The Gauss Images of Complete Minimal Surfaces of Genus Zero of Finite Total Curvature

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.