Ramification of the Gauss map of complete minimal surfaces in ℝ 3 and ℝ 4 on annular ends

Gerd Dethloff; Pham Hoang Ha

Annales de la faculté des sciences de Toulouse Mathématiques (2014)

  • Volume: 23, Issue: 4, page 829-846
  • ISSN: 0240-2963

Abstract

top
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 ([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]).

How to cite

top

Dethloff, Gerd, and 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 23.4 (2014): 829-846. <http://eudml.org/doc/275366>.

@article{Dethloff2014,
abstract = {In this article, we study the ramification 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 ([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]).},
author = {Dethloff, Gerd, Hoang Ha, Pham},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
keywords = {complete minimal surfaces; annular end; Gauss map; ramification},
language = {eng},
number = {4},
pages = {829-846},
publisher = {Université Paul Sabatier, Toulouse},
title = {Ramification of the Gauss map of complete minimal surfaces in $\{\mathbb\{R\}\}^3$ and $\{\mathbb\{R\}\}^4$ on annular ends},
url = {http://eudml.org/doc/275366},
volume = {23},
year = {2014},
}

TY - JOUR
AU - Dethloff, Gerd
AU - Hoang Ha, Pham
TI - Ramification of the Gauss map of complete minimal surfaces in ${\mathbb{R}}^3$ and ${\mathbb{R}}^4$ on annular ends
JO - Annales de la faculté des sciences de Toulouse Mathématiques
PY - 2014
PB - Université Paul Sabatier, Toulouse
VL - 23
IS - 4
SP - 829
EP - 846
AB - In this article, we study the ramification 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 ([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]).
LA - eng
KW - complete minimal surfaces; annular end; Gauss map; ramification
UR - http://eudml.org/doc/275366
ER -

References

top
  1. Ahlfors (L. V.).— An extension of Schwarz’s lemma, Trans. Amer. Math. Soc. 43, p. 359-364 (1938). Zbl64.0315.04MR1501949
  2. Chen (C. C.).— On the image of the generalized Gauss map of a complete minimal surface in ℝ 4 , Pacific J. Math. 102, p. 9-14 (1982). Zbl0498.53047MR682039
  3. Chern (S. S.), Osserman (R.).— Complete minimal surface in euclidean n - space, J. Analyse Math. 19, p. 15-34 (1967). Zbl0172.22802MR226514
  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). Zbl0629.53011MR930599
  5. Fujimoto (H.).— Modified defect relations for the Gauss map of minimal surfaces, J. Differential Geometry 29, p. 245-262 (1989). Zbl0676.53005MR982173
  6. Fujimoto (H.).— Value Distribution Theory of the Gauss map of Minimal Surfaces in ℝ m , Aspect of Math. E21, Vieweg, Wiesbaden (1993). Zbl1107.32004MR1218173
  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). Zbl1107.53042MR2272139
  8. Kao (S. J.).— On values of Gauss maps of complete minimal surfaces on annular ends, Math. Ann. 291, p. 315-318 (1991). Zbl0760.53005MR1129370
  9. Kawakami (Y.).— The Gauss map of pseudo - algebraic minimal surfaces in ℝ 4 , Math. Nachr. 282, p. 211-218 (2009). Zbl1167.53009MR2493511
  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). Zbl0666.53003MR1037404
  11. Osserman (R.).— Global properties of minimal surfaces in E 3 and E n , Ann. of Math. 80, p. 340-364 (1964). Zbl0134.38502MR179701
  12. Osserman (R.), Ru (M.).— An estimate for the Gauss curvature on minimal surfaces in ℝ m whose Gauss map omits a set of hyperplanes, J. Differential Geom. 46, p. 578-593 (1997). Zbl0918.53003MR1484891
  13. Ru (M.).— On the Gauss map of minimal surfaces immersed in ℝ n , J. Differential Geom. 34, p. 411-423 (1991). Zbl0733.53005MR1131437
  14. Ru (M.).— Gauss map of minimal surfaces with ramification, Trans. Amer. Math. Soc. 339, p. 751-764 (1993). Zbl0792.53003MR1191614
  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). Zbl0477.53007MR604048

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.