The Weierstrass-Enneper Representation using hodographic coordinates on a minimal surface

离散调和曲面的Weierstrass表示公式

A Weierstrass Representation Formula for Discrete Harmonic Surfaces

本文针对三维欧氏空间中的离散调和曲面，提出基于拓扑三价图、全纯函数与离散全纯二次型的Weierstrass表示公式，构造出收敛到经典极小曲面的离散调和曲面序列，并给出Enneper曲面的离散逼近。

Constant mean curvature surfaces via integrable dynamical system

It is shown that the equation which describes constant mean curvature surface via the generalized Weierstrass-Enneper inducing has Hamiltonian form. Its simplest finite-dimensional reduction has two degrees of freedom, integrable and its trajectories correspond to well-known Delaunay and do Carmo-Dajzcer surfaces (i.e., helicoidal constant mean curvature surfaces).

Born-Infeld solitons, Maximal surfaces and Ramanujan's identities

We show that a Born-Infeld soliton can be realised either as a spacelike minimal graph or timelike minimal graph over a timelike plane or a combination of both away from singular points. We also obtain some exact solutions of the Born-Infeld equation from already known solutions to the maximal surface equation. Further we present a method to construct a one-parameter family of complex solitons from a given one parameter family of maximal surfaces. Finally, using Ramanujan's Identities and the Weierstrass-Enneper representation of maximal surfaces, we derive further non-trivial identities.

Univalence Criteria for Lifts of Harmonic Mappings to Minimal Surfaces

A general criterion in terms of the Schwarzian derivative is given for global univalence of the Weierstrass--Enneper lift of a planar harmonic mapping. Results on distortion and boundary regularity are also deduced. Examples are given to show that the criterion is sharp. The analysis depends on a generalized Schwarzian defined for conformal metrics and on a Schwarzian introduced by Ahlfors for curves. Convexity plays a central role.

带奇异点的极大曲面的逼近与插值定理

Approximation and Interpolation Theorems for Maximal Surfaces with Singularities

本文利用Enneper-Weierstrass表示公式，将共形极小曲面的逼近与插值定理推广到Lorentz-Minkowski 3-空间中的极大曲面，并证明了具有指定奇异点和奇异集稠密像的极大曲面的存在性。

Polynomial minimal surfaces of degree five

The problem of finding all minimal surfaces presented in parametric form as polynomials of certain degree is discussed by many authors. It is known that the classical Enneper surface is (up to position in space and homothety) the only polynomial minimal surface of degree 3 in isothermal parameters. In higher degrees the problem is quite more complicated. Here we find a general form for the functions that generate a polynomial minimal surface of arbitrary degree via the Weierstrass formula and prove that any polynomial minimal surface of degree 5 in isothermal parameters may be considered as be…

Bi-quartic parametric polynomial minimal surfaces

Minimal surfaces with isothermal parameters admitting Bézier 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 we study bi-quartic isothermal minimal surfaces and establish the general form of their generating functions in the Weierstrass representation formula. We apply an approach proposed by Ganchev to compute the normal curvature and show that, in contrast to the bi-cubic case, there is a variety of bi-quartic isothermal minimal surfaces. Based on the Bezier represen…

四维空间中的高阶Henneberg型极小曲面家族

The higher-order Henneberg-type minimal surfaces family in $\mathbb{R}^4$

本文在四维空间中构造了高阶Henneberg型极小曲面家族，推导了其参数方程和几何特性，并通过投影生成可视化结果，同时研究了无积分形式的代数函数。

Parametric polynomial minimal surfaces of arbitrary degree

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 form for a class of parametric polynomial minimal surfaces of arbitrary degree. It includes the classical Enneper surface for cubic case. The proposed minimal surfaces also have some interesting properties such as symmetry, containing straight lines and self-intersections. According to the shape properties, the proposed minimal surface can be classified into fo…

A chiral family of triply-periodic minimal surfaces derived from the quartz network

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 are not in the regular class, by the Weierstrass-Enneper formalism. We identified this analytical description of the surface by generating an area-minimising mesh interface from a pair of dual graphs (qtz & qzd) using the generalised Voronoi construction of De Campo, Hyde and colleagues, followed by numerical identification of the flat point structure. This …

Schwarzian导数与经典极小曲面的次数

The Schwarzian derivative and the degree of a classical minimal surface

本文利用Schwarzian导数构造极小曲面上的亚纯微分序列，定义曲面次数，发现Enneper、螺旋面等具有小次数，并证明任意极小曲面可被高次数曲面逼近。

Lorentz-Minkowski 三维空间中的极小类时曲面及其规范参数

Minimal Timelike Surfaces in the Lorentz-Minkowski 3-space and Their Canonical Parameters

本文借助分裂复数代数中的 Weierstrass 型公式与规范参数刻画 $\mathbb R^3_1$ 极小类时曲面，建立其与全纯函数类的对应，并证明 Enneper 曲面是唯一具有三次等温多项式参数化者。

Finite Decomposition of Minimal surfaces, Maximal surfaces, Timelike Minimal surfaces and Born-Infeld solitons

We show that the height function of Scherk's second surface decomposes into a finite sum of scaled and translated versions of itself, using an Euler Ramanujan identity. A similar result appears in R. Kamien's work on liquid crystals where he shows (using an Euler-Ramanujan identity) that the Scherk's first surface decomposes into a finite sum of scaled and translated versions of itself. We give another finite decomposition of the height function of the Scherk's first surface in terms of translated helicoids and scaled and translated Scherk's first surface. We give some more examples, for insta…