Sym–Bobenko formula for minimal surfaces in Heisenberg space

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 surfaces.     

A Weierstrass Representation Formula for Minimal Surfaces in ℍ3 and ℍ2 × ℝ

We give a general setting for constructing a Weierstrass representation formula for simply connected minimal surfaces in a Riemannian manifold. Then, we construct examples of minimal surfaces in the three dimensional Heisenberg group and in the product of the hyperbolic plane with the real line. Work partially supported by RAS, INdAM, FAPESP and CNPq     

A Weierstrass Representation for Minimal Surfaces in 3-Dimensional Manifolds

In this paper we will discuss a Weierstrass type representation for minimal surfaces in Riemannian and Lorentzian 3-dimensional manifolds.     

Weierstrass Representation of Some Simply-Periodic Minimal Surfaces

We prove a desingularization result for minimal surfaces inEuclidean space using Weierstrass representation. We solve the periodproblem using the implicit function theorem at a degenerate point.     

三维Minkowski空间中的极小曲面

给出了三维Minkowski空间中类空曲面活动标价的四元素表示,并利用四元素既适合于旋转结构的侵入又适合于2×2矩阵处理极小曲面的分析特性,经研究得到了R12中的极小曲面Weierstrass表示和曲面的可积条件.     

The doubly periodic Costa surfaces

We derive global Weierstrass representations for complete minimal surfaces obtained by substituting the planar end of the Costa surface by symmetry curves. Received: 14 February 2001; in final form: 24 April 2001 / Published online: 29 April 2002     

Minimal surfaces,corners, and wires

Weierstrass representations are given for minimal surfaces that have free boundaries on two planes that meet at an arbitrary dihedral angle. The contact angles of a surface on the planes may be different. These surfaces illustrate the behavior of soapfilms in convex and nonconvex comers. They can also be used to show how a boundary wire can penetrate a soapfilm with a free end, as in the overhand knot surface. They should also cast light on the behavior of capillary 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.     

Surfaces in Three-Dimensional Lie Groups

We derive the Weierstrass (or spinor) representation for surfaces in the three-dimensional Lie groups Nil, \(\widetilde{SL}_2\), and Sol with Thurston's geometries and establish the generating equations for minimal surfaces in these groups. Using the spectral properties of the corresponding Dirac operators, we find analogs of the Willmore functional for surfaces in these geometries.     

The spiral minimal surfaces and their Legendre and Weierstrass representations

A class of spiral minimal surfaces in E³ is constructed using a symmetry reduction. The reduction leads to a cubic-nonlinear ODE whose phase portrait is described using an auxiliary Riccati's equation and the Warzewski topological principle for its solutions. The new surfaces are invariant with respect to the composition of rotation and dilation. The solutions are obtained in parametric form through the Legendre and the Weierstrass representations, and also their asymptotic behaviour is described.     

Spinorial representation of surfaces into 4-dimensional space forms

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 minimal surfaces. We also obtain as particular cases the spinorial characterizations of surfaces in $\mathbb R ^3$ and in $S^3$ given by Friedrich and by Morel.     

Multiplicative Weierstrass Points and the Jacobi Variety of a Compact Riemann Surface

In the previous papers of the present author, a general theory of multiplicative Weierstrass points on compact Riemann surfaces for arbitrary characters was developed. In the present paper, some additional relations between multiplicative Weierstrass points on a compact Riemann surface for an arbitrary character and special subsets in the Jacobi variety, the canonical embedding of a compact Riemann surface into a projective space, are established. We not distinction between classical Weierstrass points and multiplicative Weierstrass points on a compact Riemann surface.     

The density of elliptic curves having a global minimal Weierstrass equation

We show that a positive density of elliptic curves over a number field counted using their short Weierstrass equations belong to a given Weierstrass class and in particular, a positive density of elliptic curves have a global minimal Weierstrass equation. The density is given by a ratio of partial zeta functions of the number field K evaluated at 10 with some extra factors for the bad primes.     

The Geometry of Two Generator Groups: Hyperelliptic Handlebodies

A Kleinian group naturally stabilizes certain subdomains and closed subsets of the closure of hyperbolic three space and yields a number of different quotient surfaces and manifolds. Some of these quotients have conformal structures and others hyperbolic structures. For two generator free Fuchsian groups, the quotient three manifold is a genus two solid handlebody and its boundary is a hyperelliptic Riemann surface. The convex core is also a hyperelliptic Riemann surface. We find the Weierstrass points of both of these surfaces. We then generalize the notion of a hyperelliptic Riemann surface to a hyperelliptic three manifold. We show that the handlebody has a unique order two isometry fixing six unique geodesic line segments, which we call the Weierstrass lines of the handlebody. The Weierstrass lines are, of course, the analogue of the Weierstrass points on the boundary surface. Further, we show that the manifold is foliated by surfaces equidistant from the convex core, each fixed by the isometry of order two. The restriction of this involution to the equidistant surface fixes six generalized Weierstrass points on the surface. In addition, on each of these equidistant surfaces we find an orientation reversing involution that fixes curves through the generalized Weierstrass points.Mathematics Subject Classifications (2000). primary 30F10, 30F35, 30F40; secondary 14H30, 22E40.     

Lie geometry of linear Weingarten surfaces

We show how linear Weingarten surfaces appear as special Ω-surfaces and give a characterization of those linear Weingarten surfaces that allow a Weierstrass type representation.     

The Gauss maps of Demoulin surfaces with conformal coordinates

Demoulin surfaces in the real projective 3-space are investigated. Our result enables us to establish a generalized Weierstrass type representation for definite Demoulin surfaces by virtue of primitive maps into a certain semi-Riemannian 6-symmetric space.     

Parametrization of a Family of Minimal Surfaces Bounded by the Broken Lines in $$\mathbb{R}^3 $$

Consideration is given to a family of minimal surfaces bounded by the broken lines in which are locally injectively projected onto the coordinate plane. The considered family is bijectively mapped by means of the Enepper–Weierstrass representation onto a set of circular polygons of a certain type. The parametrization of this set is constructed, and the dimension of the parameter domain is established.     

Spectral transformation of constant mean curvature surfaces in H3 and Weierstrass representation

By using the method of integrable system, we study the deformation of constant mean curvature surfaces in three-dimensional hyperbolic space form H3. We also obtain a Weierstrass representation formula of the constant mean curvature surfaces with mean curvature greater than 1.     

Nonlinear integration and Weierstrass integral over a manifold: Connections with theorems on martingales

This paper is a survey on the integrals of the calculus of variations as Weierstrass integrals. While the classic integrals in terms of Lebesgue processes are valid only overAC curves, surfaces, or variety, the corresponding Weierstrass or Burkill integrals are valid in theBV case. This is the immediate advantage of the Weierstrass-Burkill form. Moreover, Cesari defined a very general integration process for set functions, in an abstract setting, which extends the Weierstrass and Burkill integrals. In more detail, we examine here some existence, approximation, representation, and semicontinuity results both in the parametric case and in the nonparametric case.     

Spectral transformation of constant mean curvature surfaces in H3 and Weierstrass representation

By using the method of integrable system, we study the deformation of constant mean curvature surfaces in three-dimensional hyperbolic space form H₃. We also obtain a Weierstrass representation formula of the constant mean curvature surfaces with mean curvature greater than 1