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Through an Euler–Hopf-type formula, we establish existence result to a Nirenberg-type problem involving the square root of the Laplacian in sphere S 2 .

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 Weitzenbock 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.

We present comparison theorems for the eigenvalues of the covariant Laplacian acting on the sections of vector bundle with the structure group U ( N ).

We study the spectral properties of the Laplacian on infinite Sierpin ski gaskets. We prove that the Laplacian with the Neumann boundary condition has pure point spectrum. Moreover, the set of eigenfunctions with compact support is complete. The same is true if the infinite Sierpin ski gasket has no boundary, but is false for the Laplacian with the Dirichlet boundary condition. In all these cases we describe the spectrum of the Laplacian and all the eigenfunctions with compact support. To obtain these results, first we prove certain new formulas for eigenprojectors of the Laplacian on finite Sierpin ski pre-gaskets. Then we prove that the spectrum of the discrete Laplacian on a Sierpin ski lattice is pure point, and the eigenfunctions are localized. 1998 Academic Press