Calculus of Variations and Geometric Measure Theory
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N. Gigli - K. Kuwada - S. I. Ohta

Heat Flow on Alexandrov spaces

created by gigli on 31 Dec 2010
modified on 02 Mar 2012


Accepted at CPAM

Inserted: 31 dec 2010
Last Updated: 2 mar 2012

Year: 2010


We prove that on compact Alexandrov spaces with curvature bounded below the gradient flow of the Dirichlet energy in the L2-space produces the same evolution as the gradient flow of the relative entropy in the L2- Wasserstein space. This means that the heat flow is well defined by either one of the two gradient flows. Combining properties of these flows, we are able to deduce the Lipschitz continuity of the heat kernel as well as Bakry-Emery gradient estimates and the \Gamma2-condition. Our identification is established by purely metric means, unlike preceding results relying on PDE techniques. Our approach generalizes to the case of heat flow with drift.


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