The heat kernel weighted Hodge Laplacian on noncompact manifolds

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.