The generalized Chen's conjecture on biharmonic submanifolds is false

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.