On the Lp independence of the spectrum of the Hodge Laplacian on non-compact manifolds

AbstractThe central aim of this paper is the study of the spectrum of the Hodge Laplacian on differential forms of any order k in Lp. The underlying space is a C∞-smooth open manifold MN with Ricci Curvature bounded below and uniformly subexponential volume growth. It will be demonstrated that on such manifolds the Lp 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 Lp independent. The proof relies on the existence of a Gaussian upper bound for the Heat kernel of the Hodge Laplacian. By considering the Lp spectra on the Hyperbolic space HN+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 L1 spectrum of the Laplacian