Abstract
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.
Highlights
Surfaces with constant mean curvature H or constant Gauss curvature K have been of particular interest in differential geometry for a long time
The investigation of constant curvature surfaces is tied to specific parametrizations, like isothermic parametrizations for constant mean curvature surfaces
In the present paper we investigate two distinct situations: (i) semidiscrete surfaces with vanishing mean curvature, and (ii) semidiscrete surfaces with constant but non-vanishing mean curvature
Summary
Surfaces with constant mean curvature H or constant Gauss curvature K have been of particular interest in differential geometry for a long time. As a first step toward this direction, Bobenko et al [6] introduced a general curvature theory for polyhedral meshes with planar faces based on mesh parallelity Their theory is capable of unifying notable previously defined classes of surfaces, such as discrete isothermic minimal or constant mean curvature surfaces. Hoffmann et al [12] presented a discrete parametrized surface theory for quadrilateral meshes equipped with unit normal vectors at the vertices, permitting non-planar faces. Their theory encompasses a remarkably large class of existing discrete special parametrizations. To the author’s knowledge, results concerning their associated families have been missing so far
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