Optimal Riemannian metric for a volumorphism and a mean ergodic theorem in complete global Alexandrov nonpositively curved spaces

In this paper we give a natural condition for when a volumorphism on a Riemannian manifold (M, g) is actually an isometry with respect to some other, optimal, Riemannian metric h. We consider the natural action of volumorphisms on the space M-mu(s), of all Riemannian metrics of Sobolev class H-s, s > n/2, with a fixed volume form mu. An optimal Riemannian metric, for a given volumorphism, is a fixed point of this action in a certain complete metric space containing M-mu(s) as an isometrically embedded subset. We show that a fixed point exists if the orbit of the action is bounded. We also generalize a mean ergodic theorem and a fixed point theorem to the nonlinear setting of complete global Alexandrov nonpositive curvature spaces.