Einstein metrics, their moduli spaces and stability

TL;DR

This survey explores the stability and deformation theory of Einstein metrics on compact manifolds, focusing on spectral analysis of the Lichnerowicz Laplacian and recent advances in the field.

Contribution

It provides a comprehensive overview of classical and recent results on Einstein metrics' stability and moduli space deformation theory.

Findings

Analysis of the spectrum and eigentensors of the Lichnerowicz Laplacian

Recent progress in understanding the moduli space of Einstein metrics

Insights into the stability criteria of Einstein metrics

Abstract

This survey deals with two closely connected topics: first, the stability of Einstein metrics under the Einstein-Hilbert functional, and second, their deformation theory and the study of the moduli space of Einstein metrics on a compact manifold. To first order, both problems reduce to studying the spectrum and eigentensors of the Lichnerowicz Laplacian. We give an introduction to the classical theory and survey recent results and advances.