Harmonic Mean Curvature Vector Research Articles

Weakly Convex Biharmonic Hypersurfaces in Nonpositive Curvature Space Forms are Minimal

A submanifold M m of a Euclidean space R m+p is said to have harmonic mean curvature vector field if $${\Delta \vec{H}=0}$$ , where $${\vec{H}}$$ is the mean curvature vector field of $${M\hookrightarrow R^{m+p}}$$ and Δ is the rough Laplacian on M. There is a famous conjecture named after Bangyen Chen which states that submanifolds of Euclidean spaces with harmonic mean curvature vector fields are minimal. In this paper we prove that weakly convex hypersurfaces (i.e. hypersurfaces whose principle curvatures are nonnegative) with harmonic mean curvature vector fields in Euclidean spaces are minimal. Furthermore we prove that weakly convex biharmonic hypersurfaces in nonpositively curved space forms are minimal.

CHEN'S CONJECTURE AND ε-SUPERBIHARMONIC SUBMANIFOLDS OF RIEMANNIAN MANIFOLDS

Chen famously conjectured that every submanifold of Euclidean space with harmonic mean curvature vector is minimal. In this note, we establish a much more general statement for a large class of submanifolds satisfying a growth condition at infinity. We discuss in particular two popular competing natural interpretations of the conjecture when the Euclidean background space is replaced by an arbitrary Riemannian manifold. Introducing the notion of ε-superbiharmonic submanifolds, which contains each of the previous notions as special cases, we prove that ε-superbiharmonic submanifolds of a complete Riemannian manifold which satisfy a growth condition at infinity are minimal.

Legendre Surfaces with Harmonic Mean Curvature Vector Field in the Unit $5$-Sphere

We obtain the explicit representation of Legendre surfaces in the unit $5$-sphere with harmonic mean curvature vector field, under the condition that the mean curvature function is constant along a certain special direction.

Screw invariant marginally trapped surfaces in Minkowski 4-space

A spacelike surface in a Lorentzian manifold whose mean curvature vector is lightlike everywhere is called marginally trapped. The classification of marginally trapped surfaces in Minkowski 4-space which are invariant under a subgroup of the Lorentz group that leaves invariant a lightlike direction, i.e. the so-called screw invariant surfaces, is obtained. As corollaries, the screw invariant marginally trapped surfaces with harmonic mean curvature vector and with prescribed Gaussian curvature are explicitly described.

Biharmonic Lorentz hypersurfaces in E14

A submanifold Mrn of pseudo-Euclidean space Es4 is said to have harmonic mean curvature vector if ?H = 0, where H denotes the mean curvature vector field and ? the Laplacian of the induced pseudo-Riemannian metric. We prove that every nondegenerate Lorentz hypersurface M13 of E14 with harmonic mean curvature vector is minimal.

Biharmonic hypersurfaces of the 4-dimensional semi-Euclidean space [formula omitted

A submanifold M r n of a semi-Euclidean space E s m is said to have harmonic mean curvature vector field if Δ H → = 0 → , where H → denotes the mean curvature vector; submanifolds with harmonic mean curvature vector are also known as biharmonic submanifolds. In this paper, we prove that every nondegenerate hypersurface of E s 4 the shape operator of which is diagonalizable, with harmonic mean curvature vector field, is minimal.

Submanifolds with harmonic mean curvature vector field in contact 3-manifolds

Biharmonic or polyharmonic curves and surfaces in 3-dimensional contact manifolds are investigated. Introduction. This paper concerns curves and surfaces in 3-dimen- sional contact manifolds whose mean curvature vector eld is in the kernel of certain elliptic dieren tial operators. First we study submanifolds whose mean curvature vector eld is in the kernel of the Laplacian (submanifolds with harmonic mean curvature vector elds). The study of such submanifolds is inspired by a conjecture of Bang-yen Chen (14):

Biharmonic Hopf Cylinders

This paper concerns curves and surfaces, into indefinite space forms, whose mean curvature vector field is in the kernel of certain elliptic differential operators. It has been inspired by the paper of M. Barros and O.J. Garay [2], where the Riemannian version of this question is solved. We first consider the Laplacian to study indefinite submanifolds with harmonic mean curvature vector field in the normal bundle. This problem is closely related to a conjecture of B.-Y. Chen [5], on Riemannian submanifolds, stated as follows: harmonicity of the mean curvature vector field implies harmonicity of the immersion. Submanifolds with harmonic mean curvature vector field were called by Chen biharmonic submanifolds. In the realm of indefinite submanifolds, counterexamples to that conjecture have been given by the two first authors (see [1]). Biharmonic submanifolds are a special class of submanifolds for which its mean curvature vector is an eigenvector of ∆, that is, ∆H = λH for some real constant λ. First we describe the family of curves whose mean curvature vector field is proper for the Laplacian. This problem has been yet solved for Euclidean curves by M. Barros and O.J. Garay [2]. We have to think of a different Laplacian if we want to characterize curves others than those of both constant curvature and torsion. Since H is a normal vector field, it seems natural to consider the Laplacian associated to the connection in the normal bundle. Then we show that the indefinite Cornu spirals are the only non standard curves in a semi-Riemannian manifold that are biharmonic in the normal bundle. As for surfaces, we deal with the semi-Riemannian Hopf cylinders we introduced in [4]. Then we show that the biharmonicity of them strongly depends on the biharmonicity of the curves to which are associated. In fact, a non standard Hopf cylinder in H1(−1) is biharmonic in the normal bundle if and only if it is associated to a Cornu spiral in Hs(−4). Then we extend the results in [2]. The second operator considered is the Jacobi operator, which was introduced by J. Simons [9] and involves the Laplacian in the normal bundle. A normal vector field is called a Jacobi field if it belongs to the kernel of the Jacobi operator. This operator appears when one studies the second variation of the area functional for compact Riemannian minimal submanifolds. It has been recently used by Barros and Garay [3] to classify Hopf cylinders into S3 with Jacobi mean curvature vector field. In [4] we have made, following [7], a qualitative description of elastic curves into indefinite space forms to be used as a tool to find Lorentzian Willmore tori in H1(−1). Now the Jacobi operator allows us to get a characterization of elastic curves, as well as a characterization of semi-Riemannian Hopf cylinders in H1(−1), in terms of elasticae in Hs(−4), s = 0, 1. We show that a curve in an indefinite real space form has Jacobi mean curvature vector field if and only if it is curvature homothetic to a free elastica. As before, this characterization leads to find Hopf cylinders into H1(−1) with Jacobi mean curvature vector field.

Hypersurfaces of E4 with Harmonic Mean Curvature Vector

AbstractA submanifold Mn of a Euclidean space Em is said to have harmonic mean curvature vector field if Δ H, where H denotes the mean curvature vector. B.‐Y. Chen conjectured that the only submanifolds of Euclidean spaces with harmonic mean curvature vector field, are the minimal ones. In this paper, we give a proof of the theorem that every hypersurface of E4 with harmonic mean curvature vector field is minimal. The method gives insight in the role of the principal curvatures.

On submanifolds with harmonic mean curvature

The classification of curves in E m {E^m} with harmonic mean curvature vector field in the normal bundle is obtained and then it is used to obtain some applications.

Hypersurfaces in E4 with Harmonic Mean Curvature Vector Field

AbstractThe minimal hypersurfaces in E4 are the only hypersurfaces possessing the following property: Its mean curvature vector field is harmonic.