Holomorphic representation of minimal surfaces in simply isotropic space

• Published in: Journal of geometry Vol. 112; no. 3
• Main Author: da Silva, Luiz C. B.
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01-12-2021; Springer Nature B.V
• Subjects: Analytic functions; Euclidean geometry; Euclidean space; Geometry; Mathematical analysis; Mathematics; Mathematics and Statistics; Minimal surfaces; Representations; 53C42; simply isotropic space; minimal surface; 53A10; 53A35; stereographic projection; holomorphic representation
• ISSN: 0047-2468; 1420-8997
• DOI: 10.1007/s00022-021-00598-z

It is known that minimal surfaces in Euclidean space can be represented in terms of holomorphic functions. For example, we have the well-known Weierstrass representation, where part of the holomorphic data is chosen to be the stereographic projection of the normal of the corresponding surface, and also the Björling representation, where it is prescribed a curve on the surface and the unit normal on this curve. In this work, we are interested in the holomorphic representation of minimal surfaces in simply isotropic space, a three-dimensional space equipped with a rank 2 metric of index zero. Since the isotropic metric is degenerate, a surface normal cannot be unequivocally defined based on metric properties only, which leads to distinct definitions of an isotropic normal. As a consequence, this may also lead to distinct forms of a Weierstrass and of a Björling representation. Here, we show how to represent simply isotropic minimal surfaces in accordance with the choice of an isotropic surface normal.