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 $$\sinh $$sinh-Gordon equation. Utilizing this theory we investigate semidiscrete Delaunay surfaces and their connection to elliptic billiards.

中文翻译：

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关于半离散常平均曲率曲面及其相关族

本论文在可积系统的框架内研究了三维欧几里得空间中的半离散曲面。特别是，我们研究了具有恒定平均曲率的半离散曲面及其相关族。本文中引入的平均曲率的概念是由最近开发的四边形网格的曲率理论激发的，该理论在顶点配备单位法向量，并扩展了之前在半离散曲面上的工作。在平均曲率消失的情况下，关联的族是通过 Weierstrass 表示定义的。对于一般的 cmc 情况，我们引入了一个 Lax 对表示，它直接定义了 cmc 曲面的相关族，并连接到半离散的 $$\sinh $$sinh-Gordon 方程。