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.In this paper, we prove that this conjecture is false by constructing a foliation 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.Date: 01/18/2011. 1991 Mathematics Subject Classification. 58E20, 53C12, 53C42.

Biharmonic hypersurfaces with constant scalar curvature in space forms

“…Recently, Ou and Tang in [34] constructed a family of counter-examples that the generalized Chen's conjecture is false when the ambient space has non-constant negative sectional curvature. However, the generalized Chen's conjecture remains open when the ambient spaces have constant sectional curvature.…”

Harmonic Maps and Biharmonic Maps

“…Theorem 3 is sharp, since one cannot weaken the assumptions. Indeed, the generalized Chen's conjecture does not hold if (M, g) is not complete (cf., the counter examples of Ou and Tang [27]). The two assumptions of finiteness of the energy and bienergy are necessary.…”

“…Notice that the generalized Chen's conjecture was solved negatively by giving a counter example by Ou and Tang [27,41]. We first give several comments on Chen's conjecture.…”

Bi-f-harmonic curves and hypersurfaces

“…But in recent years, the authors of [15] proved that the Generalized Chen's conjecture is not true by constructing examples of proper biharmonic hypersurfaces in a 5-dimensional space of non-constant negative sectional curvature. For some recent geometric studies of general biharmonic maps and biharmonic submanifolds see ( [14], [2], [12], [15], [16], [9], [18], [19]) and the references therein.…”