Rigidity of Critical Metrics for Quadratic Curvature Functionals

Abstract

In this paper we prove new rigidity results for complete, possibly non-compact, critical metrics of the quadratic curvature functionals ${\mathfrak{F}}_t^2=\int |{\mathrm{Ric}}_g{|}^{2}d{V}_g+t\int R_g^{2}d{V}_g$, $t\in \mathbb{R}$, and ${\mathfrak{S}}^2=\int R_g^{2}d{V}_g$. We show that (i) flat surfaces are the only critical points of ${\mathfrak{S}}^2$, (ii) flat three-dimensional manifolds are the only critical points of ${\mathfrak{F}}_t^2$ for every $t>-\frac{1}{3}$, (iii) three-dimensional scalar flat manifolds are the only critical points of ${\mathfrak{S}}^2$ with finite energy and (iv) $n$-dimensional, $n>4$, scalar flat manifolds are the only critical points of ${\mathfrak{S}}^2$ with finite energy and scalar curvature bounded below. In case (i), our proof relies on rigidity results for conformal vector fields and an ODE argument; in case (ii) we draw upon some ideas of M. T. Anderson concerning regularity, convergence and rigidity of critical metrics; in cases (iii) and (iv) the proofs are self-contained and depend on new pointwise and integral estimates.