Mean Curvature Vector Research Articles

Variational problems of total mean curvature of submanifolds in a sphere

Let H \mathbf {H} be the mean curvature vector of an n n -dimensional submanifold in a Riemannian manifold. The functional H = ∫ ‖ H ‖ n \mathcal {H}=\int \|\mathbf {H}\|^{n} is called the total mean curvature functional. In this paper, we present the first variational formula of H \mathcal {H} and then, for a critical surface of H \mathcal {H} in the ( 2 + p 2+p )-dimensional unit sphere S 2 + p \mathbb {S}^{2+p} , we establish the relationship between the integral of an extrinsic quantity of the surfaces and its Euler characteristic number.

Classification of totally umbilical slant submanifolds of a Kenmotsu manifold

The purpose of this paper is to classify totally umbilical slant submanifolds of a Kenmotsu manifold. We prove that a totally umbilical slant submanifold M of a Kenmotsu manifold ?M is either invariant or anti-invariant or dimM = 1 or the mean curvature vector H of M lies in the invariant normal subbundle. Moreover, we find with an example that every totally umbilical proper slant submanifold is totally geodesic.

Minimal Lagrangian submanifolds via the geodesic Gauss map

For an oriented isometric immersed submanifold of the n-sphere, the spherical Gauss map is the Legendrian immersion of its unit normal bundle into the unit sphere subbundle of the tangent bundle of the sphere, and the geodesic Gauss map projects this into the manifold of oriented geodesics (the Grassmannian of oriented 2-planes in Euclidean n+1-space), giving a Lagrangian immersion of the unit normal bundle into a Kaehler-Einstein manifold. We give expressions for the mean curvature vectors for both the spherical and geodesic Gauss maps in terms of the second fundamental form of the immersion, and show that when it has conformal shape form this depends only on its mean curvature. In particular we deduce that the geodesic Gauss map of every minimal surface in the n-sphere is minimal Lagrangian. We also give simple proofs that: deformations of the immersion always correspond to Hamiltonian deformations of its geodesic Gauss map; the mean curvature vector of the geodesic Gauss map is always a Hamiltonian vector field. This extends work of Palmer on the case when the immersion is a hypersurface.

On the geometry of complete submanifolds immersed in the hyperbolic space

We deal with $n$-dimensional complete submanifolds immersed with parallel nonzero mean curvature vector ${\bf H}$ in the hyperbolic space $\mathbb{H}^{n+p}$. In this setting, we establish sufficient conditions to guarantee that such a submanifold $M^n$ must be pseudo-umbilical, which means that ${\bf H}$ is an umbilical direction. In particular, we conclude that $M^n$ is a minimal submanifold of a small hypersphere of $\mathbb{H}^{n+p}$.

Biharmonic surfaces with parallel mean curvature in complex space forms

We classify complete biharmonic surfaces with parallel mean curvature vector field and non-negative Gaussian curvature in complex space forms.

Extremal Lorentzian surfaces with null $r$-planar geodesics in space forms

We show a congruence theorem for oriented Lorentzian surfaces with horizontal reflector lifts in pseudo-Riemannian space forms of neutral signature. As a corollary, a characterization theorem is obtained for the Lorentzian Boruvka spheres, that is, a full real analytic null $r$-planar geodesic immersion with vanishing mean curvature vector field is locally congruent to the Lorentzian Boruvka sphere in a $2r$-dimensional space form of neutral signature.

PSEUDOHERMITIAN LEGENDRE SURFACES OF SASAKIAN SPACE FORMS

From the point of view of pseudohermitian geometry, we classify Legendre surfaces of Sasakian space forms with non-minimal <TEX>${\hat{C}}$</TEX>-parallel mean curvature vector field for the Tanaka-Webster connection.

Correction to "Surfaces with parallel mean curvature vector in P2(C)"

Correction to "Surfaces with parallel mean curvature vector in P2(C)"

New characterizations of hyperbolic cylinders in semi-Riemannian space forms

Through the study of the so-called linear Weingarten spacelike submanifolds immersed with parallel normalized mean curvature vector in a semi-Riemannian space form Qpn+p(c) of index p and constant sectional curvature c∈{−1,0,1}, our aim in this paper is establish new characterizations for the hyperbolic cylinders of Qpn+p(c). Our approach is based on the use of a Simons type formula related to an appropriated Cheng–Yau modified operator jointly with the application of suitable generalized maximum principles.

SLANT AND LEGENDRE CURVES IN BERGER $su(2)$: THE LANCRET INVARIANT AND QUANTUM SPHERICAL CURVES

Slant and Legendre curves are considered on Berger $su(2)$ and are characterized through the scalar product between the normal at the curve and the vertical vector field; in the helix case they have a proper (non-harmonic) mean curvature vector field. The general expression of these curves is obtained as well as their curvature and torsion. For the slant non-Legendre case we derive a Lancret-type invariant. By using the exponential map we obtain remarkable classes of curves on $S^3(1)$; in the helix case, and taking into account a B.-Y. Chen characterization of Legendre curves, we get a $1$-parameter family of curves in relationship with the spectrum of the quantum harmonic oscillator. These curves, called by us {\it quantum spherical curves}, and their mates, provided by integer multiples of $\pi $, belong to antipodal Hopf fibres.

Mean Curvature Flow of Singular Riemannian Foliations

Given a singular Riemannian foliation on a compact Riemannian manifold, we study the mean curvature flow equation with a regular leaf as initial datum. We prove that if the leaves are compact and the mean curvature vector field is basic, then any finite time singularity is a singular leaf, and the singularity is of type I. This generalizes previous results of Liu–Terng and Koike. In particular, our results can be applied to study the orbits of an isometric action by a compact Lie group.

Marginally outer trapped surfaces in de Sitter space by low-dimensional geometries

A marginally outer trapped surface (MOTS) in de Sitter spacetime is an oriented spacelike surface whose mean curvature vector is proportional to one of the two null sections of its normal bundle. Associated with a spacelike immersed surface there are two enveloping maps into Möbius space (the conformal 3-sphere), which correspond to the two future-directed null directions of the surface normal planes. We give a description of MOTSs based on the Möbius geometry of their envelopes. We distinguish three cases according to whether both, one, or none of the fundamental forms in the normal null directions vanish. Special attention is given to MOTSs with non-zero parallel mean curvature vector. It is shown that any such a surface is generically the central sphere congruence (conformal Gauss map) of a surface in Möbius space which is locally Möbius equivalent to a non-zero constant mean curvature surface in some space form subgeometry.

Magnetic Curves in Three-Dimensional Quasi-Para-Sasakian Geometry

We study (non-geodesic) normal magnetic curves of three-dimensional normal almost paracontact manifolds. We compute their curvature and torsion as well as a Lancret invariant (in the non-Legendre case) and the mean curvature vector field. Two 1-parameter families of magnetic curves (first space like and second time like) are obtained in quasi-para-Sasakian manifolds which are not para-Sasakian; these are non-Legendre helices.

Minimal surface computation using a finite element method on an embedded surface

SummaryWe suggest a finite element method for finding minimal surfaces based on computing a discrete Laplace–Beltrami operator operating on the coordinates of the surface. The surface is a discrete representation of the zero level set of a distance function using linear tetrahedral finite elements, and the finite element discretization is carried out on the piecewise planar isosurface using the shape functions from the background three‐dimensional mesh used to represent the distance function. A recently suggested stabilized scheme for finite element approximation of the mean curvature vector is a crucial component of the method. Copyright © 2015 John Wiley &amp; Sons, Ltd.

Geometry of homogeneous polar foliations of complex hyperbolic spaces

Homogeneous polar foliations of complex hyperbolic spaces have been classified by Berndt and Diaz-Ramos. In this paper, we study geometry of leaves of such foliations: the minimality, the parallelism of the mean curvature vectors, and the congruency of orbits. In particular, we classify minimal leaves.

Isometric embeddings via heat kernel

We combine the heat kernel embedding and Gunther’s implicit function theorem to obtain isometric embeddings of compact Einstein manifolds into Euclidean spaces. As the heat ‡ow time t ! 0+, the second fundamental form of the embedded images has certain normal form, and the mean curvature vectors converge to (large) constant length. These embeddings are canonical in the sense that they are constructed by the eigenfunctions of the Laplacian and intrinsic perturbations.

Rigidity Results on Lagrangian and Symplectic Translating Solitons

In this short note, we prove that an almost calibrated Lagrangian translating soliton must be a plane if it has weighted integrable mean curvature vector or weighted quadratic area growth. Similar results are also true for symplectic translating solitons.

Minimality on biharmonic space-like submanifolds in pseudo-Riemannian space forms

In this paper, we investigate the minimality or the constraint of the mean curvature of three kinds of biharmonic space-like submanifolds in pseudo-Riemannian space forms: (1) pseudo-umbilical ones; (2) the ones with parallel mean curvature vector; (3) with constant mean curvature.

Submanifolds with Parallel Mean Curvature Vector Field in Product Spaces

By Moving Frame Method, we firstly derive some Simons type equations for an n-dimensional submanifold M n with parallel mean curvature vector field μ in \(M^{m}(c)\times \mathbb {R}\), where M m (c) is an m-dimensional space form of constant sectional curvature c and obtain a lower bound of the squared norm of the covariant differential of the second fundamental form h of M n . Then, we use these results to prove some gap theorems on |h|2 and |ϕ|2=|h|2−n|μ|2.

Reilly-type inequalities for p-Laplacian on compact Riemannian manifolds

For a compact Riemannian manifold M immersed into a higher dimensional manifold which can be chosen to be a Euclidean space, a unit sphere, or even a projective space, we successfully give several upper bounds in terms of the norm of the mean curvature vector of M for the first non-zero eigenvalue of the p-Laplacian (1 < p < +∞) on M. This result can be seen as an extension of Reilly’s bound for the first non-zero closed eigenvalue of the Laplace operator.