Curvature, Rescaling and Uniformization Problems on Bounded Symmetric Domains

Abstract

The asymptotic behavior of invariant metrics on bounded domains has been an important tool in several complex variables especially in characterization theorems. A first instance of such a re-sult is the work of B. Wong (1977) showing that a strictly pseudoconvex domain in the Euclideanspace is biholomorphic to the complex unit ball Bn whenever it admits a noncompact group of automorphisms, a result which was obtained by exploiting the asymptotic curvature behavior of the Bergman metric and by the method of rescaling. Here by rescaling we will mean the process of composing with a divergent sequence of automorphisms and extracting a convergent subsequence. We will illustrate the method of rescaling by studying special classes of algebraic subsets of bounded symmetric domains Ω in their Harish-Chandra realizations and explain how the method leads to solutions of problems on uniformization and functional transcendence on bounded symmetric domains, including (a) a differential-geometric proof of the hyperbolic Ax-Lindemann Theorem for the rank-1 case and (b) a solution of the characterization of totally geodesic subsets of XΓ= Ω/Γ as the unique bi-algebraic subvarieties without using monodromyresults of Andr ́e-Deligne. These solutions are obtained by studying the asymptotic geometry of a subvariety as it exits ∂Ω and by the method of rescaling, and they apply without assuming arithmeticity of the lattices.