Stability of Einstein metrics on homogeneous spaces

Abstract

(1) Stability of Einstein metrics on symmetric spaces of compact type: We prove the linear stability with respect to the Einstein-Hilbert action of the symmetric spaces SU(n), n ≥ 3, and E_6/F_4 . Combined with earlier results, this resolves the stability problem for irreducible symmetric spaces of compact type. (2) Coindex and rigidity of Einstein metrics on homogeneous Gray manifolds: Any 6-dimensional strict nearly Kähler manifold is Einstein with positive scalar curvature. We compute the coindex of the metric with respect to the Einstein-Hilbert functional on each of the compact homogeneous examples. Moreover, we show that the infinitesimal Einstein deformations on F_1,2 = SU(3)/T_2 are not integrable into a curve of Einstein metrics. (3) Stability of the Non-Symmetric Space E_7/PSO(8): We prove that the normal metric on the homogeneous space E_7/PSO(8) is stable with respect to the Einstein-Hilbert action, thereby exhibiting the first known example of a non-symmetric metric of positive scalar curvature with this property. (4) The Lichnerowicz Laplacian on normal homogeneous spaces: We give a new formula for the Lichnerowicz Laplacian on normal homogeneous spaces in terms of Casimir operators. We derive some practical estimates and apply them to the known list of non-symmetric, compact, simply connected homogeneous spaces G/H with G simple whose standard metric is Einstein. This yields many new examples of Einstein metrics which are stable in the Einstein-Hilbert sense, which have long been lacking in the positive scalar curvature setting.