M,g)$ is a local maximum of Perelman's shrinker entropy, any normalized Ricci flow starting close to it exists for all time and converges towards a Ricci soliton. If $g$ is not a local maximum of the shrinker entropy, we show that there exists a nontrivial normalized Ricci flow emerging from it. These theorems are analogues of results in the Ricci-flat and in the Einstein case.

23 pages, published version

References in corpus (4)

• Gaussian densities and stability for some Ricci solitons
• Existence of Gradient Kahler-Ricci Solitons
• On the uniqueness of asymptotic limits of the Ricci flow
• Kaehler-Ricci solitons on homogeneous toric bundles (I)

Cited by in corpus (2)

• Linear Instability of Sasaki Einstein and nearly parallel ${\mathrm{G}}_2$ manifolds
• Commutator formulas for gradient Ricci shrinker and their application to linear stability