Holomorphic representation of minimal surfaces in simply isotropic space

TL;DR

This paper extends the holomorphic representation of minimal surfaces from Euclidean space to simply isotropic space, addressing the challenges posed by the degenerate metric and multiple notions of surface normal.

Contribution

It introduces a framework for representing minimal surfaces in simply isotropic space using holomorphic functions, considering different definitions of isotropic normal.

Findings

Representation depends on the choice of isotropic normal

Multiple forms of Weierstrass and Björling representations are derived

Addresses the impact of degenerate metric on surface normal definition

Abstract

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 Bj\"orling 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 Bj\"orling representation. Here, we show how to represent simply isotropic minimal surfaces in accordance with the choice of an isotropic surface normal.