Minimal surfaces from circle patterns: Geometry from combinatorics

Details Minimal surfaces from circle patterns: Geometry from combinatorics Minimal surfaces from circle patterns: Geometry from combinatorics

2003-05-13

arxiv.org/abs/math/0305184v3

Ann. of Math. 164:1 (2006), 231-264

Mathematics

Differential Geometry

30 pages, many figures, some in reduced resolution. v2: Extended introduction. Minor changes in presentation. v3: revision acc

Scientific paper

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.

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