Rigidity and stability of Einstein metrics for quadratic curvature functionals

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 $g_S$.