Heat Flow on Alexandrov Spaces

Nicola Gigli, Kazumasa Kuwada, Shin Ichi Ohta

Research output: Contribution to journalArticlepeer-review

79 Citations (Scopus)


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-Émery gradient estimates and the Γ2-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.

Original languageEnglish
Pages (from-to)307-331
Number of pages25
JournalCommunications on Pure and Applied Mathematics
Issue number3
Publication statusPublished - 2013 Mar
Externally publishedYes

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics


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