Minimal surfaces from circle patterns: Geometry from combinatorics
Abstract
We suggest a new definition for discrete minimal surfaces in terms of sphere packings with orthogonally intersecting circles. These discrete minimal surfaces can be constructed from Schramm's circle patterns. We present a variational principle which allows us to construct discrete analogues of some classical minimal surfaces. The data used for the construction are purely combinatorial--the combinatorics of the curvature line pattern. A Weierstrass-type representation and an associated family are derived. We show the convergence to continuous minimal surfaces.
Cite
@article{arxiv.math/0305184,
title = {Minimal surfaces from circle patterns: Geometry from combinatorics},
author = {Alexander I. Bobenko and Tim Hoffmann and Boris A. Springborn},
journal= {arXiv preprint arXiv:math/0305184},
year = {2007}
}
Comments
30 pages, many figures, some in reduced resolution. v2: Extended introduction. Minor changes in presentation. v3: revision according to the referee's suggestions, improved & expanded exposition, references added, minor mistakes corrected