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

Tony Liimatainen*

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference article in proceedingsScientificpeer-review

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.

Original languageEnglish
Title of host publicationANALYSIS, GEOMETRY AND QUANTUM FIELD THEORY
EditorsCL Aldana, M Braverman, B Lochum, CN Jimenez
PublisherAmerican Mathematical Society
Pages163-178
Number of pages16
ISBN (Print)978-0-8218-9144-5
DOIs
Publication statusPublished - 2012
MoE publication typeA4 Conference publication
EventInternational Conference on Analysis, Geometry and Quantum Field Theory - Potsdam, Germany
Duration: 26 Sept 201130 Sept 2011

Publication series

NameContemporary Mathematics
PublisherAMER MATHEMATICAL SOC
Volume584
ISSN (Print)0271-4132

Conference

ConferenceInternational Conference on Analysis, Geometry and Quantum Field Theory
Country/TerritoryGermany
CityPotsdam
Period26/09/201130/09/2011

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