Biharmonic hypersurfaces in Riemannian manifolds

Abstract

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.

f-Biharmonic maps and f-biharmonic submanifolds II

“…It was proved in [Ou1] that a hypersurface, i.e., an isometric immersion M m ֒→ (N m+1 , h) with mean curvature H and the shape operator A is biharmonic if and only if…”

On Biharmonic Legendre curves in $S$-space forms

“…Biharmonic submanifolds have been studied by many geometers. For example, see [2], [3], [7], [8], [11], [12], [13], [14], [15], [18], [20], [21], [22], and the references therein. In a different setting, in [9], Chen defined a biharmonic submanifold M ⊂ E n of the Euclidean space as its mean curvature vector field H satisfies ∆H = 0 , where ∆ is the Laplacian.…”

Harmonic and Biharmonic Maps at Iaşi

“…After space forms, a series of non-constant sectional curvature spaces were considered as ambient spaces for the study of proper-biharmonic submanifolds (see, for example, [4,23,30,31,35,36,54,58,61]). In this respect, one of the research directions was the study of proper-biharmonic submanifolds in Sasakian space forms.…”