On complex geometry and function theory of complex unit balls and their finite volume quotients

Abstract

The complex unit ball B^n is an extensively studied object in geometry and complex analysis. The focus of this thesis is placed on two aspects of the topic related to respectively the complex geometry and the function theory of the complex unit ball.The first topic is on the boundary behavior of holomorphic functions on bounded symmetric domains of rank \geq 2. For any bounded symmetric domain of rank \geq 2, using the theory of holomorphic isometries and minimal rational curves, a Cayley projection is constructed from \Omega to a generic rank r-1 boundary component \Phi. It turns out that the Cayley projection yields a fibration from \Omega to \Phi where each fibre is holomorphically isometric to a complex unit ball. Such a construction allows us to deduce the almost everywhere existence of boundary value of bounded holomorphicfunctions with respect to any family of Cayley projections, and the construction is applied to complete the sketch of proof of a rigidity result, which is stated in [29] about finite volume quotients of bounded symmetric domains of rank \geq 2.The second topic is on the complex geometry of finite volume quotients of complex unit balls. For the Mumford compactification \overline{X} of the noncompact quotient X=B^n/\Gamma of finite volume, holomorphic sections of symmetric powers of the cotangentbundle over X vanishing at infinity are constructed. This is done by methods of L^2-estimates of the Cauchy-Riemann operator. It is well-known that such sections do not always exist. An existence result on such sections is proved under a hypothesis on canonical radii of cusps at infinity with respect to the Satake-Baily-Borel compactification. Combined with a natural requirement on injectivity radius, it has been proven that the cotangent bundle of the Mumford compactification is ample modulo the divisor at infinity.