Minimal Translation Surfaces Research Articles

The Translation Surfaces on Statistical Manifolds with Normal Distribution

In this paper, we will investigate translation surfaces on statistical manifolds. Statistical manifolds are mathematical structures that describe the geometric properties of statistical models. We will focus on minimal statistical translation surfaces and then classify statistical translation surfaces of null sectional curvature in three-dimensional hyperbolic statistical manifolds

Translation surfaces generating with some partner curve

In this article, generating curves of translation surfaces are paired with some special curve pairs. With the results obtained from these pairings, the developable and minimal translation surfaces are characterized. In addition, the surface curvatures of the translation surface are obtained. For a better understanding of the results, examples are given and their drawings are made with the help of Mathematica.

Singular Minimal Translation Surfaces in Euclidean Spaces Endowed with Semi-symmetric Connections

In this paper, we study and classify singular minimal translation surfaces in a Euclidean space of dimension $3$ endowed with a certain semi-symmetric (non-)metric connection.

On Minimal Surfaces Immersed in Three Dimensional Kropina Minkowski Space

In this paper we consider a three dimensional Kropina space and obtain a partial differential equation that characterizes minimal surfaces with the induced metric. Using this characterization equation we study various immersions of minimal surfaces. In particular, we obtain the partial differential equation that characterizes the minimal translation surfaces and show that the plane is the only such surface.

Minimal translation surfaces in Lorentz Heisenberg space (ℋ3, g 2)

The aim of this work is the classification of some types of minimal translation surfaces in the 3–dimensional Lorentzian Heisenberg space ℋ3 endowed with the left invariant metric

SPACELIKE TRANSLATION SURFACES IN MINKOWSKI 4-SPACE E_1^4

In the present paper, we consider spacelike translation surfaces in $4$-dimensio\-nal Minkowski space. We characterize such surfaces in terms of their Gaussian curvature and mean curvature functions. We classify flat and minimal spacelike translation surfaces in $\mathbb{E}_{1}^{4}$.

Classification and Construction of Minimal Translation Surfaces in Euclidean Space

A translation surface of Euclidean space $\r^3$ is the sum of two regular curves $\alpha$ and $\beta$, called the generating curves. In this paper we classify the minimal translation surfaces of $\r^3$ and we give a method of construction of explicit examples. Besides the plane and the minimal surfaces of Scherk type, it is proved that up to reparameterizations of the generating curves, any minimal translation surface is described as $\Psi(s,t)=\alpha(s)+\alpha(t)$, where $\alpha$ is a curve parameterized by arc length $s$, its curvature $\kappa$ is a positive solution of the autonomous ODE $(y')^2+y^4+c_3y^2+c_1^2y^{-2}+c_1c_2=0$ and its torsion is $\tau(s)=c_1/\kappa(s)^2$. Here $c_1\not=0$, $c_2$ and $c_3$ are constants such that the cubic equation $-\lambda^3+c_2\lambda^2-c_3\lambda+c_1=0$ has three real roots $\lambda_1$, $\lambda_2$ and $\lambda_3$.

A Note on Minimal Translation Graphs in Euclidean Space

In this note, we give a characterization of a class of minimal translation graphs generated by planar curves. Precisely, we prove that a hypersurface that can be written as the sum of n planar curves is either a hyperplane or a cylinder on the generalized Scherk surface. This result can be considered as a generalization of the results on minimal translation hypersurfaces due to Dillen–Verstraelen–Zafindratafa in 1991 and minimal translation surfaces due to Liu–Yu in 2013.

Minimal translation surfaces in the three-dimensional Heisenberg group

We define and classify minimal translation surfaces in the 3-dimensional Heisenberg group H(1,1).

Weighted minimal affine translation surfaces in Euclidean space with density

The aim of this work is to study affine translation surfaces in the Euclidean 3-space with density. We completely classify affine translation surfaces with zero weighted mean curvature.

A classification of minimal translation surfaces in Minkowski space

A classification of minimal translation surfaces in Minkowski space

Weighted minimal translation surfaces in Minkowski 3-space with density

The aim of this work is to study translation surfaces in a Minkowski 3-space [Formula: see text] with density. Translation surfaces in [Formula: see text] are defined as the two generating curves which lie in orthogonal planes. They have actually three different possible parametrizations according to the intersecting straight line of the two planes. We completely classify all translation surfaces with zero weighted mean curvature in [Formula: see text] with density [Formula: see text] by solving the second-order non-linear ODE with some smooth functions.

Weighted minimal translation surfaces in the Galilean space with density

Abstract Translation surfaces in the Galilean 3-space G3 have two types according to the isotropic and non-isotropic plane curves. In this paper, we study a translation surface in G3 with a log-linear density and classify such a surface with vanishing weighted mean curvature.

Minimal Translation Surfaces in Euclidean Space

A translation surface in Euclidean space is a surface that is the sum of two regular curves $$\alpha $$ and $$\beta $$ . In this paper we characterize all minimal translation surfaces. In the case that $$\alpha $$ and $$\beta $$ are non-planar curves, we prove that the curvature $$\kappa $$ and the torsion $$\tau $$ of both curves must satisfy the equation $$\kappa ^2 \tau = C$$ where C is constant. We show that, up to a rigid motion and a dilation in the Euclidean space and, up to reparametrizations of the curves generating the surfaces, all minimal translation surfaces are described by two real parameters $$a,b\in \mathbb {R}$$ where the surface is of the form $$\phi (s,t)=\beta _{a,b}(s)+\beta _{a,b}(t)$$ .

On affine translation surfaces in affine space

In this work we give a systemic study of affine translation surfaces in affine 3-dimensional space. Specifically, we obtain the complete classification of minimal affine translation surfaces. Moreover, we consider affine translation surfaces with some natural geometric conditions, such as constant affine mean curvature and constant Gauss–Kronecker curvature. Some characterization results with these geometric conditions are also obtained.

Affine translation surfaces in Euclidean 3-space

In this paper we define affine translation surface and classify minimal affine translation surfaces in three dimensional Euclidean space.

MINIMAL TRANSLATION SURFACES IN $\mathbb{H}^2 \times \mathbb{R}$

In this paper, we define translation surfaces in the Riemannian product space $\Bbb H^2 \times \Bbb R$ and completely classify minimal translation surfaces in $\Bbb H^2 \times \Bbb R$.

SOME TRANSLATION SURFACES IN THE 3-DIMENSIONAL HEISENBERG GROUP

In this paper, we define translation surfaces in the 3-dimensional Heisenberg group <TEX>$\mathcal{H}_3$</TEX> obtained as a product of two planar curves lying in planes, which are not orthogonal, and study minimal translation surfaces in <TEX>$\mathcal{H}_3$</TEX>.

Minimal translation surfaces in Sol3

In the homogeneous space Sol3, a translation surface is parametrized by x(s,t) = α(s) ∗ β(t), where α and β are curves contained in coordinate planes and ∗ denotes the group operation of Sol3. In this paper we study translation surfaces in Sol3 whose mean curvature vanishes.

Minimal translation surfaces in the Heisenberg group Nil3

A translation surface in the Heisenberg group $\mathrm{Nil}_3$ is a surface constructed by multiplying (using the group operation) two curves. We completely classify minimal translation surfaces in the Heisenberg group $\mathrm{Nil}_3$.