Ramification of the Gauss map of complete minimal surfaces in 3 and 4 on annular ends
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 23 (2014) no. 4, pp. 829-846.

Dans ce travail nous obtenons des théorèmes de ramification de l’application de Gauss de certaines classes de surfaces minimales complètes dans 3 et 4 .

In this article, we study the ramification of the Gauss map of complete minimal surfaces in 3 and 4 on annular ends. We obtain results which are similar to the ones obtained by Fujimoto ([4], [5]) and Ru ([13], [14]) 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 ([8]).

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     author = {Dethloff, Gerd and Hoang Ha, Pham},
     title = {Ramification of the {Gauss} map of complete minimal surfaces in ${\mathbb{R}}^3$ and ${\mathbb{R}}^4$ on annular ends},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {829--846},
     publisher = {Universit\'e Paul Sabatier, Institut de math\'ematiques},
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     volume = {Ser. 6, 23},
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Dethloff, Gerd; Hoang Ha, Pham. Ramification of the Gauss map of complete minimal surfaces in ${\mathbb{R}}^3$ and ${\mathbb{R}}^4$ on annular ends. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 23 (2014) no. 4, pp. 829-846. doi : 10.5802/afst.1426. http://archive.numdam.org/articles/10.5802/afst.1426/

[1] Ahlfors (L. V.).— An extension of Schwarz’s lemma, Trans. Amer. Math. Soc. 43, p. 359-364 (1938). | MR 1501949

[2] Chen (C. C.).— On the image of the generalized Gauss map of a complete minimal surface in 4 , Pacific J. Math. 102, p. 9-14 (1982). | MR 682039 | Zbl 0498.53047

[3] Chern (S. S.), Osserman (R.).— Complete minimal surface in euclidean n - space, J. Analyse Math. 19, p. 15-34 (1967). | MR 226514 | Zbl 0172.22802

[4] Fujimoto (H.).— On the number of exceptional values of the Gauss maps of minimal surfaces, J. Math. Soc. Japan 40, p. 235-247 (1988). | MR 930599 | Zbl 0629.53011

[5] Fujimoto (H.).— Modified defect relations for the Gauss map of minimal surfaces, J. Differential Geometry 29, p. 245-262 (1989). | MR 982173 | Zbl 0676.53005

[6] Fujimoto (H.).— Value Distribution Theory of the Gauss map of Minimal Surfaces in m , Aspect of Math. E21, Vieweg, Wiesbaden (1993). | MR 1218173 | Zbl 1107.32004

[7] Jin (L.), Ru (M.).— Values of Gauss maps of complete minimal surfaces in R m on annular ends, Trans. Amer. Math. Soc. 359, p. 1547-1553 (2007). | MR 2272139 | Zbl 1107.53042

[8] Kao (S. J.).— On values of Gauss maps of complete minimal surfaces on annular ends, Math. Ann. 291, p. 315-318 (1991). | MR 1129370 | Zbl 0760.53005

[9] Kawakami (Y.).— The Gauss map of pseudo - algebraic minimal surfaces in 4 , Math. Nachr. 282, p. 211-218 (2009). | MR 2493511 | Zbl 1167.53009

[10] Mo (X.), Osserman (R.).— On the Gauss map and total curvature of complete minimal surfaces and an extension of Fujimoto’s theorem, J. Differential Geom. 31, p. 343-355 (1990). | MR 1037404 | Zbl 0666.53003

[11] Osserman (R.).— Global properties of minimal surfaces in E 3 and E n , Ann. of Math. 80, p. 340-364 (1964). | MR 179701 | Zbl 0134.38502

[12] Osserman (R.), Ru (M.).— An estimate for the Gauss curvature on minimal surfaces in m whose Gauss map omits a set of hyperplanes, J. Differential Geom. 46, p. 578-593 (1997). | MR 1484891 | Zbl 0918.53003

[13] Ru (M.).— On the Gauss map of minimal surfaces immersed in n , J. Differential Geom. 34, p. 411-423 (1991). | MR 1131437 | Zbl 0733.53005

[14] Ru (M.).— Gauss map of minimal surfaces with ramification, Trans. Amer. Math. Soc. 339, p. 751-764 (1993). | MR 1191614 | Zbl 0792.53003

[15] Xavier (F.).— The Gauss map of a complete non-flat minimal surface cannot omit 7 points of the sphere, Ann. of Math. 113, p. 211-214 (1981). | MR 604048 | Zbl 0477.53007

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