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Linear stability of Perelman's $ν$-entropy on symmetric spaces of compact type

Huai-Dong Cao, Chenxu He

Abstract

Following Cao-Hamilton-Ilmanen, in this paper we study the linear stability of Perelman's $ν$-entropy on Einstein manifolds with positive Ricci curvature. We observe the equivalence between the linear stability restricted to the transversal traceless symmetric 2-tensors and the stability of Einstein manifolds with respect to the Hilbert action. As a main application, we give a full classification of linear stability of the $ν$-entropy on symmetric spaces of compact type. In particular, we exhibit many more linearly stable and linearly unstable examples than previously known and also the first linearly stable examples, other than the standard spheres, whose second variations are negative definite.

Linear stability of Perelman's $ν$-entropy on symmetric spaces of compact type

Abstract

Following Cao-Hamilton-Ilmanen, in this paper we study the linear stability of Perelman's -entropy on Einstein manifolds with positive Ricci curvature. We observe the equivalence between the linear stability restricted to the transversal traceless symmetric 2-tensors and the stability of Einstein manifolds with respect to the Hilbert action. As a main application, we give a full classification of linear stability of the -entropy on symmetric spaces of compact type. In particular, we exhibit many more linearly stable and linearly unstable examples than previously known and also the first linearly stable examples, other than the standard spheres, whose second variations are negative definite.

Paper Structure

This paper contains 5 sections, 9 theorems, 54 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The second variation of Perelman's $\nu$-entropy on Einstein metrics
  4. The linear stability of symmetric spaces of compact type
  5. First eigenvalues of Laplacian on functions and Lichnerowicz Laplacian on traceless transversal symmetric $2$-tensors

Key Result

Theorem 1.1

Let $(M^n, g)$ be a compact Einstein manifold other than the standard sphere, with $\mathrm{Ric} = \lambda g$ and $\lambda>0$. Then the decomposition (eqn:S2Mdecomposition) is orthogonal with respect to the second variation $\delta^2 \nu_g$ of $\nu$-entropy. Moreover

Theorems & Definitions (22)

  • Theorem 1.1: Cao-Hamilton-Ilmanen
  • Remark 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Remark 1.5
  • Remark 1.6
  • Theorem 2.1: Theorem 4.60 in Besse
  • Definition 2.2: Definition 2.7 in KoisocompactSS
  • Remark 2.3
  • Theorem 3.1: Cao-Hamilton-Ilmanen CaoHamiltonIlmanen
  • ...and 12 more