On the stability of homogeneous standard Einstein manifolds

LAURET, EMILIO AGUSTÍN; LAURET, JORGE R.

Córdoba

Workshop; Workshop on Geometric Structures and Moduli Spaces; 2022

FaMAF - Universidad Nacional de Córdoba

The stability type of a compact Einstein manifold (as a critical point of the normalized total scalar curvature functional) reduces to a condition involving the first eigenvalue of the Lichnerowicz Laplacian restricted to TT-tensors (traceless and transversal symmetric 2-tensors). Given $G/K$ a compact homogeneous manifold, Jorge Lauret has recently defined the $G$-stability types of $G$-invariant Einstein metrics on $G/K$ as critical points of the normalized scalar curvature functional restricted to $G$-invariant metrics. In analogy to the classical theory, the $G$-stability type reduces to a condition involving the first eigenvalue of the Lichnerowicz Laplacian restricted to (the finite-dimensional subspace of) $G$-invariant TT-tensors. In this talk we determine the $G$-stability type of (almost) every standard homogeneous Einstein manifolds $G/K$ with $G$ simple, classified by Wang and Ziller in 1985.While most of cases were $G$-unstable (therefore unstable in the classical sense), we found several $G$-stable examples; in this case, the standard metric is a local maximum of the normalized scalar curvature functional among $G$-invariant metrics.This is joint work with Jorge Lauret.