Heat kernel on a manifold with a local Harnack inequality. (English) Zbl 0845.58056

The author introduces a broad class of the so-called locally Harnack manifolds, which includes in particular the manifolds with \(C^k\)-bounded geometries, with (simply) bounded geometries and with weak bounded geometries. For those manifolds (i.e. not only for bounded geometries as previously) under a certain additional condition the author gives a direct proof of the fact that the heat kernel decays at least as fast as \(1/t^{1/2}\). Another sort of results for those manifolds deals with (classical and modified) isoperimetric properties. In particular a lower bound for the first Dirichlet eigenvalue is established, etc.


58J65 Diffusion processes and stochastic analysis on manifolds
58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.)
58J35 Heat and other parabolic equation methods for PDEs on manifolds
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