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Heat Content in Non-compact Riemannian Manifolds

  • M. van den Berg
Open Access
Article
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Abstract

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 \(\Omega \).

Keywords

Heat content Riemannian manifold Volume doubling Poincaré inequality 

Mathematics Subject Classification

58J32 58J35 35K20 

References

  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

Copyright information

© 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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