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Ricci flow, entropy and optimal transportation

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McCann, Robert J. and Topping, Peter (2010) Ricci flow, entropy and optimal transportation. American Journal of Mathematics, Vol.132 (No.3). pp. 711-730. doi:10.1353/ajm.0.0110 ISSN 0002-9327.

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Official URL: http://dx.doi.org/10.1353/ajm.0.0110

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Abstract

Let a smooth family of Riemannian metrics g(tau) satisfy the backwards Ricer flow equation on a compact oriented n-dimensional manifold M Suppose two families of normalized n-forms omega(tau) >= 0 and (omega) over bar(tau) >= 0 satisfy the forwaids (in tau) heat equation on M generated by the connection Laplacian Delta g(T) If these n-forms represent two evolving distributions of particles over M. the minimum root-mean-square distance W-2(omega(T). (omega) over tilde (T), t) to transport the particles of omega(T) onto those of (omega) over tilde (T) is shown to be non-increasing as a function of T, without sign conditions on the curvature of (Mg (T)) Moreover, this contractivity property is shown to characterize supersolutions to the Ricci flow.

Item Type: Journal Article
Subjects: Q Science > QA Mathematics
Divisions: Faculty of Science, Engineering and Medicine > Science > Mathematics
Journal or Publication Title: American Journal of Mathematics
Publisher: The Johns Hopkins University Press
ISSN: 0002-9327
Official Date: June 2010
Dates:
DateEvent
June 2010Published
Volume: Vol.132
Number: No.3
Number of Pages: 20
Page Range: pp. 711-730
DOI: 10.1353/ajm.0.0110
Status: Peer Reviewed
Publication Status: Published
Access rights to Published version: Restricted or Subscription Access
Funder: Leverhulme Trust (LT), Natural Sciences and Engineering Research Council of Canada (NSERC), National Science Foundation (U.S.) (NSF), Engineering and Physical Sciences Research Council (EPSRC)
Grant number: 217006-03, DMS 0354729

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