Related papers: Minimal Surfaces in the Four-Dimensional Euclidean…

It is known that any maximal space-like surface without isotropic points in the four-dimensional pseudo-Euclidean space with neutral metric admits locally geometric parameters which are special case of isothermal parameters. With respect to…

We construct a sequence of smooth minimizing surfaces in a sequence of metrics converging to the standard Euclidean metric, so that they have diverging $L^2$ norm of second fundamental form.

In this article a relation between curvature functionals for surfaces in the Euclidean space and area functionals in relative differential geometry will be given. Relative differential geometry can be described as the geometry of surfaces…

In the present paper we study normal transport surfaces in four-dimensional Euclidean space $\mathbb{E}^{4}$ which are the generalization of surface offsets in $\mathbb{E}^{3}$. We find some results of normal transport surfaces in…

This paper investigates minimal $n$-dimensional submanifolds in the Euclidean space that are $(n-2)$-umbilic, meaning they carry an umbilical distribution of rank $n-2$. We establish a correspondence between the class of minimal…

We prove a Bernstein-type theorem for two-valued minimal graphs in the four-dimensional Euclidean space $\mathbf{R}^4$. This states that two-valued functions defined on the entire $\mathbf{R}^3$, and whose graph is a minimal surface, must…

The minimal surfaces meeting in triples with equal angles along a common boundary naturally arise from soap films and other physical phenomenon. They are also the natural extension of the usual minimal surface. In this paper, we consider…

Minimal surfaces with uniform curvature (or area) bounds have been well understood and the regularity theory is complete, yet essentially nothing was known without such bounds. We discuss here the theory of embedded (i.e., without…

A surface M is called p-minimal if one of the coordinate functions is p-harmonic in the inner metric. We show that in the twodimensional case the Gaussian map of such surfaces is quasiconformal. In the case when the surface is a tube we…

In this paper we study surfaces in Euclidean 3-space that satisfy a Weingarten condition of linear type as $\kappa_1=m \kappa_2 +n$, where $m$ and $n$ are real numbers and $\kappa_1$ and $\kappa_2$ denote the principal curvatures at each…

We prove that any minimal (maximal) strongly regular surface in the three-dimensional Minkowski space locally admits canonical principal parameters. Using this result, we find a canonical representation of minimal strongly regular time-like…

In the 3-dimensional Lorentz-Minkowski space we prove that the sign of the Gaussian curvature of any timelike minimal surface is determined by the degeneracy and the orientations of the two null curves that generate the surface. Moreover,…

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 give a condition under which the findings of the paper cited above work well and determine the surfaces that were not considered before. In this paper, we show that a parallel mean curvature surface of a general type in a complex…

We investigate the minimal surface problem in the three dimensional Heisenberg group, H, equipped with its standard Carnot-Caratheodory metric. Using a particular surface measure, we characterize minimal surfaces in terms of a sub-elliptic…

We establish a curvature estimate for classical minimal surfaces with total boundary curvature less than 4\pi. The main application is a bound on the genus of these surfaces depending solely on the geometry of the boundary curve. We also…

In this paper, we study self-expanding solutions for mean curvature flows and their relationship to minimal cones in Euclidean space. In [18], Ilmanen proved the existence of self-expanding hypersurfaces with prescribed tangent cones at…

We develop an analytic theory of existence and regularity of surfaces (given by graphs) arising from the geometric minimization problem $$\min_{\mathcal{M}}\frac{1}{2}\int_{\mathcal{M}}|\nabla_{\mathcal{M}}H|^2\,dA$$ where $\mathcal{M}$…

We study the Bonnet problem for surfaces in 4-dimensional space forms, where two isometric surfaces have the same mean curvature if there exists a parallel vector bundle isometry between their normal bundles that preserves the mean…

The purpose of this article is to study the deformations of smooth surfaces $X$ of general type whose canonical map is a finite, degree 2 morphism onto a minimal rational surface or onto $\mathbf F_1$, embedded in projective space by a very…