Mean Curvature Vector Research Articles

Mean Curvature Flow of Submanifolds With Small Traceless Second Fundamental Form

Consider a family of smooth immersions F(; t) : Mn ! Rn+k of submanifolds in Rn+k moving by mean curvature ow @F @t = ~H, where ~H is the mean curvature vector for the evolving submanifold. We prove that for any n 2 and k 1, the ow starting from a closed submanifold with small L2-norm of the traceless second fundamental form contracts to a round point in nite time, and the corresponding normalized ow converges exponentially in the C1-topology, to an n-sphere in some subspace Rn+1 of Rn+k.

On the pedal surfaces of 2-d surfaces with the constant support function in E^4

In this paper, we consider the pedal surface M of a 2-d surface M with the constant support function that parameter lines of M are the lines curvatures, in 4-dimensional Euclidean Space 4 E . We investigated the relations between coefficients of the first, the second and the third fundamental forms of M and M . Also, the relations between the Gauss curvatures, the Gauss torsions, the mean curvatures, the mean curvature vectors and the principal normal curvatures of M and M were obtained. Furthermore, we proved that pedal of every Aminov surface is a curve if the parameter lines of it are the lines curvatures. Finally, we drew some special surfaces and pedals of them by using parallel projection on the Maple 17 software package programing.

Stabilized Finite Element Approximation of the Mean Curvature Vector on Closed Surfaces

The mean curvature vector of a surface is obtained by letting the Laplace--Beltrami operator act on the embedding of the surface in ${\bf R}^3$. In this contribution we develop a stabilized finite element approximation of the mean curvature vector of certain piecewise linear surfaces which enjoys first order convergence in $L^2$. The stabilization involves the jump in the tangent gradient in the direction of the outer co-normals at each edge in the surface mesh. We consider both standard meshed surfaces and so-called cut surfaces that are level sets of piecewise linear distance functions. We prove a priori error estimates and verify the theoretical results numerically.

ON THE QUASI-MINIMAL SURFACES IN THE 4-DIMENSIONAL DE SITTER SPACE WITH 1-TYPE GAUSS MAP

In this paper, we study the Gauss map of the surfaces in the de Sitter space-time S1(1). First, we prove that a space-like surface lying in the de Sitter space-time has pointwise 1-type Gauss map if and only if it has parallel mean curvature vector. Then, we obtain the complete classification of the quasi-minimal surfaces with 1-type Gauss map.

CMC biconservative surfaces in [formula omitted] and [formula omitted

We classify non-minimal biconservative surfaces with parallel mean curvature vector field in Sn×R and Hn×R. When these surfaces do not lie in Sn or Hn and they are not vertical cylinders, we find their explicit (local) equation. We also prove a result on the compactness of biconservative surfaces with constant mean curvature in Hadamard manifolds.

A classification result for helix surfaces with parallel mean curvature in product spaces

We determine all helix surfaces with parallel mean curvature vector field, which are not minimal or pseudo-umbilical, in spaces of type $M^n(c)\times\mathbb{R}$, where $M^n(c)$ is a simply-connected $n$-dimensional manifold with constant sectional curvature $c$.

Slant and Legendre curves in Bianchi-Cartan-Vranceanu geometry

We study Legendre and slant curves for Bianchi-Cartan-Vranceanu metrics. These curves are characterized through the scalar product between the normal at the curve and the vertical vector field and in the helix case they have a proper (non-harmonic) mean curvature vector field. The general expression of the curvature and torsion of these curves and the associated Lancret invariant (for the slant case) are computed as well as the corresponding variant for some particular cases. The slant (particularly Legendre) curves which are helices are completely determined.

The non-parabolicity of infinite volume ends

Let M m , with m � 3, be an m-dimensional complete noncompact manifold isometrically immersed in a Hadamard manifold ¯ M. Assume that the mean curvature vector has finite L p -norm, for some 2 � pm. We prove that each end of M must either have finite volume or be non-parabolic.

On biharmonic submanifolds in non-positively curved manifolds

In the biharmonic submanifolds theory there is a generalized Chen’s conjecture which states that biharmonic submanifolds in a Riemannian manifold with non-positive sectional curvature must be minimal. This conjecture turned out false by a counter example of Y.L. Ou and L. Tang in Ou and Tang (2012). However it remains interesting to find out sufficient conditions which guarantee this conjecture to be true. In this note we prove that: 1. Any complete biharmonic submanifold (resp. hypersurface) (M,g) in a Riemannian manifold (N,h) with non-positive sectional curvature (resp. Ricci curvature) which satisfies an integral condition: for some p∈(0,+∞), ∫M|H→|pdμg<+∞, where H→ is the mean curvature vector field of M↪N, must be minimal. This generalizes the recent results due to N. Nakauchi and H. Urakawa in Nakauchi and Urakawa (2013, 2011). 2. Any complete biharmonic submanifold (resp. hypersurface) in a Riemannian manifold of at most polynomial volume growth whose sectional curvature (resp. Ricci curvature) is non-positive must be minimal. 3. Any complete biharmonic submanifold (resp. hypersurface) in a non-positively curved manifold whose sectional curvature (resp. Ricci curvature) is smaller than −ϵ for some ϵ>0 which satisfies that ∫Bρ(x0)|H→|p+2dμg(p≥0) is of at most polynomial growth of ρ, must be minimal. We also consider ε-superbiharmonic submanifolds defined recently in Wheeler (2013) by G. Wheeler and prove similar results for ε-superbiharmonic submanifolds, which generalize the result in Wheeler (2013).

Curvature in image and shape processing

The laplacian operator applied to the coordinates of a manifold provides the mean curvature vector. Manipulating the metric of the manifold or interpreting its coordinates in various ways provide useful tools for shape and image processing and representation. We will review some of these tools focusing on scale invariant geometry, curvature flow with respect to an embedding of the image manifold in a high dimensional space, and object segmentation by active contours defined via the shape laplacian operator. Such generalizations of the curvature vector and its numerical approximation as part of an image flow or triangulated shape representation, demonstrate the omnipresence of this operator and its usefulness in imaging sciences.

Surfaces with parallel mean curvature in Sasakian space forms

We study the global geometry of surfaces in Sasakian space forms whose mean curvature vector is parallel in the normal bundle (these include the Riemannian Heisenberg space of dimension \(2n+1\)). We prove a codimension reduction theorem. We introduce two holomorphic quadratic differentials on anti-invariant such surfaces and use them to obtain classification theorems.

Uniqueness Theorems for Free Boundary Minimal Disks in Space Forms

We show that a minimal disk satisfying the free boundary condition in a constant curvature ball of any dimension is totally geodesic. We weaken the condition to parallel mean curvature vector in which case we show that the disk lies in a three dimensional constant curvature submanifold and is totally umbilic. These results extend to higher dimensions earlier three dimensional work of J. C. C. Nitsche and R. Souam.

On a Liu–Yau type inequality for surfaces

Let $\Omega$ be a compact and mean-convex domain with smooth boundary $\Sigma:=\partial\Omega$, in an initial data set $(M^3,g,K)$, which has no apparent horizon in its interior. If $\Sigma$ is spacelike in a spacetime $(\E^4,g\_\E)$ with spacelike mean curvature vector $\mathcal{H}$ such that $\Sigma$ admits an isometric and isospin immersion into $\mathbb{R}^3$ with mean curvature $H\_0$, then: \begin{eqnarray*} \int\_{\Sigma}|\mathcal{H}|d\Sigma\leq\int\_{\Sigma}\frac{H\_0^2}{|\mathcal{H}|}d\Sigma. \end{eqnarray*} If equality occurs, we prove that there exists a local isometric immersion of $\Omega$ in $\mathbb{R}^{3,1}$ (the Minkowski spacetime) with second fundamental form given by $K$. In Theorem liu-yau-minkowski, we also examine, under weaker conditions, the case where the spacetime is the $(n+2)$-dimensional Minkowski space $\mathbb{R}^{n+1,1}$ and establish a stronger rigidity result.

Helix surfaces in Euclidean spaces

We study helix surfaces with parallel mean curvature vector in Euclidean space from a local point of view. Our main result says that they are either part of a cylinder of revolution or a plane. One way to prove this is with the generalization we found about the Laplacian of a support function of a hypersurface. This allows us to study the constant mean curvature surfaces in space forms which have constant angle with respect to a closed and conformal vector field. The result we find says that these surfaces are totally umbilic.

Lagrangian immersions in the product of Lorentzian two manifolds

For Lorentzian two manifolds \((\Sigma _1,g_1)\) and \((\Sigma _2,g_2)\) we consider the two product para-Kähler structures \((G^{\epsilon },J,\Omega ^{\epsilon })\) defined on the product four manifold \(\Sigma _1\times \Sigma _2\), with \(\epsilon =\pm 1\). We show that the metric \(G^{\epsilon }\) is locally conformally flat (resp. Einstein) if and only if the Gauss curvatures \(\kappa _1,\kappa _2\) of \(g_1,g_2\), respectively, are both constants satisfying \(\kappa _1=-\epsilon \kappa _2\) (resp. \(\kappa _1=\epsilon \kappa _2\)). We give the conditions on the Gauss curvatures for which every Lagrangian surface with parallel mean curvature vector is the product \(\gamma _1\times \gamma _2\subset \Sigma _1\times \Sigma _2\), where \(\gamma _1\) and \(\gamma _2\) are geodesics. We study Lagrangian surfaces in the product \(d{\mathbb S}^2\times d{\mathbb S}^2\) with parallel mean curvature vector and finally, we explore the stability and Hamiltonian stability of certain minimal Lagrangian surfaces and \(H\)-minimal surfaces.

Four Classes of Surfaces with Constant Mean Curvature in S 3 × R and H 3 × R

Deforming rotation surfaces with constant mean curvature in S3 and H3 to S3 × R and H3 × R respectvely, we give four classes of surfaces with mean curvature vector of constant length in S3 × R and H3 × R. We have complete minimal surfaces in S3 × R and H3 × R. Also we obtain minimal 2-tori in S3 × S1, some of which are embedded.

Marginally trapped surfaces in spaces of oriented geodesics

We investigate the geometric properties of marginally trapped surfaces (surfaces which have null mean curvature vector) in the spaces of oriented geodesics of Euclidean 3-space and hyperbolic 3-space, endowed with their canonical neutral Kaehler structures. We prove that every rank one surface in these four manifolds is marginally trapped. In the Euclidean case we show that Lagrangian rotationally symmetric sections are marginally trapped and construct an explicit family of marginally trapped Lagrangian tori. In the hyperbolic case we explore the relationship between marginally trapped and Weingarten surfaces, and construct examples of marginally trapped surfaces with various properties.

Umbilical Submanifolds of Sn × R

AbstractWe give a complete classification of umbilical submanifolds of arbitrary dimension and codimension of ×ℝ, extending the classification of umbilical surfaces in ×ℝ by Souam and Toubiana as well as the local description of umbilical hypersurfaces in × ℝ by Van der Veken and Vrancken. We prove that, besides small spheres in a slice, up to isometries of the ambient space they come in a two-parameter family of rotational submanifolds whose substantial codimension is either one or two and whose profile is a curve in a totally geodesic ×ℝ or ×ℝ, respectively, the former case arising in a one-parameter family. All of them are diffeomorphic to a sphere, except for a single element that is diffeomorphic to Euclidean space. We obtain explicit parametrizations of all such submanifolds. We also study more general classes of submanifolds of × R and ℍn × ℝ. In particular, we give a complete description of all submanifolds in those product spaces for which the tangent component of a unit vector field spanning the factor ℝ is an eigenvector of all shape operators. We show that surfaces with parallel mean curvature vector in ×ℝ and ℍn×ℝ having this property are rotational surfaces, and use this fact to improve some recent results by Alencar, do Carmo, and Tribuzy. We also obtain a Dajczer-type reduction of codimension theorem for submanifolds of × ℝ and ℍn × ℝ.

Minimizing Properties of Critical Points of Quasi-Local Energy

In relativity, the energy of a moving particle depends on the observer, and the rest mass is the minimal energy seen among all observers. The Wang-Yau quasi-local mass for a surface in spacetime introduced in [7] and [8] is defined by minimizing quasi-local energy associated with admissible isometric embeddings of the surface into the Minkowski space. A critical point of the quasi-local energy is an isometric embedding satisfying the Euler-Lagrange equation. In this article, we prove results regarding both local and global minimizing properties of critical points of the Wang-Yau quasi-local energy. In particular, under a condition on the mean curvature vector we show a critical point minimizes the quasi-local energy locally. The same condition also implies that the critical point is globally minimizing among all axially symmetric embedding provided the image of the associated isometric embedding lies in a totally geodesic Euclidean 3-space.

Existence of immersed spheres minimizing curvature functionals in compact 3-manifolds

We study curvature functionals for immersed 2-spheres in a compact, three-dimensional Riemannian manifold M. Under the assumption that the sectional curvature of M is strictly positive, we prove the existence of a smoothly immersed sphere minimizing the L^{2} integral of the second fundamental form. Assuming instead that the sectional curvature is less than or equal to 2, and that there exists a point in M with scalar curvature bigger than 6, we obtain a smooth 2-sphere minimizing the integral of 1/4|H|^{2} +1, where H is the mean curvature vector.