The Hodge cohomology of a conformally compact metric

The Hodge Laplacian acting on differential fc-forms is examined for a class of complete Riemannian manifolds with negative sectional curvature near infinity.These manifolds have C°° compactifications on which the metric is conformal to one smooth up to the boundary with conformal factor p~2, p a defining function for the boundary.An example is the Poincare* ball, which serves as a model throughout.The Schwartz kernel of a parametrix for the Laplacian is described for all degrees k except those near half the dimension of the manifold.Its asymptotics are determined in sufficient detail so that we may identify the L 2 harmonic spaces with the relative and absolute cohomology of the compactification for k < (n -l)/2 and k > (n + l)/2, respectively.In addition, we locate the essential spectrum of the Laplacian in each degree.The construction relies on a calculus of pseudodifferential operators well adapted to the type of degeneracy exhibited by the Laplacian at the boundary of the compactified manifold.