2015/10/29 by Wai Yeung Lam, Lam, Wai Yeung · 1 citation
Mathematics · Physics and Astronomy · #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Mathematical Dynamics and Fractals #Metric Geometry (math.MG) #Nonlinear Waves and Solitons
paper · pdf · doi:10.48550/arxiv.1510.08788
openalex publication_date 2015/10/29 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28
We obtain a unified theory of discrete minimal surfaces based on discrete\nholomorphic quadratic differentials via a Weierstrass representation. Our\ndiscrete holomorphic quadratic differential are invariant under M "obius\ntransformations. They can be obtained from discrete harmonic functions in the\nsense of the cotangent Laplacian and Schramm's orthogonal circle patterns.\n We show that the corresponding discrete minimal surfaces unify the earlier\nnotions of discrete minimal surfaces: circular minimal surfaces via the\nintegrable systems approach and conical minimal surfaces via the curvature\napproach. In fact they form conjugate pairs of minimal surfaces.\n Furthermore, discrete holomorphic quadratic differentials obtained from\ndiscrete integrable systems yield discrete minimal surfaces which are critical\npoints of the total area.\n