Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On second variation of Perelman's Ricci shrinker entropy

Huai-Dong Cao, Meng Zhu

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 $ν$-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.

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 -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.

Paper Structure

This paper contains 3 sections, 11 theorems, 85 equations.

Table of Contents

  1. The Results
  2. The Proof of Theorem \ref{['thm1.1']}
  3. Further remarks and the proof of Theorem \ref{['thm1.3']}

Key Result

Theorem 1.1

(Cao-Hamilton-Ilmanen) Let $(M^n, g_{ij}, f)$ be a compact Ricci shrinker with the potential function $f$ and satisfying the Ricci soliton equation (1.1). For any symmetric 2-tensor $h=h_{ij}$, consider variations $g_{ij}(s)=g_{ij}+sh_{ij}$. Then the second variation $\delta^2_g\nu(h,h)$ is given by where the stability operator $\hat{N}$ is given by and $\hat{v}_h$ is the unique solution of

Theorems & Definitions (23)

  • Remark 1.1
  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.5
  • Remark 1.6
  • Lemma 2.1
  • ...and 13 more