Marginally trapped surfaces in L4 and an extended Weierstrass-Bryant representation

Loop Weierstrass 表示

Loop Weierstrass Representation

本文提出 Loop Weierstrass 表示，统一极小曲面与常平均曲率 1 曲面的关联族、对偶曲面和 Goursat 变换，引入简单因子 dressing，并分类多种例子。

Marginally trapped surfaces in ${\mathbb{L}}^{4}$ and three Weierstrass representations

We construct new integrable systems to present Weierstrass type representations for spacelike surfaces whose mean curvature vector $\mathbf{H}$ satisfies the null condition $\langle \mathbf{H}, \mathbf{H} \rangle=0$ in the four dimensional Lorentz-Minkowski space ${\mathbb{L}}^{4}$. Our new Weierstrass presentations extend simultaneously classical Weierstrass representations (of the first kind and the second kind) for maximal surfaces in ${\mathbb{L}}^{3}$ and minimal surfaces in ${\mathbb{R}}^{3}$. We solve a linear partial differental equation to construct explicit examples of marginally tra…

离散 Weierstrass 型表示

Discrete Weierstrass-type representations

本文证明已知离散 Weierstrass 型表示可视为 $Ω$-对偶变换在 Laguerre 几何类光 Gauss 映射上的应用，并据此推导出所有 Bryant 或 Bianchi 型离散线性 Weingarten曲面均可局部由离散全纯映射构造。

Moebius geometry of surfaces of constant mean curvature 1 in hyperbolic space

Various transformations of isothermic surfaces are discussed and their interrelations are analyzed. Applications to cmc-1 surfaces in hyperbolic space and their minimal cousins in Euclidean space are presented: the Umehara-Yamada perturbation, the classical and Bryant's Weierstrass type representations, and the duality for cmc-1 surfaces are interpreted in terms of transformations of isothermic surfaces. A new Weierstrass type representation is introduced and a Moebius geometric characterization of cmc-1 surfaces in hyperbolic space and minimal surfaces in Euclidean space is given.

Transformations of discrete isothermic nets and discrete cmc-1 surfaces in hyperbolic space

Using a quaternionic calculus, the Christoffel, Darboux, Goursat, and spectral transformations for discrete isothermic nets are described, with their interrelations. The Darboux and spectral transformations are used to define discrete analogs for cmc-1 surfaces in hyperbolic space and to obtain a discrete version of Bryant's Weierstrass type representation.

Supplement on Curved flats in the space of point pairs and Isothermic surfaces: A Quaternionic Calculus

A quaternionic calculus for surface pairs in the conformal 4-sphere is elaborated. This calculus is then used to discuss the relation between curved flats in the symmetric space of point pairs and Darboux and Christoffel pairs of isothermic surfaces. A new viewpoint on relations between surfaces of constant mean curvature in certain space forms is presented --- in particular, a new form of Bryant's Weierstrass type representation for surfaces of constant mean curvature 1 in hyperbolic 3-space is given.