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

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 Calderón-Zygmund inequalities on manifolds with possibly unbounded curvature and highlight consequences about the validity of the full Omori-Yau maximum principle for the Hessian.

Improved version. As a main modification, we added a final Section 8 including some additional geometric applications of our result. Furthermore, we proved in Section 7 a disturbed L^p-Sobolev-type inequality with weight more general than the previous one. 25 pages. Comments are welcome