We study biharmonic hypersurfaces in a generic Riemannian manifold. We first derive an invariant equation for such hypersurfaces generalizing the biharmonic hypersurface equation in space forms studied by Jiang, Chen, Caddeo, Montaldo, and Oniciuc. We then apply the equation to show that the generalized Chen conjecture is true for totally umbilical biharmonic hypersurfaces in an Einstein space, and construct a 2-parameter family of conformally flat metrics and a 4-parameter family of multiply warped product metrics, each of which turns the foliation of an upper-half space of ޒ m by parallel hyperplanes into a foliation with each leaf a proper biharmonic hypersurface. We also study the biharmonicity of Hopf cylinders of a Riemannian submersion.
Biharmonic maps and submanifoldsAll manifolds, maps, and tensor fields that appear in this paper are assumed to be smooth unless stated otherwise.A biharmonic map is a map ϕ : (M, g) → (N , h) between Riemannian manifolds that is a critical point of the bienergy functionalfor every compact subset of M, where τ (ϕ) = Trace g ∇dϕ is the tension field of ϕ. The Euler-Lagrange equation of this functional gives the biharmonic map equation [Jiang 1986b] (which states that ϕ is biharmonic if and only if its bitension field τ 2 (ϕ) vanishes identically. In this equation we used R N to denote the curvature operator of (N , h) MSC2000: 53C12, 58E20, 53C42.

An important theorem about biharmonic submanifolds proved independently by
Chen-Ishikawa [CI] and Jiang [Ji] states that an isometric immersion of a
surface into 3-dimensional Euclidean space is biharmonic if and only if it is
harmonic (i.e, minimal). In a later paper [CMO2], Cadeo-Monttaldo-Oniciuc shown
that the theorem remains true if the target Euclidean space is replaced by a
3-dimensional hyperbolic space form. In this paper, we prove the dual results
for Riemannian submersions, i.e., a Riemannian submersion from a 3-dimensional
space form of non-positive curvature into a surface is biharmonic if and only
if it is harmonic

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.

Inspired by the all-important conformal invariance of harmonic maps on two-dimensional domains, this article studies the relationship between biharmonicity and conformality. We first give a characterization of biharmonic morphisms, analogues of harmonic morphisms investigated by Fuglede and Ishihara, which, in particular, explicits the conditions required for a conformal map in dimension four to preserve biharmonicity and helps producing the first example of a biharmonic morphism which is not a special type of harmonic morphism. Then, we compute the bitension field of horizontally weakly conformal maps, which include conformal mappings. This leads to several examples of proper (i.e. nonharmonic) biharmonic conformal maps, in which dimension four plays a pivotal role. We also construct a family of Riemannian submersions which are proper biharmonic maps.

This paper studies conformal biharmonic immersions. We first study the transformations of Jacobi operator and the bitension field under conformal change of metrics. We then obtain an invariant equation for a conformal biharmonic immersion of a surface into Euclidean 3-space. As applications, we construct a 2-parameter family of non-minimal conformal biharmonic immersions of cylinder into R 3 and some examples of conformal biharmonic immersions of 4-dimensional Euclidean space into sphere and hyperbolic space thus provide many simple examples of proper biharmonic maps with rich geometric meanings. These suggest that there are abundant proper biharmonic maps in the family of conformal immersions. We also explore the relationship between biharmonicity and holomorphicity of conformal immersions of surfaces.

We prove that a totally umbilical biharmonic surface in any 3-dimensional Riemannian manifold has constant mean curvature. We use this to show that a totally umbilical surface in Thurston's 3-dimensional geometries is proper biharmonic if and only if it is a part of S 2 (1/ √ 2) in S 3 . We also give complete classifications of constant mean curvature proper biharmonic surfaces in 3-dimensional geometries and in 3-dimensional Bianchi-Cartan-Vranceanu spaces, and a complete classifications of proper biharmonic Hopf cylinders in 3-dimensional Bianchi-Cartan-Vranceanu spaces.

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