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 . The result is improved for 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.
Cite
@article{arxiv.1810.02578,
title = {Density problems for second order Sobolev spaces and cut-off functions on manifolds with unbounded geometry},
author = {Debora Impera and Michele Rimoldi and Giona Veronelli},
journal= {arXiv preprint arXiv:1810.02578},
year = {2019}
}
Comments
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