Mean Curvature Vector Research Articles

The Dirac Operator on Untrapped Surfaces

We establish a sharp extrinsic lower bound for the first eigenvalue of the Dirac operator of an untrapped surface in initial data sets without apparent horizon in terms of the norm of its mean curvature vector. The equality case leads to rigidity results for the constraint equations with spherical boundary as well as uniqueness results for surfaces with constant mean curvature vector field in Minkowski space.

Submanifolds with constant scalar curvature in a unit sphere

We study the submanifolds in the unit sphere ${\boldsymbol S}^{n+p}$ with constant scalar curvature and parallel normalized mean curvature vector field. In this case, we can generalize the work of the second author about hypersurfaces in Hypersurfaces with constant scalar curvature in space forms to submanifolds in a unit sphere.

Surfaces with parallel mean curvature in S3 x R and H3 x R

We prove a Simons type equation for non-minimal surfaces with par- allel mean curvature vector (pmc surfaces) in M 3 (c) × R, where M 3 (c) is a 3- dimensional space form. Then, we use this equation in order to characterize certain complete non-minimal pmc surfaces.

Hypersurfaces of type [formula omitted] in [formula omitted] with proper mean curvature vector

We show that if the mean curvature vector field of a hypersurface M23 in E24 satisfies the equation ΔH→=αH→ (α a constant) then M23 has constant mean curvature. This equation is a natural generalization of the biharmonic submanifold equation ΔH→=0→.

SLANT CURVES AND PARTICLES IN THREE-DIMENSIONAL WARPED PRODUCTS AND THEIR LANCRET INVARIANTS

AbstractSlant curves are introduced in three-dimensional warped products with Euclidean factors. These curves are characterised by the scalar product between the normal at the curve and the vertical vector field, and an important feature is that the case of constant Frenet curvatures implies a proper mean curvature vector field. A Lancret invariant is obtained and the Legendre curves are analysed as a particular case. An example of a slant curve is given for the exponential warping function; our example illustrates a proper (that is, not reducible to the two-dimensional) case of the Lancret theorem of three-dimensional hyperbolic geometry. We point out an eventuality relationship with the geometry of relativistic models.

Biharmonic PNMC submanifolds in spheres

We obtain several rigidity results for biharmonic submanifolds in $\mathbb{S}^{n}$ with parallel normalized mean curvature vector field. We classify biharmonic submanifolds in $\mathbb{S}^{n}$ with parallel normalized mean curvature vector field and with at most two distinct principal curvatures. In particular, we determine all biharmonic surfaces with parallel normalized mean curvature vector field in $\mathbb{S}^n$. Then we investigate, for (not necessarily compact) proper biharmonic submanifolds in $\mathbb{S}^n$, their type in the sense of B-Y. Chen. We prove: (i) a proper biharmonic submanifold in $\mathbb{S}^n$ is of 1-type or 2-type if and only if it has constant mean curvature ${\mcf}=1$ or ${\mcf}\in(0,1)$, respectively; (ii) there are no proper biharmonic 3-type submanifolds with parallel normalized mean curvature vector field in $\mathbb{S}^n$.

Slant Curves in 3-dimensional Normal Almost Contact Geometry

The aim of this paper is to study slant curves of three-dimensional normal almost contact manifolds as natural generalization of Legendre curves. Such a curve is characterized by means of the scalar product between its normal vector field and the Reeb vector field of the ambient space. In the particular case of a helix we show that it has a proper (non-harmonic) mean curvature vector field. The general expressions of the curvature and torsion of these curves and the associated Lancret invariant are computed as well as the corresponding variants for some particular cases, namely β-Sasakian and cosymplectic. A class of examples is discussed for a normal not quasi-Sasakian 3-manifold.

On pinching theorems for compact pseudo-umbilical submanifold

Abstract Consider submanifolds in the nested space. For a compact pseudoumbilical submanifold with parallel mean curvature vector of a Riemannian submanifold with constant curvature immersed in a quasi-constant curvature Riemannian manifold, two sufficient conditions are given to let the pseudo-umbilical submanifold become a totally umbilical submanifold.

SOME CLASSIFICATION RESULTS ON FINITE-TYPE RULED SUBMANIFOLDS IN A LORENTZ-MINKOWSKI SPACE

Ruled submanifolds of finite type in Lorentz-Minkowski space are studied. We construct a new example of ruled submanifolds with degenerate rulings called a $BS$-kind ruled submanifold, which is of finite type. Also, it is determined by the restricted minimal polynomial of the shape operator associated with the mean curvature vector field.

Biharmonic Submanifolds with Parallel Mean Curvature in $\mathbb{S}^{n}\times\mathbb{R}$

We find a Simons type formula for submanifolds with parallel mean curvature vector (pmc submanifolds) in product spaces M n (c)×ℝ, where M n (c) is a space form with constant sectional curvature c, and then we use it to prove a gap theorem for the mean curvature of certain complete proper-biharmonic pmc submanifolds, and classify proper-biharmonic pmc surfaces in $\mathbb{S}^{n}(c)\times\mathbb{R}$ .

Surfaces with parallel mean curvature vector in complex space forms

We consider surfaces with parallel mean curvature vector (pmc surfaces) in complex space forms and introduce a holomorphic differential on such surfaces. When the complex dimension of the ambient space is equal to two we find a second holomorphic differential and then determine those pmc surfaces on which both differentials vanish. We also provide a reduction of codimension theorem and prove a non-existence result for pmc 2-spheres in complex space forms.

Surfaces of prescribed mean curvature vector in semi-Riemannian manifolds

Solving a second order, quasilinear initial value problem, we prove existence and uniqueness of both space-like and time-like immersions of prescribed mean curvature vector in general semi-Riemannian manifolds. Such surfaces arise for example as critical points of generalized area functionals as well as self-similar solutions of the mean curvature flow.

Slant curves in three-dimensional [formula omitted]-Kenmotsu manifolds

The aim of this paper is to study slant curves of three-dimensional f-Kenmotsu manifolds. These curves are characterized through the scalar product between the normal at the curve and the Reeb vector field. The classification of slant curves in the hyperbolic three-dimensional space is provided as well as some remarkable cases. Slant curves with proper mean curvature vector field are characterized.

Surfaces in $${\mathbb{R}^4}$$ with constant principal angles with respect to a plane

We study surfaces in \({\mathbb{R}^4}\) whose tangent spaces have constant principal angles with respect to a plane. Using a PDE we prove the existence of surfaces with arbitrary constant principal angles. The existence of such surfaces turns out to be equivalent to the existence of a special local symplectomorphism of \({\mathbb{R}^2}\). We classify all surfaces with one principal angle equal to 0 and observe that they can be constructed as the union of normal holonomy tubes. We also classify the complete constant angles surfaces in \({\mathbb{R}^4}\) with respect to a plane. They turn out to be extrinsic products. We characterize which surfaces with constant principal angles are compositions in the sense of Dajczer-Do Carmo. Finally, we classify surfaces with constant principal angles contained in a sphere and those with parallel mean curvature vector field.

Semisimplicity of indefinite extrinsic symmetric spaces and mean curvature

Improving a result of Eschenburg and Kim we give a criterion for semisimplicity of pseudo-Riemannian extrinsic symmetric spaces in terms of the shape operator with respect to the mean curvature vector.

Linear Weingarten spacelike submanifolds in de Sitter space

Let M n be a spacelike linear Weingarten submanifold in a de Sitter space $${S^{n+p}_{p}(1)}$$ with R = aH + b, where R and H are the normalized scalar curvature and the length of the mean curvature vector respectively. In this paper, we give intrinsic and extrinsic conditions for M n to be totally umbilical, respectively.

C-parallel Mean Curvature Vector Fields along Slant Curves in Sasakian 3-manifolds

In this article, using the example of C. Camci((7)) we reconrm necessary suf- �cient condition for a slant curve. Next, wend some necessary and sufficient conditions for a slant curve in a Sasakian 3-manifold to have: (i) a C-parallel mean curvature vector

An invariant theory of marginally trapped surfaces in the four-dimensional Minkowski space

A marginally trapped surface in the four-dimensional Minkowski space is a spacelike surface whose mean curvature vector is lightlike at each point. We associate a geometrically determined moving frame field to such a surface and using the derivative formulas for this frame field we obtain seven invariant functions. Our main theorem states that these seven invariants determine the surface up to a motion in Minkowski space. We introduce meridian surfaces as one-parameter systems of meridians of a rotational hypersurface in the four-dimensional Minkowski space. We find all marginally trapped meridian surfaces.

On spacelike submanifolds with parallel mean curvature in an indefinite space form

In this work we use a Simon’s type inequality for a suitable tensor and apply it to obtain sharp estimates for the supremum of the scalar curvature for complete spacelike submanifold Mn with parallel mean curvature vector in an indefinite space form \({N^{n+p}_p(c)}\).

Convex mean curvature flow with a forcing term in direction of the position vector

A smooth, compact and strictly convex hypersurface evolving in ℝn+1 along its mean curvature vector plus a forcing term in the direction of its position vector is studied in this paper. We show that the convexity is preserving as the case of mean curvature flow, and the evolving convex hypersurfaces may shrink to a point in finite time if the forcing term is small, or exist for all time and expand to infinity if it is large enough. The flow can converge to a round sphere if the forcing term satisfies suitable conditions which will be given in the paper. Long-time existence and convergence of normalization of the flow are also investigated.