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.

arXiv comment23 pages, published version

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• Rigidity and infinitesimal deformability of Ricci solitons
• Boltzmann's Entropy and Kähler-Ricci Solitons
• Stability and moduli space of generalized Ricci solitons
• The canonical Einstein metric on $G_2$ is dynamically unstable under the Ricci flow
• Stability of Einstein metrics on fiber bundles
• Spectrally distinguishing symmetric spaces I
• Homogeneous Einstein metrics and local maxima of the Hilbert action
• On Ricci flows with closed and smooth tangent flows
• The $G_2$ geometry of $3$-Sasaki structures
• Linear Instability of Sasaki Einstein and nearly parallel ${\mathrm{G}}_2$ manifolds
• Convergence of the Ricci flow to Ricci-flat ALE manifolds and positive scalar curvature rigidity
• On the linear stability of nearly-Kähler $6$-manifolds
• Commutator formulas for gradient Ricci shrinker and their application to linear stability
• Linear stability of Perelman's $\nu$-entropy of standard Einstein manifolds