On the stability of homogeneous Einstein manifolds

TL;DR

This paper investigates the stability of G-invariant Einstein metrics on compact homogeneous spaces by analyzing the Lichnerowicz Laplacian, with special focus on naturally reductive cases, to understand their behavior as critical points of scalar curvature.

Contribution

It introduces a formula for the Lichnerowicz Laplacian at G-invariant TT-tensors and applies it to assess stability, especially in naturally reductive cases.

Findings

Derived a formula for the Lichnerowicz Laplacian on G-invariant TT-tensors.

Analyzed stability conditions for Einstein metrics on homogeneous spaces.

Provided detailed results for naturally reductive Einstein metrics.

Abstract

Let g be a G-invariant Einstein metric on a compact homogeneous space M=G/K. We use a formula for the Lichnerowicz Laplacian of g at G-invariant TT-tensors to study the stability type of g as a critical point of the scalar curvature function. The case when g is naturally reductive is studied in special detail.