Vol. 248, No. 2, 2010

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Optimal transportation and monotonic quantities on evolving manifolds

Hong Huang

Vol. 248 (2010), No. 2, 305–316
Abstract

We adapt Topping’s -optimal transportation theory for Ricci flow to a more general situation, in which a complete manifold (M,gij(t)) evolves by tgij = 2Sij, where Sij is a symmetric 2-tensor field on M. We extend some recent results of Topping, Lott and Brendle, generalize the monotonicity of the 𝒲-entropy of List (and hence also of Perelman), and recover the monotonicity of the reduced volume of Müller (and hence also of Perelman).

Keywords
optimal transportation, -length, Boltzmann–Shannon entropy, evolving manifolds
Mathematical Subject Classification 2000
Primary: 53C44
Milestones
Received: 14 September 2009
Revised: 20 January 2010
Accepted: 4 February 2010
Published: 1 December 2010
Authors
Hong Huang
School of Mathematical Sciences
Key Laboratory of Mathematics and Complex Systems
Beijing Normal University
Beijing 100875
China