On the stability of homogeneous Einstein manifolds II

TL;DR

This paper derives a formula for the Lichnerowicz Laplacian on homogeneous spaces, computes its spectrum for Einstein metrics on specific spaces, and analyzes their stability as critical points of scalar curvature.

Contribution

It provides a new formula for the Lichnerowicz Laplacian on G-invariant metrics and applies it to analyze stability of Einstein metrics on Wallach spaces and flag manifolds.

Findings

Computed G-invariant spectrum of Lichnerowicz Laplacian for Einstein metrics

Determined G-stability of Einstein metrics on studied spaces

Classified critical point types of scalar curvature functional

Abstract

For any $G$-invariant metric on a compact homogeneous space $M=G/K$, we give a formula for the Lichnerowicz Laplacian restricted to the space of all $G$-invariant symmetric $2$-tensors in terms of the structural constants of $G/K$. As an application, we compute the $G$-invariant spectrum of the Lichnerowicz Laplacian for all the Einstein metrics on most generalized Wallach spaces and any flag manifold with $b_2(M)=1$. This allows to deduce the $G$-stability and critical point types of each of such Einstein metrics as a critical point of the scalar curvature functional.