Extrema of curvature functionals on the space of metrics on 3-manifolds, II

Abstract

Several rigidity results are proved for critical points of natural Riemannian functionals on the space of metrics on 3-manifolds. Two of these results are as follows. Let (N, g) be a complete Riemannian 3-manifold, satisfying one of the following variational conditions: (i) (N, g) has non-negative scalar curvature and is a critical point for the L^2 norm of the full curvature R among compact perturbations of (N, g). (ii) (N, g) has non-negative scalar curvature, a free isometric S^1 action, and is a critical point of the L^2 norm of R among compact volume non-increasing perturbations of (N, g) with non-negative scalar curvature. In either case, (N, g) is flat. The Schwarzschild metric (on the space-like hypersurface) has an isometric S^1 action and satisfies the other assumptions in (ii), showing that this result is sharp.