The generalized Chen's conjecture on biharmonic submanifolds is false

Details The generalized Chen's conjecture on biharmonic submanifolds is false The generalized Chen's conjecture on biharmonic submanifolds is false

2010-06-09

arxiv.org/abs/1006.1838v2

Mathematics

Differential Geometry

revised version with a correction to a statement in Theorem 2.2 and corresponding changes to Lemma 3.1

Scientific paper

The generalized Chen's conjecture on biharmonic submanifolds asserts that any biharmonic submanifold of a non-positively curved manifold is minimal (see e.g., [CMO1], [MO], [BMO1], [BMO2], [BMO3], [Ba1], [Ba2], [Ou1], [Ou2], [IIU]). In this paper, we prove that this conjecture is false by constructing foliations of proper biharmonic hyperplanes in a 5-dimensional conformally flat space with negative sectional curvature. Many examples of proper biharmonic submanifolds of non-positively curved spaces are also given.

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