Stability of Compact Symmetric Spaces

Uwe Semmelmann,Gregor Weingart

doi:10.1007/s12220-021-00838-3

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.

Highlights

IntroductionM on the set M+ := ( Sym2+T ∗ M ) of Riemannian metrics on M under volumepreserving variations
Perhaps the most interesting mathematical insight gained from studying general relativity is that the Einstein metrics g on a compact-connected manifold M can be characterized variationally as critical points of the Einstein–Hilbert or total scalar curvature functionalS[ g ] := scalg | volg |M on the set M+ := ( Sym2+T ∗ M ) of Riemannian metrics on M under volumepreserving variations
Weingart of the diffeomorphism group Diff M on the set M+ via pull back, for this reason, the second variation of the Einstein–Hilbert functional in a critical point will have an infinite-dimensional space of null directions

Summary

Introduction

M on the set M+ := ( Sym2+T ∗ M ) of Riemannian metrics on M under volumepreserving variations. The original definition of Lichnerowicz spelled out the curvature term q( R ) in the form: where DerRic acts on 2-tensors by ( DerRich )( X , Y ) := h( Ric X , Y ) + h( X , Ric Y ) and: The explicit formula (3) of the Hessian S of the reduced Einstein–Hilbert functional highlights again the special role of the round spheres in the decomposition (1): According to the Theorem of Lichnerowicz–Obata [6], the Laplace–Beltrami operator satisfies. En nuce the stability problem for compact Einstein manifolds M is the question, whether the Hessian S of the reduced Einstein–Hilbert functional is negative definite on the complementary subspace ker D∗ ∩ ( Sym2◦T ∗ M ) of tt-tensors. Our main result fills this gap and clarifies the stability status for the remaining symmetric spaces of compact type: Theorem 1.1 (Stability of Quaternionic and Cayley Projective Plane) The Cayley projective plane OP2 = F4/Spin(9) is stable in the sense of Koiso.

Prototypical Differential Operators

Critical Representations of Simple Lie Algebras

Casω1 Casg

Geometry of the Cayley Projective Plane

Einstein Deformations of the Grassmannians