Discrete minimal surfaces: critical points of the area functional from integrable systems

Abstract

We obtain a unified theory of discrete minimal surfaces based on discrete holomorphic quadratic differentials via a Weierstrass representation. Our discrete holomorphic quadratic differential are invariant under M\"{o}bius transformations. They can be obtained from discrete harmonic functions in the sense of the cotangent Laplacian and Schramm's orthogonal circle patterns. We show that the corresponding discrete minimal surfaces unify the earlier notions of discrete minimal surfaces: circular minimal surfaces via the integrable systems approach and conical minimal surfaces via the curvature approach. In fact they form conjugate pairs of minimal surfaces. Furthermore, discrete holomorphic quadratic differentials obtained from discrete integrable systems yield discrete minimal surfaces which are critical points of the total area.