Several Complex Variables III

We consider the basic problems, notions and facts in the theory of entire functions of several variables, i.e. functions f (z) holomorphic in the entire space ℂ ⁿ (i.e. f∈H(ℂ ⁿ ). Such functions constitute an independent object of study and are often encountered in applications of complex analysis to other branches of mathematics. For instance, entire functions of several variables are widely used in the theory of linear p.d.e. [8], [32], in the theory of convolution equations [31], in the theory of distributions [12], [51], in probability theory [26]. There are also numerous applications of entire functions of several variables in various branches of physics.

The study of value distribution theory may be considered to have its origin in the famous Sokhotskiľ-Weierstrass theorem ¹: the set of values of a nonconstant holomorphic function w=f(z), defined in the entire z-plane, is everywhere dense in the w-plane. In fact, all values are assumed with the exception of at most one (Picard). These results and several subsequent ones, which at first sight looked so isolated, turned out to be part of a rather deep and elegant theory, known as value distribution theory or, after its founder, Nevanlinna theory. In particular it follows from the First Main Theorem of Nevanlinna that a meromorphic map not only takes almost all values but indeed takes them, in some sense, equally often. And if some values are taken too sparsely then this is compensated by the fact that, as z tends to infinity, f(z) can be approximated with these values on large subsets.

The idea to construct metrics invariant for suitable maps goes back to Bernhard Riemann. Among many others ideas which came to determine the subsequent development of the geometric approach to mathematics and its application it was expressed in his famous lecture “Über die Hypothesen, welche der Geometrie zu Grunde liegen” (1854). In a more precise form it is formulated in the so-called Erlanger program of Felix Klein in the latters inaugural lecture “Vergleichende Betrachtungen über neuere geometrische Forschungen” (1872), where invariants of various transformation groups are exhibited.

This Part is devoted to a direction of complex analysis which has its roots in the theorem of Liouville (Liouville (1844) for doubly periodic functions; Cauchy (1844) in the contemporary formulation) and Picard (1879) on the nonexistence of nonconstant holomorphic functions f: ℂ→D = {z∈ ℂ: |z| < 1} and f: ℂ→ℂ\{0, 1}. The first results on the finiteness of sets of holomorphic maps were obtained in the second half of the past century within the framework of Riemann surfaces, which subject then began to take shape. Schwarz (1879) and Poincaré (1885) proved the finiteness of the group Aut R _g of the automorphism of compact Riemann surfaces of genus g > 1. Hurwitz (1893) completed this result by establishing the explicit bound # Aut R _g ≦84(g — 1); we owe to him several other remarkable results on maps of Riemann surfaces (some of these will be set forth in § 2 and § 3 of Chap. 1). De Franchis (1913) and Severi (1926) proved the finiteness of the set Hol_*(R _g1, R _g2) of nonconstant holomorphic maps of compact Riemann surfaces R _g1→R _g2 in the hypothesis g ₂> 1. Moreover, they established that, for a fixed Riemann surface R _g1, the number of all pairs (f, R _g2), where R _g2 is a compact Riemann surface of genus g ₂>1 and f∈Hol_*(R _g1, R _g2), is finite (and admits estimates depending only on g ₁).

This Part is devoted to the problem of holomorphic equivalence. This problem consists of the following: given two domains D, G⊂ℂ ⁿ , to determine whether there exists a biholomorphic map f: D→G, or not.

The geometry of CR-manifolds goes back to Poincaré and received a great attention in the works of É. Cartan, Tanaka, Moser, Chern and others (cf. [44]). In this chapter we consider results connected with the equivalence problem for CR-manifolds in its differential geometric aspect and some applications of this.