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Discrete isothermic surfaces. (English) Zbl 0845.53005
Discrete isothermic surfaces are defined as the maps $F: \bbfZ^2\to \bbfR^3$ 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 $F: \bbfZ^2\to \bbfC$ 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.
Reviewer: A.Bobenko, U.Pinkall (Berlin)

53A05Surfaces in Euclidean space
39A12Discrete version of topics in analysis
37J35Completely integrable systems, topological structure of phase space, integration methods
37K10Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies
52C99Discrete geometry
53A30Conformal differential geometry
53A10Minimal surfaces, surfaces with prescribed mean curvature
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