Density problems on vector bundles and manifolds

TL;DR

This paper investigates the density of smooth sections and functions in differential operators and Sobolev spaces on Riemannian manifolds, establishing general conditions under which these densities hold.

Contribution

It proves the density of smooth, compactly supported sections in differential operator domains and functions in Sobolev spaces under broad geometric assumptions.

Findings

Smooth, compactly supported sections are dense in operator domains.

Smooth, compactly supported functions are dense in Sobolev spaces with Ricci curvature bounds.

Results apply to a wide class of Riemannian manifolds.

Abstract

We study some canonical differential operators on vector bundles over smooth, complete Riemannian manifolds. Under very general assumptions, we show that smooth, compactly supported sections are dense in the domains of these operators. Furthermore, we show that smooth, compactly supported functions are dense in second order Sobolev spaces on such manifolds under the sole additional assumption that the Ricci curvature is uniformly bounded from below.