Modified defect relations of the Gauss map and the total curvature of a complete minimal surface. (English) Zbl 1380.53015
Summary: In this article, we propose some conditions on the modified defect relations of the Gauss map of a complete minimal surface \(M\) to show that \(M\) has finite total curvature.
MSC:
53A10 | Minimal surfaces in differential geometry, surfaces with prescribed mean curvature |
53C42 | Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) |
30D35 | Value distribution of meromorphic functions of one complex variable, Nevanlinna theory |
32H30 | Value distribution theory in higher dimensions |
Keywords:
minimal surface; Gauss map; modified defect relation; value distribution theory; total curvatureReferences:
[1] | Ahlfors, L. V., An extension of Schwarz’s lemma, Trans. Am. Math. Soc., 43, 359-364 (1938) · JFM 64.0315.04 |
[2] | Fujimoto, H., Value distribution of the Gauss maps of complete minimal surfaces in \(R^m\), J. Math. Soc. Jpn., 35, 4, 663-681 (1983) · Zbl 0527.53004 |
[3] | Fujimoto, H., On the number of exceptional values of the Gauss maps of minimal surfaces, J. Math. Soc. Jpn., 40, 235-247 (1988) · Zbl 0629.53011 |
[4] | Fujimoto, H., Modified defect relations for the Gauss map of minimal surfaces, J. Differ. Geom., 29, 245-262 (1989) · Zbl 0676.53005 |
[5] | Fujimoto, H., Modified defect relations for the Gauss map of minimal surfaces II, J. Differ. Geom., 31, 365-385 (1990) · Zbl 0719.53005 |
[6] | Fujimoto, H., Value Distribution Theory of the Gauss map of Minimal Surfaces in \(R^m\), Aspect of Math., vol. E21 (1993), Vieweg: Vieweg Wiesbaden · Zbl 1107.32004 |
[7] | Ha, P. H., Non-integrated defect relations for the Gauss map of a complete minimal surface with finite total curvature in \(R^m\), Bull. Math. Soc. Sci. Math. Roum. (2017), in press |
[8] | Ha, P. H.; Phuong, L. B.; Thoan, P. D., Ramification of the Gauss map and the total curvature of a complete minimal surface, Topol. Appl., 199, 32-48 (2016) · Zbl 1332.53014 |
[9] | Ha, P. H.; Trao, N. V., Non-integrated defect relations for the Gauss map of complete minimal surfaces with finite total curvature, J. Math. Anal. Appl., 430, 76-84 (2015) · Zbl 1318.53009 |
[10] | Huber, A., On subhamornic functions and differential geometry in large, Comment. Math. Helv., 32, 13-72 (1961) · Zbl 0080.15001 |
[11] | Jin, L.; Ru, M., Algebraic curves and the Gauss map of algebraic minimal surfaces, Differ. Geom. Appl., 25, 701-712 (2007) · Zbl 1149.53008 |
[12] | Fang, Y., On the Gauss map of complete minimal surfaces with finite total curvature, Indiana Univ. Math. J., 42, 1389-1411 (1993) · Zbl 0794.53006 |
[13] | Kawakami, Y.; Kobayashi, R.; Miyaoka, R., The Gauss map of pseudo-algebraic minimal surfaces, Forum Math., 20, 1055-1069 (2008) · Zbl 1167.53010 |
[14] | Mo, X., Value distribution of the Gauss map and the total curvature of complete minimal surface in \(R^m\), Pac. J. Math., 163, 159-174 (1994) · Zbl 0801.53007 |
[15] | Mo, X.; Osserman, R., On the Gauss map and total curvature of complete minimal surfaces and an extension of Fujimoto’s theorem, J. Differ. Geom., 31, 343-355 (1990) · Zbl 0666.53003 |
[16] | Osserman, R., On complete minimal surfaces, Arch. Ration. Mech. Anal., 13, 392-404 (1963) · Zbl 0127.38003 |
[17] | Osserman, R., Global properties of minimal surfaces in \(E^3\) and \(E^n\), Ann. Math., 80, 340-364 (1964) · Zbl 0134.38502 |
[18] | Osserman, R., A Survey of Minimal Surfaces (1986), Dover: Dover New York · Zbl 0209.52901 |
[19] | Ru, M., On the Gauss map of minimal surfaces with finite total curvature, Bull. Aust. Math. Soc., 44, 225-232 (1991) · Zbl 0725.53010 |
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