Holomorphic Vector Fields and Quadratic Differentials on Planar Triangular Meshes

Wai Yeung Lam () and Ulrich Pinkall ()
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Wai Yeung Lam: Inst. Für Mathematik, Technische Universität Berlin
Ulrich Pinkall: Inst. Für Mathematik, Technische Universität Berlin

A chapter in Advances in Discrete Differential Geometry, 2016, pp 241-265 from Springer

Abstract: Abstract Given a triangulated region in the complex plane, a discrete vector field Y assigns a vector $$Y_i\in \mathbb {C}$$ Y i ∈ C to every vertex. We call such a vector field holomorphic if it defines an infinitesimal deformation of the triangulation that preserves length cross ratios. We show that each holomorphic vector field can be constructed based on a discrete harmonic function in the sense of the cotan Laplacian. Moreover, to each holomorphic vector field we associate in a Möbius invariant fashion a certain holomorphic quadratic differential. Here a quadratic differential is defined as an object that assigns a purely imaginary number to each interior edge. Then we derive a Weierstrass representation formula, which shows how a holomorphic quadratic differential can be used to construct a discrete minimal surface with prescribed Gauß map and prescribed Hopf differential.

Keywords: Holomorphic Vector Fields; Holomorphic Quadratic Differentials; Planar Triangular Mesh; Discrete Minimal Surfaces; Discrete Harmonic Functions (search for similar items in EconPapers)
Date: 2016
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DOI: 10.1007/978-3-662-50447-5_7

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