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A Note on a Unicity Theorem for the Gauss Maps of Complete Minimal Surfaces in Euclidean Four-space

Published online by Cambridge University Press:  20 November 2018

Pham Hoang Ha
Affiliation:
Department of Mathematics, Hanoi National University of Education, 136, XuanThuy str., Hanoi, Vietnam, e-mail: ha.ph@hnue.edu.vn
Yu Kawakami
Affiliation:
Faculty of Mathematics and Physics, Institute of Science and Engineering, Kanazawa University, Kanazawa, 920-1192, Japan, e-mail: y-kwkami@se.kanazawa-u.ac.jp
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Abstract

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The classical result of Nevanlinna states that two nonconstantmeromorphic functions on the complex plane having the same images for five distinct values must be identically equal to each other. In this paper, we give a similar uniqueness theorem for the Gauss maps of complete minimal surfaces in Euclidean four-space.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2018

References

[1] Aiyama, R., Akutagawa, K., Imagawa, S. and Kawakami, Y., Remarks on the Gauss Images of complete minimal surfaces in Euclidean four-Space. Annali di Matematica (2017), to appear.http://dx.doi.org/10.1007/s10231-017-0643-6 Google Scholar
[2] Chern, S. S., Minimal surfaces in an Euclidean Space of N dimensions. In: Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse), Princeton Univ. Press, Princeton, NJ, 1965, pp. 187198.Google Scholar
[3] Fujimoto, H., On the number of exceptional values ofthe Gauss maps of minimal surfaces. J. Math. Soc. Japan 40(1988), no. 2, 235247. http://dx.doi.org/10.2 969/jmsj704020235 Google Scholar
[4] Fujimoto, H., Value distribution theory of the Gauss map of minimal surfaces in Rm. Aspects of Mathematics, E21, Friedr. Vieweg & Sohn, Braunschweig/Wiesbaden, 1993.http://dx.doi.org/10.1007/978-3-322-80271-2 Google Scholar
[5] Fujimoto, H., Unicity theorems for the Gauss maps of complete minimal surfaces. J. Math. Soc. Japan 45(1993), no. 3, 481487.http://dx.doi.Org/10.2969/jmsj704530481 Google Scholar
[6] Fujimoto, H., Unicity theorems for the Gauss maps of complete minimal surfaces. II. Kodai Math. J. 16(1993), no. 3, 335354.http://dx.doi.org/10.2996/kmj71138039844 Google Scholar
[7] Dethloff, G. and Ha, P. H., Ramification ofthe Gauss map of complete minimal surfaces in ℝ3 and ℝ4 on annular ends. Ann. Fac. Sei. Toulouse Math. 23(2014), 829846.http://dx.doi.org/10.5802/afst.1426 Google Scholar
[8] Hoffman, D. A. and Osserman, R., The geometry of the generalized Gauss map. Mem. Amer. Math. Soc. 28(1980), no. 236.http://dx.doi.Org/1 0.1090/memo/0236 Google Scholar
[9] Hoffman, D. A. and Osserman, R., The Gauss map of surfaces in R3 and R4. Proc. London Math. Soc. (3) 50(1985), no. 1, 2756.http://dx.doi.Org/10.1112/plms/s3-50.1.27 Google Scholar
[10] Kao, S. J., On values of Gauss maps of complete minimal surfaces on annular ends. Math. Ann. 291(1991), no. 2, 315318.http://dx.doi.org/10.1007/BF01445210 Google Scholar
[11] Kawakami, Y., On the maximal number of exceptional values of Gauss maps for various classes of surfaces. Math. Z. 274(2013), no. 3-4, 12491260. http://dx.doi.org/10.1007/s00209-012-1115-8 Google Scholar
[12] Kawakami, Y., Function-theoretic properties for the Gauss maps for various classes of surfaces. Canad. J. Math. 67(2015), no. 6, 14111434. http://dx.doi.Org/10.41 53/CJM-2O1 5-008-5 Google Scholar
[13] Mo, X. and Osserman, R., On the Gauss map and total curvature of complete minimal surfaces and an extension of Fujimoto's theorem. J. Differential Geom. 31(1990), no. 2, 343355.Google Scholar
[14] Nevanlinna, R., Einige Eindeutigkeitssätze in der Theorie der Meromorphen Funktionen. Acta Math. 48(1926), no 3-4, 367391.http://dx.doi.org/10.1007/BF02565342 Google Scholar
[15] Osserman, R., Global properties of minimal surfaces in E3 and En. Ann. of Math. (2) 80(1964),340364. http://dx.doi.org/10.2307/1970396 Google Scholar
[16] Park, J. and Ru, M., Unicity results for Gauss maps of minimal surfaces immersed in ℝm. J. Geom. to appear. http://dx.doi.org/10.1007/s00022-016-0353-z Google Scholar