On the generalized Chen's conjecture on biharmonic submanifolds

Abstract

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.

revised version with a new title of the paper and journal reference. Michigan Math Journal, 2012

References in corpus (3)

• A short survey on biharmonic maps between Riemannian manifolds
• Classifications and Isolation Phenomena of Bi-Harmonic Maps and Bi-Yang-Mills Fields
• Some constructions of biharmonic maps and Chen's conjecture on biharmonic hypersurfaces

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• Biminimal properly immersed submanifolds in complete Riemannian manifolds of non-positive curvature
• Biharmonic maps in two dimensions
• Biharmonic holomorphic maps and conformally Kahler geometry
• Subelliptic biharmonic maps