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Discrete isothermic surfaces. (English) Zbl 0845.53005
Discrete isothermic surfaces are defined as the maps $F:{ℤ}^{2}\to {ℝ}^{3}$ 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 $F:{ℤ}^{2}\to ℂ$ 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 space 39A12 Discrete version of topics in analysis 37J35 Completely integrable systems, topological structure of phase space, integration methods 37K10 Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies 52C99 Discrete geometry 53A30 Conformal differential geometry 53A10 Minimal surfaces, surfaces with prescribed mean curvature