Related papers: Contact real hypersurfaces in the complex hyperbol…

There are three complete plane geometries of constant curvature: spherical, Euclidean and hyperbolic geometry. We explain how a closed oriented surface can carry a geometry which locally looks like one of these. Focussing on the hyperbolic…

We classify the hypersurfaces of dimension n >= 3 with constant sectional curvature in the product spaces R^k x S^{n-k+1} and R^k x H^{n-k+1}, for 2 <= k <= n-1. Our results provide a complete description of these hypersurfaces and extend…

In this paper, we obtain the parametric expressions of the canal hypersurfaces that are formed as the envelope of a family of pseudo hyperspheres or pseudo hyperbolic hyperspheres whose centers lie on a pseudo null, partially null or null…

Consider the Euclidean space $\mathbb{R}^3$ endowed with a canonical semi-symmetric non-metric connection determined by a vector field $\mathsf{C}\in\mathfrak{X}(\mathbb{R}^3)$. We study surfaces when the sectional curvature with respect to…

Let $M$ be a three-dimensional contact manifold and $\psi:D\setminus\{0\}\to M\times{\Bbb R}$ a finite-energy pseudoholomorphic map from a punctured disc in ${\Bbb C}$, that is asymptotic to a periodic orbit of the Reeb vector field. This…

We give the classification of constant mean curvature rotational surfaces of elliptic, hyperbolic, and parabolic type in the four-dimensional pseudo-Euclidean space with neutral metric.

We classify the isoparametric hypersurfaces and the homogeneous hypersurfaces of $\mathbb H^n\times\mathbb R$ and $\mathbb S^n\times\mathbb R$, $n\ge 2$, by establishing that any such hypersurface has constant angle function and constant…

In this paper we first introduce the full expression of the curvature tensor of a real hypersurface $M$ in complex hyperbolic two-plane Grassmannians $SU_{2,m}/S(U_2{\cdot}U_m)$, $m{\ge}2$ from the equation of Gauss. Next we derive a new…

We review recent results on classifying complete constant mean curvature 1 (CMC 1) surfaces in hyperbolic 3-space with low total curvature. There are two natural notions of "total curvature" -- one is the total absolute curvature, which is…

We classify minimal hypersurfaces in $R^n \times S^m$, $n,m \geq 2$, which are invariant by the canonical action of $O(n) \times O(m)$. We also construct compact and noncompact examples of invariant hypersurfaces of constant mean curvature.…

This paper is concerned with the completeness (with respect to the centroaffine metric) of hyperbolic centroaffine hypersurfaces which are closed in the ambient vector space. We show that completeness holds under generic regularity…

We will generalize a Maximum Principle at Infinity in the parabolic case given by De Lima [Ann. Global Anal. Geom. ${\bf 20}$, 325-343 2001] and De Lima and Meeks [Indiana Univ. Math. Journal ${\bf 53}$ 5, 1211-1223 2004], for disjoints…

A study of smooth contact quasiconformal mappings of the hyperbolic Heisenberg group is presented in this paper. Our main result is a Lifting Theorem; according to this, a symplectic quasiconformal mapping of the hyperbolic plane can be…

In this paper we investigate the convergence for the mean curvature flow of closed submanifolds with arbitrary codimension in space forms. Particularly, we prove that the mean curvature flow deforms a closed submanifold satisfying a…

We construct simply connected, complete, non-$CMC$ biconservative surfaces in the $3$-dimensional hyperbolic space $\mathbb{H}^3$ in an intrinsic and extrinsic way. We obtain three families of such surfaces, and, for each surface, the set…

We establish the existence of hypersurfaces with constant mean curvature and a prescribed boundary in Euclidean space, represented as radial graphs over domains of the unit sphere. Under the assumptions that the mean curvature of the…

Motivated by questions in detecting minimal surfaces in hyperbolic manifolds, we study the behavior of geometric flows in complete hyperbolic three-manifolds. In most cases the flows develop singularities in finite time. In this paper, we…

We get new results (and rederive some know ones) on smooth surfaces in $\mathbb{R}^n$ by unifying several view points into a coherent general view. Namely, we show and use new relations of the evolute (caustic) with the curvature ellipse,…

The asymptotic Plateau problem asks for the existence of smooth complete hypersurfaces of constant mean curvature with prescribed asymptotic boundary at infinity in the hyperbolic space $\mathbb{H}^{n+1}$. The modified mean curvature flow…

A curvature-type tensor invariant called para contact (pc) conformal curvature is defined on a paracontact manifold. It is shown that a paracontact manifold is locally paracontact conformal to the hyperbolic Heisenberg group or to a…