Nonzero Constant Mean Curvature Research Articles

On spherical submanifolds with finite type spherical Gauss map

Abstract Chen and Lue (2007) initiated the study of spherical submanifolds with finite type spherical Gauss map. In this paper, we firstly prove that a submanifold Mn of the unit sphere Sm−1 has non-mass-symmetric 1-type spherical Gauss map if and only if Mn is an open part of a small n-sphere of a totally geodesic (n + 1)- sphere Sn+1 . Sm−1. Thenwe show that a non-totally umbilical hypersurfaceM of Sn+1 with nonzero constant mean curvature in Sn+1 has mass-symmetric 2-type spherical Gauss map if and only if the scalar curvature curvature of M is constant. Finally, we classify constant mean curvature surfaces in S3 with mass-symmetric 2-type spherical Gauss map.

On regular algebraic surfaces of $\mathbb{R}^3$ with constant mean curvature

We consider regular surfaces $M$ that are given as the zeros of a polynomial function $p : \mathbb{R}^3 \to \mathbb{R}$, where the gradient of $p$ vanishes nowhere. We assume that $M$ has non-zero constant mean curvature and prove that there exist only two examples of such surfaces, namely the sphere and the circular cylinder.

SOME CLASSIFICATIONS OF LORENTZIAN SURFACES WITH FINITE TYPE GAUSS MAP IN THE MINKOWSKI 4-SPACE

In this paper we study the Lorentzian surfaces with finite type Gauss map in the four-dimensional Minkowski space. First, we obtain the complete classification of minimal surfaces with pointwise 1-type Gauss map. Then, we get a classification of Lorentzian surfaces with nonzero constant mean curvature and of finite type Gauss map. We also give some explicit examples.

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.

Hyperbolic submanifolds with finite type hyperbolic Gauss map

We study submanifolds of hyperbolic spaces with finite type hyperbolic Gauss map. First, we classify the hyperbolic submanifolds with 1-type hyperbolic Gauss map. Then we prove that a non-totally umbilical hypersurface Mn with nonzero constant mean curvature in a hyperbolic space [Formula: see text] has 2-type hyperbolic Gauss map if and only if M has constant scalar curvature. We also classify surfaces with constant mean curvature in the hyperbolic space [Formula: see text] having 2-type hyperbolic Gauss map. Moreover we show that a horohypersphere in [Formula: see text] has biharmonic hyperbolic Gauss map.

New Examples of Constant Mean Curvature Surfaces in S2×R and H2×R

We construct non-zero constant mean curvature H surfaces in the product spaces $\mathbb{S}^2 \times \mathbb{R}$ and $\mathbb{H}^2\times \mathbb{R}$ by using suitable conjugate Plateau constructions. The resulting surfaces are complete, have bounded height and are invariant under a discrete group of horizontal translations. In $\mathbb{S}^2\times\mathbb{R}$ (for any $H > 0$) or $\mathbb{H}^2\times\mathbb{R}$ (for $H > 1/2$), a 1-parameter family of unduloid-type surfaces is obtained, some of which are shown to be compact in $\mathbb{S}^2\times\mathbb{R}$. Finally, in the case of $H = 1/2$ in $\mathbb{H}^2 \times \mathbb{R}$, the constructed examples have the symmetries of a tessellation of $\mathbb{H}^2$ by regular polygons.

Classification Results for λ-Biminimal Surfaces in 2-Dimensional Complex Space Forms

We completely classify λ-biharmonic slant surfaces and λ-biminimal Lagrangian surfaces in 2-dimensional complex space forms, under the condition that the mean curvature is nonzero constant. In addition, we provide some examples of λ-biminimal slant surfaces with nonzero constant mean curvature, which are neither Lagrangian nor λ-biharmonic.

Biminimal Lagrangian H-umbilical submanifolds in complex space forms

All biminimal Lagrangian surfaces of nonzero constant mean curvature in 2-dimensional complex space forms have been determined in Sasahara (Differ Geom Appl 27:647–652, 2009). In this paper, we completely determine biminimal Lagrangian H-umbilical submanifolds of nonzero constant mean curvature in complex space forms of dimension ≥ 3.

Hypersurfaces with constant mean curvature in a hyperbolic space

Let M n be an n-dimensional complete connected and oriented hypersurface in a hyperbolic space H n+1 ( c) with non-zero constant mean curvature H and two distinct principal curvatures. In this paper, we show that (1) if the multiplicities of the two distinct principal curvatures are greater than 1, then M n is isometric to the Riemannian product S k ( r) × H n−k (−1/( r 2 + ρ 2)), where r > 0 and 1 < k < n − 1; (2) if H 2 > −c and one of the two distinct principal curvatures is simple, then M n is isometric to the Riemannian product S n−1 ( r) × H 1 (−1/( r 2 + ρ 2)) or S 1( r) × H n−1 (−1/( r 2 + ρ 2)) r > 0, if one of the following conditions is satisfied (i) S ≤ ( n − 1 ) t 2 2 + c 2 t 2 − 2 on M n or (ii) S ≥ ( n − 1 ) t 1 2 + c 2 t 1 − 2 on M n or (iii) ( n − 1 ) t 2 2 + c 2 t 2 − 2 ≤ ( n − 1 ) t 1 2 + c 2 t 1 − 2 on M n , where t 1 and t 2 are the positive real roots of (1.5).

SPHERE-FOLIATED MINIMAL AND CONSTANT MEAN CURVATURE HYPERSURFACES IN PRODUCT SPACES

In this paper, we prove that minimal hypersurfaces when n <TEX>$\geq$</TEX> 3 and nonzero constant mean curvature hypersurfaces when n <TEX>$\geq$</TEX> 2 foliated by spheres in parallel horizontal hyperplanes in <TEX>$\mathbb{H}^n{\times}\mathbb{R}$</TEX> must be rotationally symmetric.

Affine Translation Surfaces with Constant Gaussian Curvature

Abstract. We study a–ne translation surfaces in R 3 and get a complete classiflcation ofsuch surfaces with constant Gauss-Kronecker curvature. 1. IntroductionA surface in E 3 is called a translation surface if it is obtained as a graph of a func-tion F ( x;y ) = p ( x )+ q ( y ), where p ( x ) and q ( y ) are difierentiable functions. It’s wellknown that a minimal translation surface in the Euclidean space E 3 must be a planeor a Scherk surface, which is the graph of the function F ( x;y ) = ln(cos x= cos y ),the only doubly periodic minimal translation surface.In this note, we study nondegenerate translation surfaces in a–ne space R 3 .This class of surfaces has been studied previously by many geometers. F. Manhart[3] classifled all the nondegenerate a–ne minimal translation surfaces in a–ne spaceR 3 . Further treatments are due to H. F. Sun [5], who classifled the nondegeneratea–ne translation surface with nonzero constant mean curvature in R 3 . Later on,Sun and Chen extended this into the case of hypersurfaces [6]. On the other hand,Binder [1] classifled locally symmetric a–ne translation surfaces in R

The dynamics theorem for CMC surfaces in $R^3$

In this paper, we study the space of translational limits T(M) of a surface M properly embedded in R^3 with nonzero constant mean curvature and bounded second fundamental form. There is a natural map T which assigns to any surface M' in T(M), the set T(M') in T(M). Among various dynamics type results we prove that surfaces in minimal T-invariant sets of T(M) are chord-arc. We also show that if M has an infinite number of ends, then there exists a nonempty minimal T-invariant set in T(M) consisting entirely of surfaces with planes of Alexandrov symmetry. Finally, when M has a plane of Alexandrov symmetry, we prove the following characterization theorem: M has finite topology if and only if M has a finite number of ends greater than one.

Complete hypersurfaces in a 4-dimensional hyperbolic space

This paper gives a classification of complete hypersurfaces with nonzero constant mean curvature and constant quasi-Gauss-Kronecker curvature in the hyperbolic space H 4 ( 1), whose scalar curvature is bounded from below.

Lower bounds for eigenvalues of the Dirac-Witten operator

We get optimal lower bounds for the eigenvalues of the Dirac-Witten operator of compact (with or without boundary) spacelike hypersurfaces of Lorentian manifold satisfying certain conditions, just in terms of the mean curvature and the scalar curvature and the spinor energy-momentum tensor. In the limiting case, the spacelike hypersurface is either maximal and Einstein manifold with positive scalar curvature or Ricci-flat manifold with nonzero constant mean curvature.

Biminimal Lagrangian surfaces of constant mean curvature in complex space forms

We determine all biminimal Lagrangian surfaces of non-zero constant mean curvature in 2-dimensional complex space forms.

Lagrangian spheres in the 2-dimensional complex space forms

By constructing a holomorphic cubic form for Lagrangian surfaces with nonzero constant length mean curvature vector in a 2-dimensional complex space form \(\tilde M\)(4c), we characterize the Lagrangian pesudosphere as the only branched Lagrangian immersion of a sphere in \(\tilde M\)(4c) with nonzero constant length mean curvature vector. When c = 0, our result reduces to Castro-Urbano’s result in [1].

On a new class of embedded hypersurfaces in with nonzero constant mean curvature

Abstract We build a new class of embedded hypersurfaces in which have nonzero constant mean curvature and are O(m)×O(n)-invariant. For this we consider a 1-parameter family of planar vector fields associated to the differential equation of mean curvature.

Curvatures of complete hypersurfaces in space forms

In this paper we investigate three-dimensional complete minimal hypersurfaces with constant Gauss-Kronecker curvature in a space form M4(c) (c ≤ 0). We prove that if the scalar curvature of a such hypersurface is bounded from below, then its Gauss-Kronecker curvature vanishes identically. Examples of complete minimal hypersurfaces which are not totally geodesic in the Euclidean space E4 and the hyperbolic space H4(c) with vanishing Gauss-Kronecker curvature are also presented. It is also proved that totally umbilical hypersurfaces are the only complete hypersurfaces with non-zero constant mean curvature and with zero quasi-Gauss-Kronecker curvature in a space form M4(c) (c ≤ 0) if the scalar curvature is bounded from below. In particular, we classify complete hypersurfaces with constant mean curvature and with constant quasi-Gauss-Kronecker curvature in a space form M4(c) (c ≤ 0) if the scalar curvature r satisfies r≥ ⅔c.

Surfaces of revolution with periodic mean curvature

The surfaces of revolution with constant mean curvature in 3 are classified by Delaunay [2] in 1841. They are locally plane, catenoid, sphere, circular cylinder, unduloid, and nodoid up to isometries of 3. On these 10 years, new and interesting examples of non-zero constant mean curvature surfaces are discovered. In the global study of complete surfaces with constant mean curvature, unduloids and nodoids play important role as the models of ends of such surfaces (see [5], [6]). The work by Delaunay is now revived after 150 years of his discovery. The purpose of this paper is to study surfaces of revolution with periodic mean curvature in order to extend the theory of constant mean curvature surfaces. In general such a surface is not periodic, because the catenoid gives the counter-example. First we show the criterion for a periodic function to be the mean curvature of a periodic surface of revolution and second describe a method how to construct these periodic surfaces of revolution whose mean curvatures are periodic functions satisfying the criterion. I thank Professor Yusuke Sakane for his interest and plotting of the beautiful pictures by the computer of Osaka University which I used in the seminar talk of Granada University on March, 2001, and in the AMS meeting held in Hoboken on April, 2001. The figures of this paper are all programmed by the author using the symbolic manipulation program Mathematica. I also thank my graduate student Shinya Hirakawa for his help to simplify the programing.

Sphere-Foliated Minimal and Constant Mean Curvature Hypersurfaces in Space Forms and Lorentz-Minkowski Space

We prove that a sphere-foliated minimal or constant mean curvature hypersurface in hyperbolic space of dimension > 5 is one of the following: hypersurface of rotation around a geodesic, geodesic hyperplane, horosphere, equidistant hypersurface, or a geodesic sphere in the upper half-space model. And we show that a sphere-foliated minimal or constant mean curvature hypersurface in sphere of dimension > 5 is either a hypersurface of rotation or a hypersphere. We also show that a hypersurface of nonzero constant mean curvature in Lorentz-Minkowski space foliated by spheres in space-like hyperplanes is either a hypersurface of rotation or a pseudo-hyperbolic space and that maximal space-like hypersurfaces foliated by spheres in hyperplanes are rotational if the ambient space has dimension > 4.