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I'd like to understand some issues about the heat problem related to the Laplacian of a Riemannian manifold especially when the manifold is noncompact. So first recall the heat equation on a Riemannian $C^{\infty}$-Manifold $(M,g)$. When $M$ is compact, there is a unique function, the heat kernel, $K \in C^\infty (M \times M \times \mathbb{R^+})$ satisfying

$(\partial_t + \Delta_x)K(x,y,t) = 0$

and

$ \lim_{t \rightarrow 0} \ \int\limits_{M} K(x,y,t)f(y)dvol_y = f(x) \ \ \forall f \in L^2(M) $

And as far as I know this fact stays true If $\partial M \neq \varnothing$ when one inserts an appropriate boundary condition (i.e. Dirichlet-condition) for the Problem.

When $(M,g)$ is considered to be noncompact (the examples I want to examine are mainly conformally compact metrics) there is, as far as I know, no such $C^{\infty}$-Kernel but a distributional solution of the heat problem defining (as a distributional Schwartz kernel) a kind of (pseudodifferential ?) operator, that "has a conormal singularity" on the diagonal $D:= \{(x,x,0) | x \in M \} \subset M\times M \times \mathbb{R}$. My first question is: Is that correct so far? (1)

edit: After further investigation I got confused about the question, if a $C^{\infty}$-heat-kernel for a noncompact manifold exists. In Isaac Chavel's book "Eigenvalues in Riemannian Geometry" there is a theorem on page 188 in chapter XIII, that says: There IS a a $C^{\infty}$-kernel. Is that right or did I miss something? (Here is the Google-books link: http://books.google.de/books?id=0v1VfTWuKGgC&printsec=frontcover&hl=de#v=onepage&q&f=false)

Until now I'm not very familiar with pseudodifferential operator calculus (but I want to improve my knowledge) and so I ask myself the following question:

(2) Why can't there be an $C^{\infty}$-Solution like in the compact case? How can one prove that there can't be? Or can anybody refer to a book/paper/lecture note where this question is concerned?

(3) What does "conormal singularity" with respect to some submanifold $Y$ exactly mean? I just vague know that it means to have a singularity much stronger in normal direction than in tangential direction and it is related to the Fourier transform. Nevertheless it stays mysterious to me. How can the occurence of the conormal singularity of the heat kernel for noncompact manifolds be seen?

I would be happy for any comments or small hints, too! Thank you for your help.

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1 Answer 1

up vote 3 down vote accepted

You have answered the question on your own:

Theorem 4 in Chapter VIII (on page 188) in the book Eigenvalues in Riemannian Geometry by Isaac Chavel (which is the main reference for such questions, as it seems) states the existence of a smooth heat kernel on non-compact manifolds.

Though there are some issues with uniqueness of the heat kernel, there seems to be a unique minimal heat kernel.

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