The heat kernel weighted Hodge Laplacian on noncompact manifolds

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.