Rigidity and stability of Einstein metrics for quadratic curvature functionals

TL;DR

This paper studies the rigidity and stability of Einstein metrics as critical points of quadratic curvature functionals, showing conditions under which they are isolated or local minimizers, with implications for volume comparison.

Contribution

It introduces a gauge fixing for the Euler-Lagrange equations to analyze the moduli space of critical metrics and provides examples of rigidity and local minimality.

Findings

Infinitesimal rigidity of certain compact critical metrics.

Existence of critical metrics that are strict local minimizers.

A local reverse Bishop's inequality relating volume and Ricci curvature.

Abstract

We investigate rigidity and stability properties of critical points of quadratic curvature functionals on the space of Riemannian metrics. We show it is possible to "gauge" the Euler-Lagrange equations, in a self-adjoint fashion, to become elliptic. Fredholm theory may then be used to describe local properties of the moduli space of critical metrics. We show a number of compact examples are infinitesimally rigid, and consequently, are isolated critical points in the space of unit-volume Riemannian metrics. We then give examples of critical metrics which are strict local minimizers (up to diffeomorphism and scaling). A corollary is a local "reverse Bishop's inequality" for such metrics. In particular, any metric $g$ in a $C^{2,\alpha}$-neighborhood of the round metric $(S^n,g_S)$ satisfying $Ric(g) \leq Ric(g_S)$ has volume $Vol(g) \geq Vol(g_S)$, with equality holding if and only if $g$ is isometric to $g_S$.