The Gauss images of complete minimal surfaces of genus zero of finite total curvature

TL;DR

This paper systematically studies the Gauss images of complete minimal surfaces of genus zero with finite total curvature, revealing new examples with specific ramification properties and discussing open problems.

Contribution

It constructs new minimal surfaces with Gauss maps having two omitted values and one ramified value, advancing understanding of their geometric properties.

Findings

Constructed minimal surfaces with Gauss maps having 2 omitted values.

Identified surfaces with Gauss maps having 1 totally ramified value of order 2.

Discussed open problems in the classification of Gauss images.

Abstract

This paper aims to present a systematic study on the Gauss images of complete minimal surfaces of genus 0 of finite total curvature in Euclidean 3-space and Euclidean 4-space. We focus on the number of omitted values and the total weight of the totally ramified values of their Gauss maps. In particular, we construct new complete minimal surfaces of finite total curvature whose Gauss maps have 2 omitted values and 1 totally ramified value of order 2, that is, the total weight of the totally ramified values of their Gauss maps are 5/2 (=2.5) in Euclidean 3-space and Euclidean 4-space, respectively. Moreover we discuss several outstanding problems in this study.