On second variation of Perelman's Ricci shrinker entropy

Abstract

In this paper we provide a detailed proof of the second variation formula, essentially due to Richard Hamilton, Tom Ilmanen and the first author, for Perelman's $\nu$-entropy. In particular, we correct an error in the stability operator stated in Theorem 6.3 of [2]. Moreover, we obtain a necessary condition for linearly stable shrinkers in terms of the least eigenvalue and its multiplicity of certain Lichnerowicz type operator associated to the second variation.

13 pages; final version; to appear in Math. Ann

References in corpus (3)

• Gaussian densities and stability for some Ricci solitons
• Generic mean curvature flow I; generic singularities
• On gradient Ricci solitons

Cited by in corpus (2)

• The total second variation of Perelman's $\mathcal{W}$-functional
• A second variation formula for Perelman's $\mathcal{W}$-functional along the modified Kähler-Ricci flow