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

Gerd Dethloff (1) , Pham Hoang Ha (1)

1 LMBA - Laboratoire de Mathématiques de Bretagne Atlantique

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.

Mots clés: Minimal surface; Gauss map; Ramification; Value distribution theory

Domaines: Variables complexes [math.CV]; Géométrie différentielle [math.DG]; Géométrie algébrique [math.AG]

Gerd Dethloff, Pham Hoang Ha. Ramification of the Gauss Map of Complete Minimal Surfaces in R^3 and R^4 on Annular Ends. Annales de la Faculté de Sciences de Toulouse, 2014, 23, pp.829-846. ⟨hal-00818081v2⟩