Conformal Structures and Period Matrices of Polyhedral Surfaces

Conformal Structures and Period Matrices of Polyhedral Surfaces Alexander Bobenko? Christian Mercat† Markus Schmies? March 7, 2008 Abstract We recall the theory of linear discrete Riemann surfaces and show how to use it in order to interpret a surface embedded in R3 as a discrete Riemann surface and compute its basis of holomorphic forms on it. We present numerical examples, recovering known results to test the numerics and giving the yet unknown period matrix of the Lawson genus-2 surface. 1 Introduction Finding a conformal parameterization for a surface and computing its period matrix is useful in a lot of contexts, from statistical mechanics to computer graphics. The 2D-Ising model [18, 8, 9] for example takes place on a cellular decomposi- tion of a surface whose edges are decorated by interaction constants, understood as a discrete conformal structure. In certain configurations, called critical tem- perature, the model exhibits conformal invariance properties in the thermody- namical limit and certain statistical expectations become discrete holomorphic at the finite level. The computation of the period matrix of higher genus sur- faces built from the rectangular and triangular lattices from discrete Riemann theory has been addressed in the cited papers by Costa-Santos and McCoy. Global conformal parameterization of a surface is important in computer graphics [16, 12, 2, 25, 17, 26] in issues such as texture mapping of a flat picture onto a curved surface in R3.

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• conformal equivalence


• linear discrete


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• riemann surface


• cauchy-riemann equation


• holomorphic forms