Heat Content in Non-compact Riemannian Manifolds

Open Access


Let \(\Omega \) be an open set in a complete, smooth, non-compact, m-dimensional Riemannian manifold M without boundary, where M satisfies a two-sided Li-Yau gaussian heat kernel bound. It is shown that if \(\Omega \) has infinite measure, and if \(\Omega \) has finite heat content \(H_{\Omega }(T)\) for some \(T>0\), then \(H_{\Omega }(t)<\infty \) for all \(t>0\). Comparable two-sided bounds for \(H_{\Omega }(t)\) are obtained for such \(\Omega \).


Heat content Riemannian manifold Volume doubling Poincaré inequality 

Mathematics Subject Classification

58J32 58J35 35K20 


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© The Author(s) 2018

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.School of MathematicsUniversity of BristolBristolUK

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