Complete hypersurfaces with constant mean curvature and finite L p norm curvature in Euclidean spaces

摘要

In this paper, we consider complete hypersurfaces in R n+1 with constant mean curvature H and prove that M n is a hyperplane if the L 2 norm curvature of M n satisfies some growth condition and M n is stable. It is an improvement of a theorem proved by H. Alencar and M. do Carmo in 1994. In addition, we obtain that M n is a hyperplane (or a round sphere) under the condition that M n is strongly stable (or weakly stable) and has some finite L p norm curvature.