Hypersurfaces In Hyperbolic Space Research Articles

Bifurcations of immersed constant mean curvature hypersurfaces in hyperbolic space

In this paper, we prove the existence of new branches of hypersurfaces with constant mean curvature which bifurcate from the rotationally invariant immersed constant mean curvature hypersurfaces in the hyperbolic space.

On Complete Spacelike Hypersurfaces in a Semi-Riemannian Warped Product

By applying Omori-Yau maximal principal theory and supposing an appropriate restriction on the norm of gradient of height function, we obtain some new Bernstein-type theorems for complete spacelike hypersurfaces with nonpositive constant mean curvature immersed in a semi-Riemannian warped product. Furthermore, some applications of our main theorems for entire vertical graphs in Robertson-Walker spacetime and for hypersurfaces in hyperbolic space are given.

Curvature flow of complete hypersurfaces in hyperbolic space

In this paper we continue our study of finding the curvature flow of complete hypersurfaces in hyperbolic space with a prescribed asymptotic boundary at infinity. Our main results are proved by deriving a priori global gradient estimates and C 2 estimates.

Slant geometry of spacelike hypersurfaces in hyperbolic space and de Sitter space

We consider a one-parameter family of new extrinsic differential geometries on hypersurfaces in hyperbolic space. Recently, the second author and his collaborators have constructed a new geometry which is called horospherical geometry on hyperbolic space. There is another geometry which is the famous Gauss–Bolyai–Robechevski geometry (i.e., the hyperbolic geometry) on hyperbolic space. The slant geometry is a one-parameter family of geometries which connect these two geometries. Moreover, we construct a one-parameter family of geometries on spacelike hypersurfaces in de Sitter space.

Linear combinations of hypersurfaces in hyperbolic space

In this article we introduce and investigate linear combinations of hypersurfaces in hyperbolic space. For this purpose we use some linear structure in the space of horospheres.

Curvature Flow of Complete Convex Hypersurfaces in Hyperbolic Space

We investigate the existence, convergence, and uniqueness of modified general curvature flow (MGCF) of convex hypersurfaces in hyperbolic space with a prescribed asymptotic boundary.

Phase Transitions and Minimal Hypersurfaces in Hyperbolic Space

The purpose of this paper is to investigate the Cahn–Hillard approximation for entire minimal hypersurfaces in the hyperbolic space. Combining comparison principles with minimization and blow-up arguments, we prove existence results for entire local minimizers with prescribed behavior at infinity. Then, we study the limit as the length scale tends to zero through a Γ-convergence analysis, obtaining existence of entire minimal hypersurfaces with prescribed boundary at infinity. In particular, we recover some existence results proved in [3, 21] using geometric measure theory.

Minimal Hypersurfaces in Hyperbolic Spaces

Minimal Hypersurfaces in Hyperbolic Spaces

Hypersurfaces in hyperbolic space associated with the conformal scalar curvature equation [formula omitted

We consider a class of oriented hypersurfaces in hyperbolic space satisfying ∑ r = 0 n ( c − n + 2 r ) ( n r ) H r = 0 , where H r is the rth mean curvature and c is a real constant. We show how this class is characterized by a harmonic map derived from the two hyperbolic Gauss maps. By looking at hypersurfaces as orthogonal to a congruence of geodesics, we also show the relation of such hypersurfaces with solutions of the equation Δ u + k u n + 2 n − 2 = 0 , where k ∈ { − 1 , 0 , 1 } . Finally, we apply the relation mentioned above to obtain examples and geometrical results for the hypersurfaces.

The lightlike flat geometry on spacelike submanifolds of codimension two in Minkowski space

We introduce the notion of the lightcone Gauss-Kronecker curvature for a spacelike submanifold of codimension two in Minkowski space, which is a generalization of the ordinary notion of Gauss curvature of hypersurfaces in Euclidean space. In the local sense, this curvature describes the contact of such submanifolds with lightlike hyperplanes. We study geometric properties of such curvatures and show a Gauss-Bonnet type theorem. As examples we have hypersurfaces in hyperbolic space, spacelike hypersurfaces in the lightcone and spacelike hypersurfaces in de Sitter space.

The horospherical Gauss-Bonnet type theorem in hyperbolic space

We introduce the notion horospherical curvatures of hypersurfaces in hyperbolic space and show that totally umbilic hypersurfaces with vanishing cur- vatures are only horospheres. We also show that the Gauss-Bonnet type theorem holds for the horospherical Gauss-Kronecker curvature of a closed orientable even dimensional hypersurface in hyperbolic space. + (i1) by using the model in Minkowski space. We introduced the notion of hyperbolic Gauss indicatrices slightly modified the definition of hyperbolic Gauss maps. The notion of hyperbolic indicatrices is independent of the choice of the model of hyperbolic space. Using the hyperbolic Gauss indicatrix, we defined the principal hyperbolic curvatures ¯ • § and the hyperbolic Gauss-Kronecker curvature K § h by exactly the same way as the definition of those of classical Gaussian dif- ferential geometry in Euclidean space. Totally umbilic hypersurfaces with respect to the above curvatures are equidistant hypersurfaces, hyperspheres or hyperhorospheres which are called model hypersurfaces in hyperbolic space. The hyperbolic Gauss-Kronecker curvature is a hyperbolic invariant which describes the contact of hypersurfaces with such model hypersurfaces. We remark that Kobayashi (13), (14) had already defined the notion of hyperbolic Gauss-Kronecker curvature under a dierent framework and studied some basic properties of it from the view point of the theory of Fourier transformations. In this paper we introduce the principal horospherical curvature e § (cf., x3). This new curvature is not a hyperbolic invariant but an SO(n)-invariant, where we consider the canonical SO(n)-subgroup in the group of hyperbolic motions. However, we can show that e § (p) is invariant under hyperbolic motions if and only if e § (p) = 0. We can

Symmetry of Stationary Hypersurfaces in Hyperbolic Space

We deposit a prescribed amount of liquid on an umbilical hypersurfaceof the hyperbolic space H n+1 . Under the presence of a uniform gravity vector field directed towards � , we seek the shape of such a liquid drop in a state of equilibrium of the mechanical system. The liquid-air interface is then modeled by a hypersurface under the condition that its mean curvature is a function of the distance from � , together with the fact that the angle that makes withalong its boundary is constant. We show that the hypersurface is rotational symmetric with respect to a geodesic orthogonal to � .W e extend this result to other configurations, for example, liquid bridges trapped between two umbilical hypersurfaces. Finally, we obtain a result which says that, under some assump- tions on the mean curvature, an embedded hypersurface inherits a certain symmetry from its boundary.

Hadamard-type theorems for hypersurfaces in hyperbolic spaces

We prove that a bounded, complete hypersurface in hyperbolic space with normal curvatures greater than −1 is diffeomorphic to a sphere. The completeness condition is relaxed when the normal curvatures are bounded away from −1. The diffeomorphism is constructed via the Gauss map of some parallel hypersurface. We also give bounds for the total curvature of this parallel hypersurface.

Minimizing Constant Mean Curvature Hypersurfaces in Hyperbolic Space

We study the constant mean curvature (CMC) hypersurfaces in $$\mathbb{H}^{n+1}$$ whose asymptotic boundaries are closed codimension-1 submanifolds in $$S^{n}_{\infty}(\mathbb{H}^{n+1})$$ . We consider CMC hypersurfaces as generalizations of minimal hypersurfaces. We naturally generalize some notions of minimal hypersurfaces like being area-minimizing, convex hull property, exchange roundoff trick to CMC hypersurface context. We also give a generic uniqueness result for CMC hypersurfaces in hyperbolic space.

Integral geometry of equidistants in hyperbolic space

We generalize the classical formulas of integral geometry, by getting integral geometric formulas for the intersection of a fixed compact hypersurface of hyperbolic space and a moving totally umbilical hypersurface. In particular we compute the mean value of the volume, the total mean curvatures and the Euler characteristic of these intersections when the totally umbilical hypersurface moves over all the intersecting positions. Analogous formulas are given for totally umbilical hypersurfaces contained in totally geodesic planes of ℍn.

Integral geometry and geometric inequalities in hyperbolic space

Using results from integral geometry, we find inequalities involving mean curvature integrals of convex hypersurfaces in hyperbolic space. Such inequalities generalize the Minkowski formulas for euclidean convex sets.

SOME ESTIMATES FOR COMPACT HYPERSURFACES IN HYPERBOLIC SPACE

Let M be a compact hypersurface in hyperbolic space and let A be the area of M and V be the volume of the compact domain bounded by M. In this paper, we find a lower bound for <TEX>$\frac{A}{V}$</TEX> in two cases, M has constant scalar curvature and M has constant mean curvature.

A remark on the mean curvature of a graph-like hypersurface in hyperbolic space

In this paper we investigate the mean curvature H of a radial graph in hyperbolic space H n + 1 . We obtain an integral inequality for H, and find that the lower limit of H at infinity is less than or equal to 1 and the upper limit of H at infinity is more than or equal to −1. As a byproduct we get a relation between the n-dimensional volume of a bounded domain in an n-dimensional hyperbolic space and the ( n − 1 )-dimensional volume of its boundary. We also sharpen the main result of a paper by P.-A. Nitsche dealing with the existence and uniqueness of graph-like prescribed mean curvature hypersurfaces in hyperbolic space.

SINGULARITIES OF EVOLUTES OF HYPERSURFACES IN HYPERBOLIC SPACE

AbstractWe study the differential geometry of hypersurfaces in hyperbolic space. As an application of the theory of Lagrangian singularities, we investigate the contact of hypersurfaces with families of hyperspheres or equidistant hyperplanes.AMS 2000 Mathematics subject classification: Primary 53A35. Secondary 58C27

SINGULARITIES OF HYPERBOLIC GAUSS MAPS

In this paper we adopt the hyperboloid in Minkowski space as the model of hyperbolic space. We define the hyperbolic Gauss map and the hyperbolic Gauss indicatrix of a hypersurface in hyperbolic space. The hyperbolic Gauss map has been introduced by Ch. Epstein [J. Reine Angew. Math. 372 (1986) 96–135] in the Poincaré ball model, which is very useful for the study of constant mean curvature surfaces. However, it is very hard to perform the calculation because it has an intrinsic form. Here, we give an extrinsic definition and we study the singularities. In the study of the singularities of the hyperbolic Gauss map (indicatrix), we find that the hyperbolic Gauss indicatrix is much easier to calculate. We introduce the notion of hyperbolic Gauss–Kronecker curvature whose zero sets correspond to the singular set of the hyperbolic Gauss map (indicatrix). We also develop a local differential geometry of hypersurfaces concerning their contact with hyperhorospheres.2000 Mathematical Subject Classification: 53A25, 53A05, 58C27.