Gauss map of minimal surfaces with ramification

Ru, Min

doi:10.1090/S0002-9947-1993-1191614-3

Gauss map of minimal surfaces with ramification
HTML articles powered by AMS MathViewer

by Min Ru

Trans. Amer. Math. Soc. 339 (1993), 751-764

DOI: https://doi.org/10.1090/S0002-9947-1993-1191614-3

PDF | Request permission

Abstract:

We prove that for any complete minimal surface $M$ immersed in ${R^n}$, if in $C{P^{n - 1}}$ there are $q > n(n + 1)/2$ hyperplanes ${H_j}$ in general position such that the Gauss map of $M$ is ramified over ${H_j}$ with multiplicity at least ${e_j}$ for each $j$ and \[ \sum \limits _{j = 1}^q {\left ({1 - \frac {{(n - 1)}} {{{e_j}}}} \right ) > n(n + 1)/2} \] , then $M$ must be flat.

References

Lars V. Ahlfors, Conformal invariants: topics in geometric function theory, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. MR 0357743
Wan Xi Chen, Defect relations for degenerate meromorphic maps, Trans. Amer. Math. Soc. 319 (1990), no. 2, 499–515. MR 1010882, DOI 10.1090/S0002-9947-1990-1010882-9
Shiing-shen Chern and Robert Osserman, Complete minimal surfaces in euclidean $n$-space, J. Analyse Math. 19 (1967), 15–34. MR 226514, DOI 10.1007/BF02788707

The Kobayashi metric on

hyperplanes

Michael Cowen and Phillip Griffiths, Holomorphic curves and metrics of negative curvature, J. Analyse Math. 29 (1976), 93–153. MR 508156, DOI 10.1007/BF02789977
Hirotaka Fujimoto, On the Gauss map of a complete minimal surface in $\textbf {R}^{m}$, J. Math. Soc. Japan 35 (1983), no. 2, 279–288. MR 692327, DOI 10.2969/jmsj/03520279
Hirotaka Fujimoto, On the number of exceptional values of the Gauss maps of minimal surfaces, J. Math. Soc. Japan 40 (1988), no. 2, 235–247. MR 930599, DOI 10.2969/jmsj/04020235
Hirotaka Fujimoto, Modified defect relations for the Gauss map of minimal surfaces. II, J. Differential Geom. 31 (1990), no. 2, 365–385. MR 1037406
Xiaokang Mo and Robert Osserman, 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, 343–355. MR 1037404
E. I. Nočka, Uniqueness theorems for rational functions on algebraic varieties, Bul. Akad. Štiince RSS Moldoven. 3 (1979), 27–31, 93 (Russian). MR 567787

On the theory of meromorphic functions

27

Robert Osserman, A survey of minimal surfaces, 2nd ed., Dover Publications, Inc., New York, 1986. MR 852409
Min Ru, On the Gauss map of minimal surfaces immersed in $\textbf {R}^n$, J. Differential Geom. 34 (1991), no. 2, 411–423. MR 1131437
Min Ru, On the Gauss map of minimal surfaces with finite total curvature, Bull. Austral. Math. Soc. 44 (1991), no. 2, 225–232. MR 1126360, DOI 10.1017/S0004972700029658
Min Ru and Pit-Mann Wong, Integral points of $\textbf {P}^n-\{2n+1\;\textrm {hyperplanes\;in\;general\;position}\}$, Invent. Math. 106 (1991), no. 1, 195–216. MR 1123379, DOI 10.1007/BF01243910
Fumio Sakai, Degeneracy of holomorphic maps with ramification, Invent. Math. 26 (1974), 213–229. MR 355118, DOI 10.1007/BF01418950
B. V. Shabat, Distribution of values of holomorphic mappings, Translations of Mathematical Monographs, vol. 61, American Mathematical Society, Providence, RI, 1985. Translated from the Russian by J. R. King; Translation edited by Lev J. Leifman. MR 807367, DOI 10.1090/mmono/061

Defect relations for maps on parabolic spaces and Kobayashi metric on projective spaces omitting hyperplanes

Frederico Xavier, The Gauss map of a complete nonflat minimal surface cannot omit $7$ points of the sphere, Ann. of Math. (2) 113 (1981), no. 1, 211–214. MR 604048, DOI 10.2307/1971139