Stability of Einstein metrics on symmetric spaces of compact type

Paul Schwahn

doi:10.1007/s10455-021-09810-4

Abstract

We prove the linear stability with respect to the Einstein-Hilbert action of the symmetric spaces {text {SU}}(n), nge 3, and E_6/F_4. Combined with earlier results, this resolves the stability problem for irreducible symmetric spaces of compact type.

Highlights

Let M be a closed manifold of dimension n > 2
The stability problem is to decide whether they vanish for a given Einstein manifold (M, g)
A different approach for proving the stability of both spaces that involves explicit computations of the divergence operator can be found in the Appendix

Summary

Introduction

Let M be a closed manifold of dimension n > 2. SU(n)/ SO(n), SU(2n)/ Sp(n) (n ≥ 3), SU( p + q)/S(U( p) × U(q)) ( p ≥ q ≥ 2), which shows that they are linearly stable This did not fully settle the stability problem on irreducible symmetric spaces of compact type. Koiso has proved the infinitesimal non-deformability of a large class of such manifolds: Theorem 1.4 Let (M, g) be a locally symmetric Einstein manifold of compact type. This critical eigenvalue is significantly smaller than the one from stability with respect to the Einstein-Hilbert action, and even negative for n > 9 As it turns out, all irreducible symmetric spaces of compact type are physically stable (see [5]). A different approach for proving the stability of both spaces that involves explicit computations of the divergence operator can be found in the Appendix

Preliminaries

Invariant differential operators

Proofs of general statements