Modified defect relations of the Gauss map of a complete minimal surface on annular ends

2020 ◽  
Vol 97 (3-4) ◽  
pp. 265-287
Author(s):  
Pham Hoang Ha ◽  
Nguyen Hoang Trang
Keyword(s):  
Minimal Surface ◽  
Gauss Map ◽  
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].

scholarly journals Local minima of the Gauss curvature of a minimal surface

1991 ◽  
Vol 44 (3) ◽  
pp. 397-404
Author(s):  
Shinji Yamashita
Keyword(s):  
Minimal Surface ◽  
Gauss Curvature ◽  
Gauss Map ◽  
Local Minima ◽  
Jordan Curves ◽  
The Third ◽  
Isolated Points ◽  
Set Of Points

Let D be a domain in the complex ω-plane and let x: D → R3 be a regular minimal surface. Let M(K) be the set of points ω0 ∈ D where the Gauss curvature K attains local minima: K(ω0) ≤ K(ω) for |ω – ω0| < δ(ω0), δ(ω0) < 0. The components of M(K) are of three types: isolated points; simple analytic arcs terminating nowhere in D; analytic Jordan curves in D. Components of the third type are related to the Gauss map.

scholarly journals On the Gauss map of minimal surfaces with finite total curvature

1991 ◽  
Vol 44 (2) ◽  
pp. 225-232 ◽  
Author(s):  
Min Ru
Keyword(s):  
Minimal Surface ◽  
Minimal Surfaces ◽  
General Position ◽  
Total Curvature ◽  
Gauss Map ◽  
Finite Total Curvature

We prove that if a nonflat complete regular minimal surface immersed in Rn is of finite total curvature, then its Gauss map can omit at most (n – 1)(n + 2)/2 hyperplanes in general position in Pn–1 (ℂ).

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