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

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.