Discrete holomorphic geometry I. Darboux transformations and spectral curves - Mathematics > Differential GeometryReportar como inadecuado




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Abstract: Finding appropriate notions of discrete holomorphic maps and, more generally,conformal immersions of discrete Riemann surfaces into 3-space is an importantproblem of discrete differential geometry and computer visualization. Wepropose an approach to discrete conformality that is based on the concept ofholomorphic line bundles over -discrete surfaces-, by which we mean the vertexsets of triangulated surfaces with bi-colored set of faces. The resultingtheory of discrete conformality is simultaneously Moebius invariant and basedon linear equations. In the special case of maps into the 2-sphere we obtain areinterpretation of the theory of complex holomorphic functions on discretesurfaces introduced by Dynnikov and Novikov. As an application of our theory weintroduce a Darboux transformation for discrete surfaces in the conformal4-sphere. This Darboux transformation can be interpreted as the space- andtime-discrete Davey-Stewartson flow of Konopelchenko and Schief.



Autor: Christoph Bohle, Franz Pedit, Ulrich Pinkall

Fuente: https://arxiv.org/







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