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.
