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Bernstein type theorems for complete submanifolds in space forms
Author(s) -
Fu HaiPing
Publication year - 2012
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201000039
Subject(s) - mathematics , hyperplane , affine space , euclidean space , norm (philosophy) , type (biology) , mean curvature , affine transformation , second fundamental form , space (punctuation) , pure mathematics , euclidean geometry , combinatorics , mathematical analysis , curvature , geometry , ecology , linguistics , philosophy , political science , law , biology
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.