Parabolic Revolution Surfaces of Finite Type in Simply Isotropic 3-spaces

2020 ◽  
pp. 2150034
Author(s):  
Alev Kelleci Akbay
Keyword(s):  
Finite Type ◽  
Laplace Operator ◽  
Fundamental Form ◽  
Three Dimensional ◽  
Second Fundamental Form ◽  
Real Numbers ◽  
Isotropic Space ◽  
Simply Isotropic Space ◽  
Revolution Surfaces ◽  
The Laplace Operator

In this paper, we classify parabolic revolution surfaces in the three-dimensional simply isotropic space [Formula: see text] under the condition [Formula: see text] where [Formula: see text] is the Laplace operator with respect to first and second fundamental form and [Formula: see text], [Formula: see text] are some real numbers. Also, as an application, we give some explicit examples for these surfaces.

scholarly journals Classification of Some Special Types Ruled Surfaces in Simply Isotropic 3-Space

2017 ◽  
Vol 55 (1) ◽  
pp. 87-98 ◽  
Author(s):  
Murat Kemal Karacan ◽  
Dae Won Yoon ◽  
Nural Yuksel
Keyword(s):  
Real Number ◽  
Laplace Operator ◽  
Fundamental Form ◽  
Three Dimensional ◽  
Ruled Surfaces ◽  
Isotropic Space ◽  
Simply Isotropic Space ◽  
The Laplace Operator ◽  
First Fundamental Form

AbstractIn this paper, we classify two types ruled surfaces in the three dimensional simply isotropic space I13under the condition ∆xi= λixiwhere ∆ is the Laplace operator with respect to the first fundamental form and λ is a real number. We also give explicit forms of these surfaces.

Translation surfaces in the three-dimensional simply isotropic space 𝕀31

2016 ◽  
Vol 13 (07) ◽  
pp. 1650088 ◽  
Author(s):  
Murat Kemal Karacan ◽  
Dae Won Yoon ◽  
Bahaddin Bukcu
Keyword(s):  
Real Number ◽  
Laplace Operator ◽  
Three Dimensional ◽  
Translation Surfaces ◽  
Isotropic Space ◽  
Simply Isotropic Space ◽  
The Laplace Operator

In this paper, we classify translation surfaces in the three-dimensional simply isotropic space [Formula: see text] under the condition [Formula: see text] where [Formula: see text] is the Laplace operator with respect to the first and second fundamental forms and [Formula: see text] is a real number. We also give explicit forms of these surfaces.

scholarly journals Helicoidal Surfaces in the three dimensional simply isotropic space I₃¹

2017 ◽  
Vol 48 (2) ◽  
pp. 123-134
Author(s):  
Murat Kemal Karacan ◽  
Dae Won Yoon ◽  
Sezai Kiziltug
Keyword(s):  
Fundamental Form ◽  
Three Dimensional ◽  
Algebraic Equations ◽  
The Third ◽  
Isotropic Space ◽  
Simply Isotropic Space ◽  
Helicoidal Surfaces ◽  
Coordinate Functions ◽  
Third Fundamental Form ◽  
Laplacian Operators

In this paper, we classify helicoidal surfaces in the three dimensional simply isotropic space  I₃¹ satisfying some algebraic equations in terms of the coordinate functions and the Laplacian operators with respect to the first, the second and the third fundamental form of the surface. We also give explicit forms of these surfaces.

scholarly journals Evolution of the first eigenvalue of the Laplace operator and the p-Laplace operator under a forced mean curvature flow

2020 ◽  
Vol 18 (1) ◽  
pp. 1518-1530
Author(s):  
Xuesen Qi ◽  
Ximin Liu
Keyword(s):  
Laplace Operator ◽  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Second Fundamental Form ◽  
Coefficient Function ◽  
First Eigenvalue ◽  
Initial Hypersurface ◽  
The Mean ◽  
The Laplace Operator

Abstract In this paper, we discuss the monotonicity of the first nonzero eigenvalue of the Laplace operator and the p-Laplace operator under a forced mean curvature flow (MCF). By imposing conditions associated with the mean curvature of the initial hypersurface and the coefficient function of the forcing term of a forced MCF, and some special pinching conditions on the second fundamental form of the initial hypersurface, we prove that the first nonzero closed eigenvalues of the Laplace operator and the p-Laplace operator are monotonic under the forced MCF, respectively, which partially generalize Mao and Zhao’s work. Moreover, we give an example to specify applications of conclusions obtained above.

scholarly journals GEOMETRIC MOMENTUM IN THE MONGE PARAMETRIZATION OF TWO-DIMENSIONAL SPHERE

2013 ◽  
Vol 10 (03) ◽  
pp. 1220031 ◽  
Author(s):  
D. M. XUN ◽  
Q. H. LIU
Keyword(s):  
Quantum Mechanics ◽  
Laplace Operator ◽  
Three Dimensional ◽  
Dimensional Sphere ◽  
Two Dimensional ◽  
Constrained Motion ◽  
Geometric Invariant ◽  
Gradient Operator ◽  
The Laplace Operator

A two-dimensional (2D) surface can be considered as three-dimensional (3D) shell whose thickness is negligible in comparison with the dimension of the whole system. The quantum mechanics on surface can be first formulated in the bulk and the limit of vanishing thickness is then taken. The gradient operator and the Laplace operator originally defined in bulk converges to the geometric ones on the surface, and the so-called geometric momentum and geometric potential are obtained. On the surface of 2D sphere the geometric momentum in the Monge parametrization is explicitly explored. Dirac's theory on second-class constrained motion is resorted to for accounting for the commutator [xi, pj] = iℏ(δij - xixj/r2) rather than [xi, pj] = iℏδij that does not hold true anymore. This geometric momentum is geometric invariant under parameters transformation, and self-adjoint.

scholarly journals Characterization of Clifford Torus in Three-Spheres

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 718
Author(s):  
Dong-Soo Kim ◽  
Young Ho Kim ◽  
Jinhua Qian
Keyword(s):  
Laplace Operator ◽  
Unit Sphere ◽  
Three Dimensional ◽  
Gauss Map ◽  
Dimensional Sphere ◽  
Clifford Torus ◽  
Closed Surfaces ◽  
Dimensional Unit ◽  
The Laplace Operator

We characterize spheres and the tori, the product of the two plane circles immersed in the three-dimensional unit sphere, which are associated with the Laplace operator and the Gauss map defined by the elliptic linear Weingarten metric defined on closed surfaces in the three-dimensional sphere.

scholarly journals Characterizations of the sphere by the mean II-curvature

1981 ◽  
Vol 23 (2) ◽  
pp. 249-253 ◽  
Author(s):  
George Stamou
Keyword(s):  
Euclidean Space ◽  
Fundamental Form ◽  
Gaussian Curvature ◽  
Convex Surface ◽  
Three Dimensional ◽  
Second Fundamental Form ◽  
Positive Definite ◽  
Dimensional Euclidean Space ◽  
The Second Fundamental Form ◽  
The Mean

The notion of “mean II-curvature” of a C4-surface (without parabolic points) in the three-dimensional Euclidean space has been introduced by Ekkehart Glässner. The aim of this note is to give some global characterizations of the sphere related to the above notion.In the three-dimensional Euclidean space E3 we consider a sufficiently smooth ovaloid S (closed convex surface) with Gaussian curvature K > 0 . The ovaloid S possesses a positive definite second fundamental form II, if appropriately oriented. During the last years several authors have been concerned with the problem of characterizations of the sphere by the curvature of the second fundamental form of S. In this paper we give some characterizations of the sphere using the concept of the mean II-curvatureHII (of S), defined by Ekkehart Glässner.

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