Let $M$ be a non-compact Riemannian manifold which is "nice enough", and $D$ a generalized Laplacian on it. The construction of the heat kernel for the Laplace-Beltrami operator on $M$ seems to be generalizable to construct a heat kernel for $D$. Has this already been done or tried?
In my setupDetails of the Setup
The two sources I found most informative about the problem are "Eigenvalues in Riemannian Geometry" (Chavel), that constructs the heat kernel for the Laplace-Beltrami operator, and "Heat Kernels and Dirac Operators" (Berline, Getzler, Vergne), that constructs the heat kernel for generalized Laplacians in the compact case.
I assume $M$ isto be a complete non-compact Riemannian manifold with Ricci curvature bounded from below;below. This is the setting of Chavel; the boundedness of the Ricci curvature is only used to prove uniqueness of the heat kernel, and $D_L$ is a generalized Laplacianlikely not to be enough for $M$generalized Laplacians. This means that
A $D_L$generalized Laplacian is a linear second-order differential operator with smooth coefficients, whose second-order part equals the second-order part of the Laplace-Beltrami operator on $M$.
Given that:Overview of the construction and why I think it can be generalized
- In "Eigenvalues in Riemannian Geometry" (Chavel), the minimal positive heat kernel for the Laplace-Beltrami operator on $M$ is constructed as the limit of Dirichlet heat kernels on a sequence of regular domains exhausting $M$. Moreover this heat kernel is proved to be well-behaved: it is a strictly positive $\mathcal{C}^\infty$ function, it is symmetric in the space variables and it is the unique heat kernel on $M$.
- In "Heat Kernels and Dirac Operators" (Berline, Getzler, Vergne), the authors construct the heat kernel associated to a Generalized Laplacian on a compact Riemannian manifold, and it's proven to be $\mathcal{C}^\infty$.
- The maximum principle for solutions of the heat equation, which seems to be central in the proof of Chavel, also holds for general elliptic operators; and the bound on the Ricci curvature, whose implication on the behavior of solutions of a generalized Laplacian are now somehow limited, is only used to prove the uniqueness of the heat kernel.
I suspect that one could construct a "reasonably good" heat kernel for $D_L$ on $M$, proceeding along Chavel builds the lines ofheat kernel in the proofnon-compact case as the limit of Chavel. Sadly I couldn't find anywhereDirichlet heat kernels on a treatisesequence of regular domains exhausting $M$, and he constructs the heat kernels on existence or properties associated to these objectsregular domains by heat kernels on compact manifolds. Has this already been done somewhere? Am I failing to see some crucial obstructionThe main ingredient that prevent to carry out such a program? More generallyChavel uses for the construction of the heat kernel on regular domains is that the heat kernel in the compact case is smooth and almost Euclidean, whatand the main ingredient he uses to pass from regular domains to non-compact manifolds is the statestrong maximum principle for solutions of the art?heat equation. But Berline-Getzer-Vergne construct a smooth heat kernel for generalized Laplacians, that I think to be also almost Euclidean, and the strong maximum principle holds in general for elliptic operators.
Additional notesBackground
A related problem, which doesn't fall in the setting I outlined though, is when the manifold $M$ is compact, and $D_L$ is a generalized Laplacian whose first-order part is mildly singular at some point of $M$.
The notation $D_L$ comes from the fact that I think $D_L$ to be the Laplacian acting on smooth sections of the metrized line bundle $L$ on $M$.
This question is somehow related, but it has a different level of generality and it didn't receive particular attention.
For me $M$ is a modular curve equipped with the hyperbolic metric, and $D$ is the Laplacian acting on the space of smooth modular forms of a fixed positive weight on $M$, i.e. smooth sections of powers of the Hodge bundle equipped with the Petersson metric. Thanks to a regularization procedure I would also be happy to have heat kernels for a generalized Laplacian with "mildly singular" first order coefficients on a compact Riemannian manifold.
A notable achievement is the construction of the heat kernel in this situation by Fay in "Fourier coefficients of the resolvent for a Fuchsian subgroup", while this is enough to indicate that such an heat kernel should in general exists, the construction as in Chavel would be better for me, since I aim to prove some convergence results on it via metric regularization.