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

Abstract: Recently, Paul Schwahn showed that, contrary to previous expectations, there exist many compact, simply connected standard Einstein manifolds that are not symmetric spaces and 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 restricted to $TT$-tensors, and $E$ is the Einstein constant.

In this talk, we will see how Lie-theoretic methods can be used to estimate the first positive eigenvalue $\lambda_1$ of the Laplace-Beltrami operator on compact, simply connected non-symmetric standard Einstein manifolds $(G/H,g_{st})$, where $G$ is a compact, connected, simple Lie group. Our main result shows that $\lambda_1>2E$ for all such manifolds, with only seven exceptions.

These estimates imply that all the Einstein manifolds proved stable by Schwahn are, in fact, linearly stable with respect to Perelman's $\nu$-entropy functional. This complements previous work of Cao and He on compact irreducible symmetric spaces.

This talk is based on joint work with Emilio Lauret (Universidad Nacional del Sur, Argentina).

differential geometry

Audience: researchers in the topic

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Geometry Webinar AmSur /AmSul

Series comments: The Geometry Webinar AmSur/AmSul is promoted by differential geometry groups from Universities in Argentina and Brazil. The webinar happens weekly on Fridays at 14h00 (GMT-3) via Google Meet. Talks will be in Spanish, Portuguese or English. This virtual seminar aims to establish the contact with several research groups and mathematicians from Latin America. Everybody is invited to participate writing to the contact e-mail: geodif@unicamp.br

For each talk, the Google Meet link will be sent.

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