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

## TL;DR

This paper extends the Hodge decomposition to the Sobolev space $H^1$ for differential forms on non-compact manifolds with nonpositive constant curvature, including Euclidean space, broadening its applicability.

## Contribution

It generalizes the Hodge decomposition to the Sobolev space $H^1$ on non-compact, nonpositively curved manifolds, including Euclidean space, for all $k$-forms.

## Key findings

- Hodge decomposition extended to $H^1$ on non-compact manifolds
- Decomposition applies to $
^N$
- Results encompass general $k$-forms

## 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 $\mathbb R^N$.