Related papers: Minimal Surfaces via Complex Quaternions
This paper establishes the basis of the quaternionic differential geometry ($\mathbbm H$DG) initiated in a previous article. The usual concepts of curves and surfaces are generalized to quaternionic constraints, as well as the curvature and…
We consider a higher-order Henneberg-type minimal surfaces family using the generalized Weierstrass--Enneper representation in four-dimensional space $\mathbb{R}^4$. We derive explicit parametric equations for the surface and determine its…
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…
This paper is devoted to a study of the connection between the immersion functions of two-dimensional surfaces in Euclidean or hyperbolic spaces and classical orthogonal polynomials. After a brief description of the soliton surfaces…
Minimal surfaces are among the most natural objects in Differential Geometry, and have been studied for the past 250 years ever since the pioneering work of Lagrange. The subject is characterized by a profound beauty, but perhaps even more…
Variational principles are developed within the framework of a spinor representation of the surface geometry to examine the equilibrium properties of a membrane or interface. This is a far-reaching generalization of the Weierstrass-Enneper…
Minimal surfaces with isothermal parameters admitting B\'{e}zier representation were studied by Cosin and Monterde. They showed that, up to an affine transformation, the Enneper surface is the only bi-cubic isothermal minimal surface. Here…
Harmonic mappings have long intrigued researchers due to their intrinsic connection with minimal surfaces. In this paper, we investigate shearing of two distinct classes of univalent conformal mappings which are convex in horizontal…
We introduce on any smooth oriented minimal surface in Euclidean $3$-space a meromorphic quadratic differential, $P$, which we call the entropy differential. This differential arises naturally in a number of different contexts. Of…
In this paper we obtain the general solution to the minimal surface equation, namely its local Weierstrass-Enneper representation, by using a system of hodographic coordinates. This is done by using the method of solving the Born-Infeld…
This work provides explicit characterizations and formulae for the minimal polynomials of a wide variety of structured $4\times 4$ matrices. These include symmetric, Hamiltonian and orthogonal matrices. Applications such as the complete…
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…
We study minimal Lagrangian surfaces in the complex hyperbolic quadric. We show that minimality of a Lagrangian surface is characterized by a loop of flat connections, which yields an associated $\mathbb S^1$-family of isometric…
We consider surfaces immersed in three-dimensional pseudohermitian manifolds. We define the notion of (p-)mean curvature and of the associated (p-)minimal surfaces, extending some concepts previously given for the (flat) Heisenberg group.…
This is the first comprehensive introduction to the authors' recent attempts toward a better understanding of the global concepts behind spinor representations of surfaces in 3-space. The important new aspect is a quaternionic-valued…
A fake quadric is a smooth minimal surface of general type with the same invariants as the quadric in P^3, i.e. K^2=2c_2=8 and q=p_g=0. We study here quaternionic fake quadrics i.e. fake quadrics constructed arithmetically by using some…
In this paper, we study totally real minimal surfaces in the quaternionic projective space $\mathbb{H}P^n$. We prove that the linearly full totally real flat minimal surfaces of isotropy order $n$ in $\mathbb{H}P^n$ are two surfaces in…
We describe a new family of triply-periodic minimal surfaces with hexagonal symmetry, related to the quartz (qtz) and its dual (the qzd net). We provide a solution to the period problem and provide a parametrisation of these surfaces, that…
Minimal surfaces and Einstein manifolds are among the most natural structures in differential geometry. Whilst minimal surfaces are well understood, Einstein manifolds remain far less so. This exposition synthesises together a set of…
In this note, we initiate a study of the finite-dimensional representation theory of a class of algebras that correspond to noncommutative deformations of compact surfaces of arbitrary genus. Low dimensional representations are investigated…