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 $\mathfrak{F}^{2}_t = \int |\operatorname{Ric}_g|^{2} dV_g + t \int R^{2}_g dV_g$, $t\in\mathbb{R}$, and $\mathfrak{S}^2 = \int R_g^{2} dV_g$. 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 $\mathfrak{F}^{2}_t$ 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.