Einstein metrics, their moduli spaces and stability
Paul Schwahn, Uwe Semmelmann
TL;DR
This survey analyzes stability and deformation theory of Einstein metrics on compact manifolds, framing the problem through the Einstein–Hilbert functional and its second variation via the Lichnerowicz Laplacian $\Delta_L$. It connects local stability to the Yamabe functional, scalar curvature rigidity, and Ricci-flow dynamics, and develops a detailed deformation-theoretic picture including premoduli spaces, obstructions, and integrability. The text further synthesizes results across products, warped products, Riemannian submersions, manifolds with parallel spinors, and special holonomy (Kähler–Einstein, quaternion–Kähler, Killing spinors), and then specializes to homogeneous spaces where representation theory yields concrete stability classifications and coindex computations. The work highlights both pervasive instability in many positive-curvature and non-symmetric settings and the rare stable cases (notably certain symmetric spaces and special holonomy manifolds), while outlining key open questions about moduli-space structure, integrability obstructions, and the full scope of stability in various geometric contexts.
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.