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Bernstein type theorems for complete submanifolds in space forms

✍ Scribed by Hai-Ping Fu

Publisher
John Wiley and Sons
Year
2011
Tongue
English
Weight
119 KB
Volume
285
Category
Article
ISSN
0025-584X

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✦ Synopsis

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 \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbb {R}^{n+p},(n\le 5)$\end{document} 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 \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbb {R}^{n+1},(n\le 5)$\end{document} with constant mean curvature and finite L^d^ (d = 1, 2, 3) norm of traceless second fundamental form must be hyperplanes.

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