Related papers: The Topological Classification of Minimal Surfaces…

We present in this article a survey of recent results in value distribution theory for the Gauss maps of several classes of immersed surfaces in space forms, for example, minimal surfaces in Euclidean $n$-space ($n$=3 or 4), improper affine…

We provide the full classification of equidistant decomposition of a two-dimensional Euclidean plane and a two-dimensional sphere.

Surfaces with concentric $K$-contours and parallel $K$-contours in Euclidean $3$-space are defined. Crucial examples are presented and characterization of them are given.

For an immersed minimal surface in $\mathbb{R}^3$, we show that there exists a lower bound on its Morse index that depends on the genus and number of ends, counting multiplicity. This improves, in several ways, an estimate we previously…

First, we classify proper biharmonic Hopf real hypersurfaces in $\mathbb{C}P^2$. Next, we classify proper biharmonic real hypersurfaces with two distinct principal curvatures in $\mathbb{C}P^n$, where $n\geq 2$. Finally, we prove that…

This} paper presents relations between least area and normal surfaces, embedded in either a Euclidean or hyperbolic $3$-manifold. A relaxed version of normal surfaces, termed quasi-normal, is introduced, and it is shown that under…

Special class of surfaces in five-dimensional sphere in $C^3$ is considered. Immersion equations for minimal tori of that class are shown to be reducible to the equation $u_{z\bar z}=e^u-e^{-2u}$ which is integrable by means of inverse…

Methodology is provided towards the solution of the minimum enclosing ball problem. This problem concerns the determination of the unique spherical surface of smallest radius enclosing a given bounded set in the d-dimensional Euclidean…

In this paper, we solve affirmatively B.-Y. Chen's conjecture for hypersurfaces in the Euclidean space, under a generic condition. More precisely, every biharmonic hypersurface of the Euclidean space must be minimal if their principal…

We construct a weakly complete flat surface in hyperbolic 3-space having a pair of hyperbolic Gauss maps both of whose images are contained in an arbitrarily given open disc in the ideal boundary of H^3. This construction is accomplished as…

In this paper, we study closed embedded minimal hypersurfaces in a Riemannian $(n+1)$-manifold ($2\le n\le 6$) that minimize area among such hypersurfaces. We show they exist and arise either by minimization techniques or by min-max…

We survey Bernstein-type theorems for graphical surfaces in the Euclidean space and the Lorentz-Minkowski space. More specifically, we explain several proofs of the Bernstein theorem for minimal graphs in the Euclidean 3-space. Furthermore,…

The topological index of a surface was previously introduced by the first author as the topological analogue of the index of an unstable minimal surface. Here we show that surfaces of arbitrarily high topological index exist.

In this paper we survey with complete proofs some well--known, but hard to find, results about constructing closed embedded minimal surfaces in a closed 3-dimensional manifold via min--max arguments. This includes results of J. Pitts, F.…

We study compact stable embedded minimal surfaces whose boundary is given by two collections of closed smooth Jordan curves in close planes of Euclidean 3-space. Our main result is a classification of these minimal surfaces, under certain…

We prove that a surface in Euclidean $3$-space has Maslovian normal bundle if and only if it is a part of a round sphere, a circular cylinder, or a circular cone.

In the present paper, we discuss the singular minimal surfaces in a Euclidean 3-space R^{3} which are minimal. In fact, such a surface is nothing but a plane, a trivial outcome. However, a non-trivial outcome is obtained when we modify the…

We state several sufficient conditions for compact spacelike surface in the3-dimensional de Sitter space to be totally geodesic or spherical.

In this paper, we prove that every conformal minimal immersion of a compact bordered Riemann surface $M$ into a minimally convex domain $D\subset \mathbb{R}^3$ can be approximated, uniformly on compacts in $\mathring M=M\setminus bM$, by…

We found a class of triangulated surfaces in Euclidean space which have similar properties as isothermic surfaces in Differential Geometry. We call a surface isothermic if it admits an infinitesimal isometric deformation preserving the mean…