Constant Mean Curvature Surfaces Based on Fundamental Quadrilaterals

We describe the construction of CMC surfaces with symmetries in $\mathbb {S}^{3}$ S 3 and $\mathbb {R}^{3}$ ℝ 3 using a CMC quadrilateral in a fundamental tetrahedron of a tessellation of the space. The fundamental piece is constructed by the generalized Weierstrass representation using a geometric flow on the space of potentials.

SOLITON DEFORMATION OF INVERTED CATENOID

The minimal surface (see [1]) is determined using the Weierstrass representation in three-dimensional space. The solution of the Dirac equation [2] in terms of spinors coincides with the representations of this surface with conservation of isothermal coordinates. The equation represented through the Dirac operator, which is included in the Manakov’s L, A, B triple [3] as equivalent to the modified Veselov-Novikov equation (mVN) [4]. The potential 𝑈 of the Dirac operator is the potential of representing a minimal surface. New solutions of the mVN equation are constructed using the pre-known potentials of the Dirac operator and this algorithm is said to be Moutard transformations [5]. Firstly, the geometric meaning of these transformations which found in [6], [7], gives us the definition of the inversion of the minimal surface, further after finding the exact solutions of the mVN equation, we can represent the inverted surfaces. And these representations of the new potential determine the soliton deformation [8], [9]. In 2014, blowing-up solutions to the mVN equation were obtained using a rigid translation of the initial Enneper surface in [6]. Further results were obtained for the second-order Enneper surface [10]. Now the soliton deformation of an inverted catenoid is found by smooth translation along the second coordinate axis. In this paper, in order to determine catenoid inversions, it is proposed to find holomorphic objects as Gauss maps and height differential [11]; the soliton deformation of the inverted catenoid is obtained; particular solution of modified Karteweg-de Vries (KdV) equation is found that give some representation of KdV surface [12],[13].

Implicit Equations of the Henneberg-Type Minimal Surface in the Four-Dimensional Euclidean Space

Considering the Weierstrass data as ( ψ , f , g ) = ( 2 , 1 - z - m , z n ) , we introduce a two-parameter family of Henneberg-type minimal surface that we call H m , n for positive integers ( m , n ) by using the Weierstrass representation in the four-dimensional Euclidean space E 4 . We define H m , n in ( r , θ ) coordinates for positive integers ( m , n ) with m ≠ 1 , n ≠ - 1 , - m + n ≠ - 1 , and also in ( u , v ) coordinates, and then we obtain implicit algebraic equations of the Henneberg-type minimal surface of values ( 4 , 2 ) .

Nambu-Goto Strings viaWeierstrass Representation

A description of string model of gauge theory are related to minimal surfaces. notations of minimal surface and related mean and Gauss curvature discussed. The Weierstrass representation for a surface conformally which immersed in R used to represent Nambu- Goto action, action of Nambu Goto is calculated usingWeierstrass representation which can be used to calculate the Partion Function and potential, then a non-perturbative solution for action is aimed and fulfilled and a consequences of that are investigated and its mathematical and physical properties are discussed.