标题： 完整全局亚纯非正曲率空间中的体积微分同胚的最优黎曼度量和平均遍历定理 显示英文标题

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

摘要： 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. 显示英文摘要

摘要： 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.