Bernstein type theorems for complete submanifolds in space forms

Abstract

AbstractWe 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 \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbb {R}^{n+p}\,(n\le 5)$\end{document} with finite L1 norm of the second fundamental form must be affine n‐dimensional planes. We also prove that any complete noncompact weakly stable hypersurfaces in \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbb {R}^{n+1}\,(n\le 5)$\end{document} with constant mean curvature and finite Ld (d = 1, 2, 3) norm of traceless second fundamental form must be hyperplanes.