Modified defect relations of the Gauss map of complete minimal surfaces on annular ends

TL;DR

This paper investigates the modified defect relations of the Gauss map for complete minimal surfaces on annular ends in three and four-dimensional spaces, providing improved results over previous studies.

Contribution

It extends and refines the understanding of Gauss map defect relations specifically on annular ends of minimal surfaces, building upon Fujimoto's earlier work.

Findings

Derived new defect relation results for Gauss maps on annular ends.

Improved upon previous theorems for minimal surfaces in $\

a0obtained results applicable to surfaces in $\

Abstract

In this article, we study the modified defect relations of the Gauss map of complete minimal surfaces in $\mathbb R^3$ and $ \mathbb R^4$ on annular ends. We obtain results which are similar to the ones obtained by Fujimoto~[J. Differential Geometry \textbf{29} (1989), 245-262] for (the whole) complete minimal surfaces. We thus give some improvements of the previous results for the Gauss maps of complete minimal surfaces restricted on annular ends.