A Note on Pseudoconvexity and Proper Holomorphic Mappings

In this paper we discuss some connections between proper holomorphic mappings between domains in Cn and the boundary behaviors of certain canonical invariant metrics. A compactness theorem has been proved. This generalizes slightly an earlier result proved by the second author. Introduction. A continuous mapping f:Xx —* X2 between two topological spaces is called proper if f~l(K) c X\ is compact whenever K c X2 is compact. Proper holomorphic mappings between analytic spaces stand out for their beauty and simplicity. For instance, if g: Dx —» D2 is a proper holomorphic mapping between two bounded domains in C", a theorem of Remmert says that (Dx,g,D2) is a finite branching cover. The branching locus in Di is described by {z G D\ \ det(dg(z)) = 0}. For the past ten years, there has been a great amount of activity in characterizing the proper holomorphic mappings between pseudoconvex domains. It has been known for a long time that there are numerous proper holomorphic maps between unit disks in C1. The simplest example is g: A = {z G C1 | \z\ < 1} —♦ A, g(z) = zn, where n is any positive integer. Nevertheless, such a phenomenon is no longer true in higher-dimensional cases. H. Alexander was able to verify the following interesting fact. THEOREM 1 [1]. LetBn = {(zi,z2,...,zn)\Y!i=i\zi\2 2. Suppose f : Bn —* Bn is a proper holomorphic mapping. Then f must be a biholomorphism. The following result due to S. Pincuk is an extension of Alexander's theorem. THEOREM 2 [5]. Let Dx and D2 be two strongly pseudoconvex bounded domains with smooth boundaries in CTM, n > 2. Suppose f : Dx —► D2 is a proper holomorphic mapping. Then f is a covering. In [7] the second author proved the following result concerning biholomorphic groups of strongly pseudoconvex domains. THEOREM 3 [7]. Let D be a strongly pseudoconvex bounded domain with smooth boundary in C". Then Aut(L>) is noncompact iff D is biholomorphic to Bn, n = dime D. Received by the editors June 3, 1986 and, in revised form, December 30, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 53C55; Secondary 32A17.