On Complete Hypersurfaces with Constant Mean Curvature and Finite L p -norm Curvature in R n+1

Abstract

By using curvature estimates, we prove that a complete non-compact hypersurface M with constant mean curvature and finiteL n -norm curvature in R n+1 must be minimal, so that M is a hyperplane if it is strongly stable. This is a generalization of the result on stable complete minimal hypersurfaces of R n+1. Moreover, complete strongly stable hypersurfaces with constant mean curvature and finite L 1-norm curvature in R n+1 are considered.