On the [formula omitted] independence of the spectrum of the Hodge Laplacian on non-compact manifolds

Abstract

The central aim of this paper is the study of the spectrum of the Hodge Laplacian on differential forms of any order k in L p . The underlying space is a C ∞ -smooth open manifold M N with Ricci Curvature bounded below and uniformly subexponential volume growth. It will be demonstrated that on such manifolds the L p spectrum of the Hodge Laplacian on differential k -forms is independent of p for 1 ⩽ p ⩽ ∞ , whenever the Weitzenböck Tensor on k -forms is also bounded below. It follows as a corollary that the isolated eigenvalues of finite multiplicity are L p independent. The proof relies on the existence of a Gaussian upper bound for the Heat kernel of the Hodge Laplacian. By considering the L p spectra on the Hyperbolic space H N + 1 we conclude that the subexponential volume growth condition is necessary in the case of one-forms. As an application, we will show that the spectrum of the Laplacian on one-forms has no gaps on certain manifolds with a pole or that are in a warped product form. This will be done under less strict curvature restrictions than what has been known so far and it was achieved by finding the L 1 spectrum of the Laplacian.