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

TL;DR

This paper establishes conditions under which volumorphisms become isometries for an optimal Riemannian metric and extends ergodic and fixed point theorems to nonpositively curved Alexandrov spaces.

Contribution

It introduces a criterion for volumorphisms to be isometries under an optimal metric and generalizes ergodic and fixed point theorems to Alexandrov nonpositive curvature spaces.

Findings

Fixed point existence when the orbit is bounded.

Generalization of mean ergodic theorem to Alexandrov spaces.

Extension of fixed point theorem to nonpositively curved spaces.

Abstract

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.