Rigidity and stability of Einstein metrics for quadratic curvature functionals

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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 C²,α-neighborhood of the round metric (Sⁿ,gₛ) satisfying Ric(g) ≤ Ric(gₛ) has volume Vol(g) ≥ Vol(gₛ), with equality holding if and only if g is isometric to gₛ.