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Several rigidity results are proved for critical points of natural Riemannianfunctionals on the space of metrics on 3-manifolds. Two of these results are asfollows. Let (N, g) be a complete Riemannian 3-manifold, satisfying one of thefollowing variational conditions:  (i) (N, g) has non-negative scalar curvature and is a critical point for theL^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 volumenon-increasing perturbations of (N, g) with non-negative scalar curvature.  In either case, (N, g) is flat. The Schwarzschild metric (on the space-likehypersurface) has an isometric S^1 action and satisfies the other assumptionsin (ii), showing that this result is sharp.

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