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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



The heat kernel weighted Hodge Laplacian
on noncompact manifolds

Author: Edward L. Bueler
Journal: Trans. Amer. Math. Soc. 351 (1999), 683-713
MSC (1991): Primary 58A14, 35J10, 58G11
MathSciNet review: 1443866
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Abstract: On a compact orientable Riemannian manifold, the Hodge Laplacian $\triangle $ has compact resolvent, therefore a spectral gap, and the dimension of the space $\mathcal{H}^{p} = \ker \triangle ^{p}$ of harmonic $p$-forms is a topological invariant. By contrast, on complete noncompact Riemannian manifolds, $\triangle $ is known to have various pathologies, among them the absence of a spectral gap and either ``too large'' or ``too small'' a space $\mathcal{H}^{p}$. In this article we use a heat kernel measure $d\mu $ to determine the space of square-integrable forms and to construct the appropriate Laplacian $\triangle _{\mu }$. We recover in the noncompact case certain results of Hodge's theory of $\triangle $ in the compact case. If the Ricci curvature of a noncompact connected Riemannian manifold $M$ is bounded below, then this ``heat kernel weighted Laplacian'' $\triangle _{\mu }$ acts on functions on $M$ in precisely the manner we would wish, that is, it has a spectral gap and a one-dimensional kernel. We prove that the kernel of $\triangle _{\mu }$ on $n$-forms is zero-dimensional on $M$, as we expect from topology, if the Ricci curvature is nonnegative. On Euclidean space, there is a complete Hodge theory for $\triangle _{\mu }$. Weighted Laplacians also have a duality analogous to Poincaré duality on noncompact manifolds. Finally, we show that heat kernel-like measures give desirable spectral properties (compact resolvent) in certain general cases. In particular, we use measures with Gaussian decay to justify the statement that every topologically tame manifold has a strong Hodge decomposition.

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Edward L. Bueler
Affiliation: Department of Mathematics Sciences, University of Alaska, Fairbanks, Alaska 99775

Keywords: Hodge theory, heat kernels, weighted cohomology, Schr\"{o}dinger operators
Received by editor(s): July 29, 1996
Received by editor(s) in revised form: February 5, 1997
Article copyright: © Copyright 1999 American Mathematical Society