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We prove the triviality of the first L 2 cohomology class of based path spaces of Riemannian manifolds furnished with Brownian motion measure, and the consequent vanishing of L 2 harmonic one-forms. We give explicit formulae for closed and co-closed one-forms expressed as differentials of functions and co-differentials of L 2 two-forms, respectively; these are considered as extended Clark–Ocone formulae. A feature of the proof is the use of the temporal structure of path spaces to relate a rough exterior derivative operator on one-forms to the exterior differentiation operator used to construct the de Rham complex and the self-adjoint Laplacian on L 2 one-forms. This Laplacian is shown to have a spectral gap.

We establish the Strong Lp-Hodge decomposition theorem and the Lp-Poincaré inequalities on differential forms over complete non-compact Riemannian manifolds. As applications, we prove some Lp-estimates and existence theorems for the de Rham operator as well as some vanishing theorems for the Lp-cohomology and the Lp-torsion over complete non-compact Riemannian manifolds with suitable geometric conditions.

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 consider two basic potential theoretic problems in Riemannian manifolds: Hodge decompositions and Maxwell's equations. Here we are concerned with smoothness and integrability assumptions. In the context of L p forms in Lipschitz domains, we show that both are well posed provided that 2− e p e , for some e >0, depending on the domain. Our approach is constructive (in the sense that we produce integral representation formulas for the solutions) and emphasizes the intimate connections between the two problems at hand. Applications to other related PDEs, such as boundary problems for the Hodge Dirac operator, are also presented.

A Hodge decomposition theorem is proved for the leafwise Laplacian of a Riemannian foliation on a closed Riemannian manifold. © 1991.

We discuss the theory of infinite-dimensional manifolds from the point of view of establishing a widely applicable framework for generalization of the finite-dimensional Hodge theory. The principal result is the development of an exterior algebra based on a weakened definition of differentiation, so that “ C ∞ ” partitions of unity are available for paracompact manifolds modelled on arbitrary real separable Banach spaces. We prove a Poincare lemma for our new notion of exterior differentiation, and go on to discuss the relationship of the exterior derivative with current research efforts toward the definition of an infinite-dimensional Laplace-Beltrami operator.

An L-2 theory of differential forms is proposed for the Banach manifold of continuous paths on a Riemannian manifold M furnished with its Brownian motion measure. Differentiation must be restricted to certain Hilbert space directions, the H-tangent vectors. To obtain a closed exterior differential operator the relevant spaces of differential forms, the H-forms, are perturbed by the curvature of M. A Hodge decomposition is given for L-2 H-one-forms, and the structure of H-two-forms is described. The dual operator d* is analysed in terms of a natural connection on the H-tangent spaces. Malliavin calculus is a basic tool. (c) 2007 Elsevier Inc. All rights reserved.