Rigidity of critical metrics for quadratic curvature functionals

In this paper we prove new rigidity results for complete, possibly non-compact, t f R2 critical metrics of the quadratic curvature functionals a2t = f | Ricg |2dVg + gdVg, t is an element of R, and 672 = f R2gdVg. We show that (i) flat surfaces are the only critical points of 672, (ii) flat three-dimensional manifolds are the only critical points of a2t for every t > -13, (iii) three-dimensional scalar flat manifolds are the only critical points of 672 with finite energy and (iv) n-dimensional, n > 4, scalar flat manifolds are the only critical points of 672 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.