Stability of Compact Symmetric Spaces

TL;DR

This paper investigates the stability of Einstein metrics on compact symmetric spaces, classifying relevant Lie algebra representations and establishing stability or instability results for specific spaces.

Contribution

It classifies certain Lie algebra representations and proves stability of Einstein metrics on quaternionic and Cayley projective planes, advancing understanding of geometric stability.

Findings

Stable Einstein metrics on quaternionic and Cayley projective planes.

Unstable Einstein metrics on quaternionic Grassmannians not isomorphic to projective spaces.

Complete classification of Lie algebra representations with low Casimir eigenvalues.

Abstract

In this article we study the stability problem for the Einstein-Hilbert functional on compact symmetric spaces following and completing the seminal work of Koiso on the subject. We classify in detail the irreducible representations of simple Lie algebras with Casimir eigenvalue less than the Casimir eigenvalue of the adjoint representation, and use this information to prove the stability of the Einstein metrics on both the quaternionic and Cayley projective plane. Moreover we prove that the Einstein metrics on quaternionic Grassmannians different from projective spaces are unstable.