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Heat flow on Finsler manifolds
Author(s) -
Ohta Shinichi,
Sturm KarlTheodor
Publication year - 2009
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.20273
Subject(s) - mathematics , finsler manifold , heat equation , riemannian manifold , manifold (fluid mechanics) , mathematical analysis , minkowski space , norm (philosophy) , balanced flow , heat kernel , tangent space , flow (mathematics) , space (punctuation) , regular polygon , semigroup , pure mathematics , scalar curvature , curvature , mathematical physics , geometry , mechanical engineering , linguistics , philosophy , political science , law , engineering
This paper studies the heat flow on Finsler manifolds. A Finsler manifold is a smooth manifold M equipped with a Minkowski norm F ( x , ·) : T x M → ℝ + on each tangent space. Mostly, we will require that this norm be strongly convex and smooth and that it depend smoothly on the base point x . The particular case of a Hilbert norm on each tangent space leads to the important subclasses of Riemannian manifolds where the heat flow is widely studied and well understood. We present two approaches to the heat flow on a Finsler manifold: as gradient flow on L 2 ( M, m ) for the energy$$ \varepsilon(u) = {1 \over 2} \int \limits_{M} F^{2}(\nabla u)dm $$as gradient flow on the reverse L 2 ‐Wasserstein space 2 ( M ) of probability measures on M for the relative entropy$$ {\rm Ent}(u) = \int \limits_{M} u \log u \,dm. $$Both approaches depend on the choice of a measure m on M and then lead to the same nonlinear evolution semigroup. We prove 1, α regularity for solutions to the (nonlinear) heat equation on the Finsler space ( M, F, m ). Typically solutions to the heat equation will not be 2 . Moreover, we derive pointwise comparison results à la Cheeger‐Yau and integrated upper Gaussian estimates à la Davies. © 2008 Wiley Periodicals, Inc.