Linear stability of Perelman's $\nu$-entropy of standard Einstein manifolds

TL;DR

This paper proves that most standard Einstein manifolds are linearly stable under Perelman's $ u$-entropy, extending previous stability results for certain Einstein manifolds and providing eigenvalue estimates for the Laplace-Beltrami operator.

Contribution

It provides eigenvalue bounds for the Laplace-Beltrami operator on standard Einstein manifolds, showing their linear stability with respect to Perelman's $ u$-entropy, except for seven specific spaces.

Findings

Most standard Einstein manifolds satisfy $ ext{lambda}_1 > 2E$

All Schwahn's stable Einstein manifolds are linearly stable under $ u$-entropy

Seven spaces do not satisfy the eigenvalue inequality

Abstract

Paul Schwahn recently exhibited 112 non-symmetric, connected, simply connected, compact Einstein manifolds that are stable with respect to the total scalar curvature functional restricted to the space of Riemannian metrics with constant scalar curvature and fixed volume. This stability follows from the inequality $\lambda_L > 2E$, where $\lambda_L$ denotes the smallest eigenvalue of the Lichnerowicz Laplacian on TT-tensors and $E$ is the corresponding Einstein factor. In this paper, we estimate the smallest positive eigenvalue $\lambda_1$ of the Laplace-Beltrami operator for connected, simply connected, non-symmetric standard Einstein manifolds $(G/H,g_{\operatorname{st}})$ with $G$ a compact and connected simple Lie group. We obtain that $\lambda_1>2E$ for all of them excepting $7$ spaces. As a consequence of our estimates, we establish that all stable Einstein manifolds found by Schwahn are in fact linear stable with respect to Perelman's $\nu$-entropy.