Stability of Einstein metrics under Ricci flow

Abstract

We prove dynamical stability and instability theorems for compact Einstein metrics under the Ricci flow. We give a nearly complete charactarization of dynamical stability and instability in terms of the conformal Yamabe invariant and the Laplace spectrum. In particular, we prove dynamical stability of some classes of Einstein manifolds for which it was previously not known. Additionally, we show that the complex projective space with the Fubini-Study metric is surprisingly dynamically unstable.

29 pages, final version with a modified title, to appear in Comm. Anal. Geom

References in corpus (2)

• Ricci flow with surgery on three-manifolds
• Gaussian densities and stability for some Ricci solitons

Cited by in corpus (13)

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• Lojasiewicz-Simon gradient inequalities for analytic and Morse-Bott functions on Banach spaces
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• Rigidity and infinitesimal deformability of Ricci solitons
• Linear stability of compact shrinking Ricci solitons
• The canonical Einstein metric on $G_2$ is dynamically unstable under the Ricci flow
• Stability and moduli space of generalized Ricci solitons
• Dynamic instability of ${\mathbb{CP}}^N$ under Ricci flow
• Spectra, rigidity and stability of sine-cones
• Construction of initial data sets for Lorentzian manifolds with lightlike parallel spinors
• Local and global scalar curvature rigidity of Einstein manifolds
• Convergence of the Ricci flow to Ricci-flat ALE manifolds and positive scalar curvature rigidity
• Linear stability of Perelman's $\nu$-entropy of standard Einstein manifolds