Higher dimensional analogue of Ahlfors covering surface theory

Question

It is well known that Ahlfors covering surface theory in one dimensional is very powerful in dealing with many problems. I wonder whether there exists some generalization of this theory into higher dimension.

Answer 1

Yes, there exists such a generalization. It was done by Marie-Helene Schwartz.

Formules apparentées à la formule de Gauss-Bonnet pour certaines applications d’une variété à n dimensions dans une autre. (French) Zbl 0057.38102 Acta Math. 91, 189-244 (1954).

Formules apparentées à celles de Nevanlinna-Ahlfors pour certaines applications d’une variété à n dimensions dans une autre. (French) Zbl 0057.31602 Bull. Soc. Math. Fr. 82, 317-360 (1954).

However, on my opinion, this generalization was much less successful than the original Ahlfors theory.

On the other hand, Ahlfors's one-dimensional theory has several interesting APPLICATIONS to several complex variables, the key word is "Ahlfors currents", see, for example,

Henry de Thélin, Ahlfors' currents in higher dimension. Ann. Fac. Sci. Toulouse Math. (6) 19 (2010), no. 1, 121–133.

Remark. In general, there is (at least) two very different generalizations of classical complex analysis to several dimensions. 1. Several complex variables, and 2. Quasiregular maps between real n-dimensional manifolds. The geometric part of the theory (where the main notion is conformity) better generalizes in the setting 2 (for analytic functions of SCV, conformality is meaningless). The work of M-H Schwartz belongs to the second direction.