Density problems for second order Sobolev spaces and cut-off functions on manifolds with unbounded geometry

TL;DR

This paper establishes the density of smooth compactly supported functions in second order Sobolev spaces on certain non-compact manifolds with unbounded curvature, using new cut-off functions and geometric estimates.

Contribution

It introduces new cut-off functions and proves density results for Sobolev spaces on manifolds with sub-quadratic curvature growth, extending previous results.

Findings

Existence of distance-like functions with bounded gradient and controlled Hessian.

Density of smooth compactly supported functions in $W^{2,p}$ spaces under new geometric conditions.

New Sobolev and Calderón-Zygmund inequalities on manifolds with unbounded curvature.

Abstract

We consider complete non-compact manifolds with either a sub-quadratic growth of the norm of the Riemann curvature, or a sub-quadratic growth of both the norm of the Ricci curvature and the squared inverse of the injectivity radius. We show the existence on such a manifold of a distance-like function with bounded gradient and mild growth of the Hessian. As a main application, we prove that smooth compactly supported functions are dense in $W^{2,p}$. The result is improved for $p=2$ avoiding both the upper bound on the Ricci tensor, and the injectivity radius assumption. As further applications we prove new disturbed Sobolev and Calder\'on-Zygmund inequalities on manifolds with possibly unbounded curvature and highlight consequences about the validity of the full Omori-Yau maximum principle for the Hessian.