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This paper aims to present a systematic study on the Gauss images of completeminimal surfaces of genus 0 of finite total curvature in Euclidean 3-space andEuclidean 4-space. We focus on the number of omitted values and the totalweight of a number of totally ramified values of their Gauss maps. Inparticular, we construct new complete minimal surfaces of finite totalcurvature whose Gauss maps have 2 omitted values and 1 totally ramified valueof order 2, that is, the total weight of a number of totally ramified values oftheir Gauss maps are 2.5 in Euclidean 3-space and Euclidean 4-space,respectively. Moreover we discuss several outstanding problems in this study.

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