scholarly journals $d$-Minimal Surfaces in Three-Dimensional Singular Semi-Euclidean Space $\mathbb{R}^{0,2,1}$

2021 ◽  
Vol 52 (1) ◽  
pp. 37-67
Author(s):  
Yuichiro Sato
Keyword(s):  
Euclidean Space ◽  
Minkowski Space ◽  
Minimal Surfaces ◽  
Mean Curvature ◽  
Three Dimensional ◽  
Representation Formula ◽  
Degenerate Metric ◽  
Dimensional Minkowski Space ◽  
One To One ◽  
Zero Mean Curvature

In this paper, we investigate surfaces in singular semi-Euclidean space $\mathbb{R}^{0,2,1}$ endowed with a degenerate metric. We define $d$-minimal surfaces, and give a representation formula of Weierstrass type. Moreover, we prove that $d$-minimal surfaces in $\mathbb{R}^{0,2,1}$ and spacelike flat zero mean curvature (ZMC) surfaces in four-dimensional Minkowski space $\mathbb{R}^{4}_{1}$ are in one-to-one correspondence.

SOME CLASSIFICATIONS OF LORENTZIAN SURFACES WITH FINITE TYPE GAUSS MAP IN THE MINKOWSKI 4-SPACE

2015 ◽  
Vol 99 (3) ◽  
pp. 415-427 ◽  
Author(s):  
NURETTIN CENK TURGAY
Keyword(s):  
Minkowski Space ◽  
Finite Type ◽  
Minimal Surfaces ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Gauss Map ◽  
Complete Classification ◽  
Nonzero Constant ◽  
Dimensional Minkowski Space

In this paper we study the Lorentzian surfaces with finite type Gauss map in the four-dimensional Minkowski space. First, we obtain the complete classification of minimal surfaces with pointwise 1-type Gauss map. Then, we get a classification of Lorentzian surfaces with nonzero constant mean curvature and of finite type Gauss map. We also give some explicit examples.

scholarly journals On the Schrödinger map for regular helical polygons in the hyperbolic space

Nonlinearity ◽  
2021 ◽  
Vol 35 (1) ◽  
pp. 84-109
Author(s):  
Sandeep Kumar
Keyword(s):  
Euclidean Space ◽  
Minkowski Space ◽  
Three Dimensional ◽  
Smooth Solutions ◽  
Straight Line ◽  
Dimensional Minkowski Space ◽  
Polygonal Curve ◽  
Schrödinger Map ◽  
Theoretical Results ◽  
Nonzero Torsion

Abstract The main purpose of this article is to understand the evolution of X t = X s ∧− X ss , with X(s, 0) a regular polygonal curve with a nonzero torsion in the three-dimensional Minkowski space. Unlike in the case of the Euclidean space, a nonzero torsion now implies two different helical curves. This generalizes recent works by the author with de la Hoz and Vega on helical polygons in the Euclidean space as well as planar polygons in the Minkowski space. Numerical experiments in this article show that the trajectory of the point X(0, t) exhibits new variants of Riemann’s non-differentiable function whose structure depends on the initial torsion in the problem. As a result, we observe that the smooth solutions (helices, straight line) in the Minkowski space show the same instability as displayed by their Euclidean counterparts and curves with zero-torsion. These numerical observations are in agreement with some recent theoretical results obtained by Banica and Vega.

Harmonic evolutes of B-scrolls with constant mean curvature in Lorentz–Minkowski space

2019 ◽  
Vol 16 (05) ◽  
pp. 1950076 ◽  
Author(s):  
Rafael López ◽  
Željka Milin Šipuš ◽  
Ljiljana Primorac Gajčić ◽  
Ivana Protrka
Keyword(s):  
Euclidean Space ◽  
Minkowski Space ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Ruled Surfaces ◽  
Null Curve ◽  
Base Curve

In this paper, we study harmonic evolutes of [Formula: see text]-scrolls, that is, of ruled surfaces in Lorentz–Minkowski space having no Euclidean counterparts. Contrary to Euclidean space where harmonic evolutes of surfaces are surfaces again, harmonic evolutes of [Formula: see text]-scrolls turn out to be curves. In particular, we show that the harmonic evolute of a [Formula: see text]-scroll of constant mean curvature together with its base curve forms a null Bertrand pair. This allows us to characterize [Formula: see text]-scrolls of constant mean curvature and reconstruct them from a given null curve which is their harmonic evolute.

scholarly journals The Dirichlet Problem of the Constant Mean Curvature in Equation in Lorentz-Minkowski Space and in Euclidean Space

Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1211 ◽  
Author(s):  
Rafael López
Keyword(s):  
Dirichlet Problem ◽  
Euclidean Space ◽  
Minkowski Space ◽  
Mean Curvature ◽  
Elliptic Equations ◽  
Constant Mean Curvature ◽  
Euclidean Case ◽  
Curvature Equation ◽  
Mean Curvature Equation ◽  
The Mean

We investigate the differences and similarities of the Dirichlet problem of the mean curvature equation in the Euclidean space and in the Lorentz-Minkowski space. Although the solvability of the Dirichlet problem follows standards techniques of elliptic equations, we focus in showing how the spacelike condition in the Lorentz-Minkowski space allows dropping the hypothesis on the mean convexity, which is required in the Euclidean case.

scholarly journals The Algebraic Surfaces of the Enneper Family of Maximal Surfaces in Three Dimensional Minkowski Space

Axioms ◽  
2021 ◽  
Vol 11 (1) ◽  
pp. 4
Author(s):  
Erhan Güler
Keyword(s):  
Minkowski Space ◽  
Three Dimensional ◽  
Algebraic Surfaces ◽  
Cartesian Coordinates ◽  
Maximal Surfaces ◽  
Dimensional Minkowski Space ◽  
The Family

We consider the Enneper family of real maximal surfaces via Weierstrass data (1,ζm) for ζ∈C, m∈Z≥1. We obtain the irreducible surfaces of the family in the three dimensional Minkowski space E2,1. Moreover, we propose that the family has degree (2m+1)2 (resp., class 2m(2m+1)) in the cartesian coordinates x,y,z (resp., in the inhomogeneous tangential coordinates a,b,c).

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