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Nevanlinna theory for minimal surfaces of parabolic type. (English) Zbl 0859.32013

There are close connections between the theory of minimal surfaces and Nevanlinna theory of meromorphic functions. In the present paper the author extends some results of Beckenbach and collaborators on the first and second main theorem for minimal surfaces. For complete minimal surfaces in \(\mathbb{R}^m\) with finite total curvature the Second Main Theorem is equivalent to the formula of Gauß-Bonnet type \[ C= \chi-\sum^k_{\ell =1} I_\ell, \] which was proved by the referee for surfaces in \(\mathbb{R}^3\) and \(\mathbb{R}^m\) in 1974 and 1975 and by Jorge and Meeks for \(\mathbb{R}^3\) in 1980.

MSC:

32H30 Value distribution theory in higher dimensions
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
32A30 Other generalizations of function theory of one complex variable
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