Hodge decomposition of the Sobolev space $H^1$ on a space form of nonpositive curvature

Abstract

The Hodge decomposition is well-known for compact manifolds. The result has been extended by Kodaira to include non-compact manifolds and $L^2$ forms. We further extend the Hodge decomposition to the Sobolev space $H^1$ for general $k$-forms on non-compact manifolds of nonpositive constant sectional curvature. As a result, we also obtain a decomposition on $\mathbb R^N$.