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## Discrete holomorphic geometry I. Darboux transformations and spectral curves.

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Author(s) : Christoph Bohle , Franz Pedit , Ulrich Pinkall

Preprint series of the Institute of Mathematics, Technische Universität Berlin
Preprint 25-2007

MSC 2000

53C42 Immersions
53A30 Conformal differential geometry
53A05 Surfaces in Euclidean space
37K35 Lie-Bäcklund and other transformations
30G25 Discrete analytic functions

Abstract :
Finding appropriate notions of discrete holomorphic maps and, more generally, conformal immersions of discrete Riemann surfaces into 3-space is an important problem of discrete differential geometry and computer visualization. We investigate an approach to discrete conformality based on the concept of holomorphic line bundles over ``discrete surfaces'', that is, over vertex sets of triangulated surfaces with bi-colored set of faces. In the special case of maps into the 2-sphere our approach reinterprets the theory of complex holomorphic functions on discrete surfaces proposed by Dynnikov and Novikov. This reinterpretation reveals invariance under Möbius transformations and shows that the proposed theory of discrete holomorphicity is simultaneously based on linear equations and Möbius invariant. As an application of quaternionic holomorphic line bundles we introduce a Darboux transformation for discrete surfaces in the conformal 4-sphere. We show that this Darboux transformation can be interpreted as the space-and time-discrete Davey-Stewartson flow introduced by Konopelchenko and Schief. For generic maps of discrete tori with regular combinatorics, we prove that the space of all Darboux transforms has the structure of a compact Riemann surface, the spectral curve. This makes contact to the theory of algebraically completely integrable systems and is the starting point of a soliton theory for triangulated tori in 3- and 4-space devoid of special assumptions on the geometry of the surface.