A Nonorientable Complete Minimal Surface in R 3 Between Two Parallel Planes

1988 ◽  
Vol 103 (3) ◽  
pp. 913 ◽  
Author(s):  
Francisco J. Lopez
Keyword(s):  
Minimal Surface ◽  
Complete Minimal Surface

scholarly journals Modified defect relations for the gauss map of minimal surfaces, III

1991 ◽  
Vol 124 ◽  
pp. 13-40 ◽  
Author(s):  
Hirotaka Fujimoto
Keyword(s):  
Minimal Surface ◽  
Complex Projective Space ◽  
Meromorphic Functions ◽  
Total Curvature ◽  
Gauss Map ◽  
Holomorphic Maps ◽  
Complete Minimal Surface ◽  
Defect Relation ◽  
Definition Of ◽  
Complex Projective

In [5], the author proved that the Gauss map of a nonflat complete minimal surface immersed in R3 can omit at most four points of the sphere, and in [7] he revealed some relations between this result and the defect relation in Nevanlinna theory on value distribution of meromorphic functions. Afterwards, Mo and Osserman obtained an improvement of these results in their paper [11], which asserts that if the Gauss map of a nonflat complete minimal surface M immersed in R3 takes on five distinct values only a finite number of times, then M has finite total curvature. The author also gave modified defect relations for holomorphic maps of a Riemann surface with a complete conformai metric into the n-dimensional complex projective space Pn(C) and, as its application, he showed that, if the (generalized) Gauss map G of a complete minimal surface M immersed in Rm is nondegenerate, namely, the image G(M) is not contained in any hyperplane in Pm − 1(C), then it can omit at most m(m + 1)/2 hyperplanes in general position ([8]). Here, the number m(m + 1)/2 is best-possible for arbitrary odd numbers and some small even numbers m (see [6]). Recently, Ru showed that the “nondegenerate” assumption of the above result can be dropped ([13]). In this paper, we shall introduce a new definition of modified defect and prove a refined Modified defect relation. As its application, we shall give some improvements of the above-mentioned results in [5], [7], [8], [11] and [13].

A Complete Minimal Surface in R 3 Between two Parallel Planes

1980 ◽  
Vol 112 (1) ◽  
pp. 203 ◽  
Author(s):  
Luquesio P. de M. Jorge ◽  
Frederico Xavier
Keyword(s):  
Minimal Surface ◽  
Complete Minimal Surface

scholarly journals A Jordan Curve Spanned by a Complete Minimal Surface

2006 ◽  
Vol 184 (2) ◽  
pp. 285-301 ◽  
Author(s):  
Francisco Martín ◽  
Nikolai Nadirashvili
Keyword(s):  
Minimal Surface ◽  
Jordan Curve ◽  
Complete Minimal Surface
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