Applications of Bergman Geometry

Abstract

In this chapter, results will be presented that arise by combining geometric arguments with the asymptotic curvature constancy at the boundary (discussed in the previous chapter) and other aspects of the geometry of the Bergman metric. The completeness of the Bergman metric of strongly pseudoconvex domains (Theorem 3.4.2) fits the whole situation into the framework of global Riemannian geometry, the basic idea of which is that the global geometry of a complete Riemannian manifold is controlled by curvature. Without completeness, this property fails entirely (cf. [Gromov 1969]). But, with completeness in hand, one expects curvature information to control the geometry in many respects.