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On the number of exceptional values of the Gauss map of minimal surfaces. (English) Zbl 0629.53011

Let M be a (connected, oriented immersed) minimal surface in \(R^ 3\). By definition the Gauss map G of M maps each point p of M to the unit normal vector G(p) of M at p. In 1961, R. Ossermann showed that, if M is non- flat and complete, then the Gauss map of M cannot omit a set of positive logarithmic capacity. Afterwards, F. Xavier proved that the Gauss map of such a surface can omit at most six points of the sphere. In this paper, the author proves that the number six in the result of F. Xavier can be replaced by four, namely, the Gauss map of a complete non-flat minimal surface in \(R^ 3\) can omit at most four points of the sphere.
It is well-known that there are many examples of non-flat complete minimal surfaces in \(R^ 3\) whose Gauss maps omit four points. So, the number four is best-possible. Moreover, he shows that, for a not necessarily complete minimal surface M in \(R^ 3\), if the Gauss map of M omits five points, then there exists a positive constant C depending only on these points such that \(| K(p)| \leq C/d(p)^ 2\) for an arbitrary point p of M, where K(p) and d(p) denote the Gaussian curvature of M at p and the distance from p to the boundary of M respectively. He also studies a complete non-flat minimal surface M in \(R^ 4\). The Gauss map of M may be naturally identified with a pair of meromorphic functions \(g=(g_ 1,g_ 2)\). He proves that either \(g_ 1\) or \(g_ 2\) can omit at most four values if \(g_ 1\neq const. and\) \(g_ 2\neq const.\), and \(g_ 1\) can omit at most three values if \(g_ 2=const.\).

MSC:

53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
32H30 Value distribution theory in higher dimensions
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