Carl Wolfgang
Institute of Geometry, TU Graz, Kopernikusgasse 24, 8010 Graz, Austria.
Mon Hefte Math. 2017;182(3):537-563. doi: 10.1007/s00605-016-0929-6. Epub 2016 May 23.
The present paper studies semidiscrete surfaces in three-dimensional Euclidean space within the framework of integrable systems. In particular, we investigate semidiscrete surfaces with constant mean curvature along with their associated families. The notion of mean curvature introduced in this paper is motivated by a recently developed curvature theory for quadrilateral meshes equipped with unit normal vectors at the vertices, and extends previous work on semidiscrete surfaces. In the situation of vanishing mean curvature, the associated families are defined via a Weierstrass representation. For the general cmc case, we introduce a Lax pair representation that directly defines associated families of cmc surfaces, and is connected to a semidiscrete [Formula: see text]-Gordon equation. Utilizing this theory we investigate semidiscrete Delaunay surfaces and their connection to elliptic billiards.
本文在可积系统的框架下研究三维欧几里得空间中的半离散曲面。特别地，我们研究具有常平均曲率的半离散曲面及其相关族。本文引入的平均曲率概念是受最近为在顶点配备单位法向量的四边形网格发展的曲率理论所推动，并扩展了先前关于半离散曲面的工作。在平均曲率为零的情况下，相关族通过魏尔斯特拉斯表示来定义。对于一般的常平均曲率情况，我们引入一个拉克斯对表示，它直接定义常平均曲率曲面的相关族，并与一个半离散的[公式：见原文]-戈登方程相联系。利用这一理论，我们研究半离散德劳内曲面及其与椭圆台球的联系。
