On semidiscrete constant mean curvature surfaces and their associated families
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- 作者：Wolfgang Carl
- 关键词：Semidiscrete surface ; Constant mean curvature ; Associated family ; Weierstrass representation ; Lax pair representation
- 刊名：Monatshefte für Mathematik
- 出版年：2017
- 出版时间：March 2017
- 年：2017
- 卷：182
- 期：3
- 页码：537-563
- 全文大小：1222KB
- 刊物主题：Mathematics, general;
- 出版者：Springer Vienna
- ISSN：1436-5081
- 卷排序：182
文摘
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 \(\sinh \)-Gordon equation. Utilizing this theory we investigate semidiscrete Delaunay surfaces and their connection to elliptic billiards.