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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
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: |
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| 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 |
Data sourced from Thomson Reuters' Web of Knowledge
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