Bernstein type theorems for complete submanifolds in space forms

Abstract

We study the Bernstein type problem for complete submanifolds in the space forms. In particular, we prove that any complete super stable minimal submanifolds in an (n + p)-dimensional Euclidean space R n + p (n ≤ 5) with finite L 1 norm of the second fundamental form must be affine n-dimensional planes. We also prove that any complete noncompact weakly stable hypersurfaces in R n + 1 (n ≤ 5) with constant mean curvature and finite L d (d = 1, 2, 3) norm of traceless second fundamental form must be hyperplanes.

Lpharmonic 1-forms and first eigenvalue of a stable minimal hypersurface

“…One crucial step in the proofs of our theorems is to obtain an inequality of Simons' type for |φ| p rather than |φ|, where φ is a geometric quantity which we want to analyze. This kind of inequalities has been used in [Deng 2008;Fu 2012;Shen and Zhu 2005]. Equipped with this Simons' type inequality, we extend the original Bochner technique to our cases.…”