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 Ft2=∫|Ricg|2dVg+t∫Rg2dVg, t∈R, and S2=∫Rg2dVg. We show that (i) flat surfaces are the only critical points of S2, (ii) flat three-dimensional manifolds are the only critical points of Ft2 for every t>−13, (iii) three-dimensional scalar flat manifolds are the only critical points of S2 with finite energy and (iv) n-dimensional, n>4, scalar flat manifolds are the only critical points of S2 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.