Vol. 28, No. 1, 1969

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Meromorphic minimal surfaces

E. F. Beckenbach and Gerald Andrew Hutchison

Vol. 28 (1969), No. 1, 17–47

Meromorphic minimal surfaces are defined in this paper, and some of their differential-geometric properties are noted. The first fundamental theorem of Nevanlinna for meromorphic functions of a complex variable is extended so as to apply to these surfaces, as is the Ahlfors-Shimizu spherical version of this theorem. For these results, the classical proximity and enumerative functions of complex-variable theory are generalized, and a new visibility function is introduced. Convexity properties of some of these functions are established.

For plane meromorphic maps, the visibility function vanishes at all points on the plane but is positive at all other points of space. In general, in the present development, the sum of the enumerative function and the visibility function corresponds to the enumerative function in the classical theory.

Mathematical Subject Classification
Primary: 53.04
Received: 8 September 1967
Published: 1 January 1969
E. F. Beckenbach
Gerald Andrew Hutchison