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In "Heat Kernels and Spectral Theory" Davies constructs upper and lower bounds for the kernels associated to Dirichlet elliptic operators on regular domains of $\mathbb R^n$. Has anybody done the same thing for (the Dirichlet Laplacian of) regular domains in (complete) Riemannian manifolds?

While Davies' book does have a chapter about Riemannian manifolds, it is not about domains with boundary in a manifold - which is what I am looking for.

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    $\begingroup$ Have you read the paper by Ray and Singer on analytic torsion, their first paper? In that paper they have done estimates for heat kernel for manifold with boundary with Dirichlet/Newmann conditions. I think maybe chapter 5-6. $\endgroup$
    – Bombyx mori
    Commented Jun 16, 2018 at 2:18
  • $\begingroup$ @Bombyxmori: Excellent reference, indeed. Unfortunately, it is not exactly what I am looking for, because in the exponent of the Gaussian bound $\rho(x,y)$ is $d(x,y)$ only in a neighbourhood of the diagonal of $M \times M$ (in order to avoid the cut locus, I believe) and not everywhere, as I want (see formulae 5.2 and 5.3 at pages 176-177). $\endgroup$
    – Alex M.
    Commented Jun 17, 2018 at 14:05

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Have you tried: Grigor’yan, Alexander, Heat kernel and analysis on manifolds, AMS/IP Studies in Advanced Mathematics 47. Providence, RI: American Mathematical Society (AMS); Somerville, MA: International Press (ISBN 978-0-8218-4935-4/hbk). xvii, 482 p. (2009). ZBL1206.58008.

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    $\begingroup$ Yes, I know that book and it doesn't have what I am asking about. $\endgroup$
    – Alex M.
    Commented Jun 15, 2018 at 18:01

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