Complete minimal hypersurfaces in complex hyperbolic space

We construct new examples of embedded, complete, minimal hypersurfaces in complex hyperbolic space, including deformations of bisectors and some minimal foliations. Received: 20 March 2000 / Revised version: 21 July 2000     

On some real hypersurfaces of a complex hyperbolic space

Given a real hypersurface of a complex hyperbolic space #x2102;?H ⁿ ,we construct a principal circle bundle over it which is a Lorentzian hypersurface of the anti-De Sitter space H ₁ ²ⁿ⁺¹ .Relations between the respective second fundamental forms are obtained permitting us to classify a remarkable family of real hypersurfaces of ?H ⁿ .     

On convexity of hypersurfaces in the hyperbolic space

In the Hyperbolic space \({\mathbb{H}^n}\) (n ≥ 3) there are uncountably many topological types of convex hypersurfaces. When is a locally convex hypersurface in \({\mathbb{H}^n}\) globally convex, that is, when does it bound a convex set? We prove that any locally convex proper embedding of an (n ? 1)-dimensional connected manifold is the boundary of a convex set whenever the complement of (n ? 1)-flats of the resulting hypersurface is connected.     

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.     

Homogeneous hypersurfaces in complex hyperbolic spaces

We study the geometry of homogeneous hypersurfaces and their focal sets in complex hyperbolic spaces. In particular, we provide a characterization of the focal set in terms of its second fundamental form and determine the principal curvatures of the homogeneous hypersurfaces together with their multiplicities.      

Weakly horospherically convex hypersurfaces in hyperbolic space

In Bonini et al. (Adv Math 280:506–548, 2015), the authors develop a global correspondence between immersed weakly horospherically convex hypersurfaces \(\phi :M^n \rightarrow \mathbb {H}^{n+1}\) and a class of conformal metrics on domains of the round sphere \(\mathbb {S}^n\). Some of the key aspects of the correspondence and its consequences have dimensional restrictions \(n\ge 3\) due to the reliance on an analytic proposition from Chang et al. (Int Math Res Not 2004(4):185–209, 2004) concerning the asymptotic behavior of conformal factors of conformal metrics on domains of \(\mathbb {S}^n\). In this paper, we prove a new lemma about the asymptotic behavior of a functional combining the gradient of the conformal factor and itself, which allows us to extend the global correspondence and embeddedness theorems of Bonini et al. (2015) to all dimensions \(n\ge 2\) in a unified way. In the case of a single point boundary \(\partial _{\infty }\phi (M)=\{x\} \subset \mathbb {S}^n\), we improve these results in one direction. As an immediate consequence of this improvement and the work on elliptic problems in Bonini et al. (2015), we have a new, stronger Bernstein type theorem. Moreover, we are able to extend the Liouville and Delaunay type theorems from Bonini et al. (2015) to the case of surfaces in \(\mathbb {H}^{3}\).     

Complete minimal hypersurfaces in quaternionic hyperbolic space

We construct new examples of embedded, complete minimal hypersurfaces in quaternionic hyperbolic space and also some minimal foliations. We introduce fans and construct analytic deformations of bisectors.     

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.     

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 H4（-1）,whose scalar curvature is bounded from below.     

Eigenvalue estimates for minimal hypersurfaces in hyperbolic space

Recently Candel [A. Candel, Eigenvalue estimates for minimal surfaces in hyperbolic space, Trans. Amer. Math. Soc. 359 (2007) 3567-3575] proved that if M is a simply-connected stable minimal surface isometrically immersed in H³, then the first eigenvalue of M satisfies 1/4?λ(M)?4/3 and he asked whether the bound is sharp and gave an example such that the lower bound is attained. In this note, we prove that the upper bound can never be attained. Also we extend the result by proving that if M is compact stable minimal hypersurface isometrically immersed in Hn+1 where n?3 such that its smooth Yamabe invariant is negative, then (n−1)/4?λ(M)?n²(n−2)/(7n−6).     

On the geometry of linear Weingarten hypersurfaces in the hyperbolic space

In this paper, we apply some forms of generalized maximum principles in order to study the geometry of complete linear Weingarten hypersurfaces with nonnegative sectional curvature immersed in the hyperbolic space. In this setting, under the assumption that the mean curvature attains its maximum, we prove that such a hypersurface must be either totally umbilical or isometric to a hyperbolic cylinder.     

Lie geometry of flat fronts in hyperbolic space

We propose a Lie geometric point of view on flat fronts in hyperbolic space as special Ω-surfaces and discuss the Lie geometric deformation of flat fronts.     

Real hypersurfaces in the complex hyperbolic quadric with harmonic curvature

We give a classification of real hypersurfaces in the complex hyperbolic quadric ${Q^m}^{?}=S{O}_{2,m}^o/S{O}_{2}S{O}_m$ that have constant mean curvature and harmonic curvature.     

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 the Gauss map of complete CMC hypersurfaces in the hyperbolic space

In this paper, as suitable applications of the so-called Omori–Yau generalized maximum principle, we obtain rigidity results concerning to complete hypersurfaces with constant mean curvature in the hyperbolic space, under appropriated restrictions on their Gauss image. Furthermore, by supposing a linear dependence between support functions naturally attached to such hypersurfaces, we establish a characterization theorem.     

Stability of constant mean curvature hypersurfaces of revolution in hyperbolic space

In this article, by solving a nonlinear differential equation, we prove the existence of a one parameter family of constant mean curvature hypersurfaces in the hyperbolic space with two ends. Then, we study the stability of these hypersurfaces.