On second variation of Perelman's Ricci shrinker entropy
Huai-Dong Cao, Meng Zhu
TL;DR
This paper rigorously derives the second variation formula for Perelman's 0-entropy, correcting previous errors and establishing stability conditions for Ricci shrinkers based on eigenvalues of related operators.
Contribution
It provides a detailed proof of the second variation formula for Perelman's 0-entropy and corrects an existing error in the stability operator, offering new stability criteria.
Findings
Corrected the second variation formula for 0-entropy.
Established a necessary condition for linear stability of Ricci shrinkers.
Linked stability to eigenvalues of a Lichnerowicz-type operator.
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 -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.
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Taxonomy
TopicsGeometric Analysis and Curvature Flows · Geometry and complex manifolds · Nonlinear Partial Differential Equations