Ramification of the Gauss Map of Complete Minimal Surfaces in R^3 and R^4 on Annular Ends

TL;DR

This paper investigates how the Gauss map of complete minimal surfaces in R^3 and R^4 behaves on annular ends, showing that restrictions on branching are similar to those on entire surfaces, refining previous results.

Contribution

It extends known results about the Gauss map's ramification from entire minimal surfaces to their annular ends, providing sharper bounds and insights.

Findings

Gauss map restrictions on annular ends mirror those on entire surfaces

No additional branching occurs on annular ends compared to the whole surface

Improves previous bounds on the ramification of Gauss maps for minimal surfaces

Abstract

In this article, we study the ramification of the Gauss map of complete minimal surfaces in R^3 and R^4 on annular ends. We obtain results which are similar to the ones obtained by Fujimoto and Ru for (the whole) complete minimal surfaces, thus we show that the restriction of the Gauss map to an annular end of such a complete minimal surface cannot have more branching (and in particular not avoid more values) than on the whole complete minimal surface. We thus give an improvement of the results on annular ends of complete minimal surfaces of Kao.