Heat Content in Non-compact Riemannian Manifolds

  • M. van den Berg
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 


  1. 1.
    Davies, E.B.: Heat Kernels and Spectral Theory. Cambridge University Press, Cambridge (1989)CrossRefzbMATHGoogle Scholar
  2. 2.
    Grigor’yan, A.: Analytic and geometric backgroud of recurrence and non-explosion of the Brownian motion on Riemannian manifolds. Bull. Am. Math. Soc. 36, 135–249 (1999)CrossRefzbMATHGoogle Scholar
  3. 3.
    Saloff-Coste, L.: Aspects of Sobolev-Type Inequalities, London Mathematical Society Lecture Note Series 289. Cambridge University Press, Cambridge (2002)Google Scholar
  4. 4.
    Strichartz, R.S.: Analysis of the Laplacian on the complete Riemannian manifold. J. Funct. Anal. 52, 48–79 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    van den Berg, M., Gilkey, P.: Heat flow out of a compact manifold. J. Geom. Anal. 25, 1576–1601 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Miranda Jr., M., Pallara, D., Paronetto, F., Preunkert, M.: On a characterisation of perimeters in \({\mathbb{R}}^N\) via heat semigroup. Ric. Mat. 44, 615–621 (2005)zbMATHGoogle Scholar
  7. 7.
    Miranda Jr., M., Pallara, D., Paronetto, F., Preunkert, M.: Short-time heat flow and functions of bounded variation in \({\mathbb{R}}^N\). Ann. Fac. Sci. Toulouse 16, 125–145 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Preunkert, M.: A semigroup version of the isoperimetric inequality. Semigroup Forum 68, 233–245 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    van den Berg, M.: Heat flow and perimeter in \(\mathbb{R}^{m}\). Potential Anal. 39, 369–387 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    van den Berg, M., Gittins, K.: Uniform bounds for the heat content of open sets in Euclidean space. Differ. Geom. Appl. 40, 67–85 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    van den Berg, M., Gittins, K.: On the heat content of a polygon. J. Geom. Anal. 26, 2231–2264 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Grigor’yan, A.: Heat kernel and Analysis on Manifolds, AMS-IP Studies in Advanced Mathematics, 47, American Mathematical Society, Providence, RI. International Press, Boston, MA (2009)Google Scholar
  13. 13.
    Grigor’yan, A.: The heat equation on noncompact Riemannian manifolds. Math. USSR-Sb. 72, 47–77 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Saloff-Coste, L.: Parabolic Harnack inequality for divergence-form second-order differential operators. Potential Anal. 4, 429–467 (1995)MathSciNetCrossRefzbMATHGoogle Scholar

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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 (, 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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