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

2018/10/05 by Impera, Debora, Rimoldi, Michele, Veronelli, Giona
#Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1810.02578

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.

Related

• Higher order distance-like functions and Sobolev spaces
• The Calderón-Zygmund inequality and Sobolev spaces on noncompact Riemannian manifolds
• Hessian heat kernel estimates and Calderón-Zygmund inequalities on complete Riemannian manifolds
• Sequences of Laplacian cut-off functions
• Geometry of manifolds with densities