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

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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:	Date Event June 2010 Published
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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