Related papers: Minimal surfaces associated with orthogonal polyno…
In this paper, following Sullivan, Kusner, and Schmitt, we study conformal immersions of Riemann surfaces into the three-dimensional Euclidean space. Regarding such immersions as special bundle maps from the tangent bundle of the surface to…
The main objective of this paper is to derive the Enneper-Weierstrass representation of minimal surfaces in $\mathbb{E}^3$ using the soliton surface approach. We exploit the Bryant-type representation of conformally parametrized surfaces in…
This paper is devoted to a study of the connections between three different analytic descriptions for the immersion functions of 2D-surfaces corresponding to the following three types of symmetries: gauge symmetries of the linear spectral…
A conformal map from a Riemann surface to a Euclidean space of dimension greater than or equal to three is explained by using the Clifford algebra, in a similar fashion to quaternionic holomorphic geometry of surfaces in the Euclidean…
The main aim of this paper is to study soliton surfaces immersed in Lie algebras associated with ordinary differential equations (ODE's) for elliptic functions. That is, given a linear spectral problem for such an ODE in matrix Lax…
We study certain new properties of 2D surfaces associated with the $\mathbb{C}P^{N-1}$ models and the wave functions of the corresponding linear spectral problem. We show that $su(N)$-valued immersion functions expressed in terms of rank-1…
Weierstrass representation is a classical parameterization of minimal surfaces. However, two functions should be specified to construct the parametric form in Weierestrass representation. In this paper, we propose an explicit parametric…
Minimal surfaces play a fundamental role in differential geometry, with applications spanning physics, material science, and geometric design. In this paper, we explore a novel quaternionic representation of minimal surfaces, drawing an…
We will investigate the local geometry of the surfaces in the $7$-dimensional Euclidean space associated to harmonic maps from a Riemann surface $\Sigma$ into $S^6$. By applying methods based on the use of harmonic sequences, we will…
We study minimal graphs in the homogeneous Riemannian 3-manifold $\widetilde{PSL_2(\mathbb{R})}$ and we give examples of invariant surfaces. We derive a gradient estimate for solutions of the minimal surface equation in this space and…
In this paper, we consider both differential and algebraic properties of surfaces associated with sigma models. It is shown that surfaces defined by the generalized Weierstrass formula for immersion for solutions of the CP^{N-1} sigma model…
The main aim of this paper is to discuss recent results on the adaptation of the Fokas-Gel'fand procedure for constructing soliton surfaces in Lie algebras, which was originally derived for PDEs [Grundland, Post 2011], to the case of…
In this paper we give a geometrically invariant spinorial representation of surfaces in four-dimensional space forms. In the Euclidean space, we obtain a representation formula which generalizes the Weierstrass representation formula of…
In this paper, the Weierstrass technique for harmonic maps S^2 -> CP^(N-1) is employed in order to obtain surfaces immersed in multidimensional Euclidean spaces. It is shown that if the CP^(N-1) model equations are defined on the sphere S^2…
The main aim of this paper is to introduce a new version of the Fokas-Gel'fand formula for immersion of soliton surfaces in Lie algebras. The paper contains a detailed exposition of the technique for obtaining exact forms of 2D-surfaces…
We consider a surface $M$ immersed in $\mathbb{R}^3$ with induced metric $g=\psi\delta_2$ where $\delta_2$ is the two dimensional Euclidean metric. We then construct a system of partial differential equations that constrain $M$ to lift to a…
This article surveys the Weierstrass representation of surfaces in the three- and four-dimensional spaces, with an emphasis on its relation to the Willmore functional. We also describe an application of this representation to constructing a…
We investigate certain properties of $\mathfrak{su}(N)$-valued two-dimensional soliton surfaces associated with the integrable $\mathbb{C}P^{N-1}$ sigma models constructed by the orthogonal rank-one Hermitian projectors, which are defined…
We give an immersion formula, the Sym-Bobenko formula, for minimal surfaces in the 3-dimensional Heisenberg space. Such a formula can be used to give a generalized Weierstrass type representation and construct explicit examples of minimal…
A discrete harmonic surface is a trivalent graph which satisfies the balancing condition in the 3-dimensional Euclidean space and achieves energy minimizing under local deformations. Given a topological trivalent graph, a holomorphic…