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Discrete isothermic surfaces. (English) Zbl 0845.53005
Discrete isothermic surfaces are defined as the maps $$F: \mathbb{Z}^2\to \mathbb{R}^3$$ such that all elementary quadrilaterals of the surface have cross ratio $$-1$$. It is shown that these discrete surfaces possess properties which are characteristic for smooth isothermic surfaces (Möbius invariance, dual surface). Quaternionic zero-curvature loop group representations for smooth and discrete isothermic surfaces are presented. Discrete holomorphic maps are defined as the maps $$F: \mathbb{Z}^2\to \mathbb{C}$$ such that all elementary quadrilaterals have cross ratio $$-1$$. A Weierstrass type representation for the discrete minimal isothermic surfaces (which are a special class of discrete isothermic surfaces) in terms of discrete holomorphic maps is obtained and the discrete catenoid and the Enneper surface are constructed.

MSC:
 53A05 Surfaces in Euclidean and related spaces 39A12 Discrete version of topics in analysis 37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests 37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) 52C99 Discrete geometry 53A30 Conformal differential geometry (MSC2010) 53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
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