Some biholomorphic invariants of bounded domains

Abstract

To study the geometric properties of bounded domains in Cn, Deng, Guan and Zhang introduced the notion of squeezing function in 2012. The theory of squeezing function has been studied intensively in the recent years. Many results about it focus on relationship between boundary behaviors of squeezing function and geometry of the domain. However, few result concerning the explicit formula of squeezing function has been derived. Therefore, the list of domains of which the squeezing function is well understood is quite short. In chapter 3, we focus on estimating the value of squeezing function on some special domains in high dimensional spaces, especially the symmetrized bidisc, which is the first known domain that the Kobayashi metric and Caratheodory metric coincide but cannot be exhausted by domains that are biholomorphic to convex domains. Furthermore, we also derive some estimations for domains that are constructed by taking away a subdomain of some classical domains. In chapter 4, we focus on the relationship between the automorphism group of a domain and the squeezing function associated with it. A general belief is that the automorphism group can determine the domain to some extent under biholomorphism. Furthermore, one may observe that the squeezing function has similar behaviors as the automorphism group in some cases. Therefore, it is reasonable to conjecture that the homogeneity of a domain would force the squeezing function to become a constant over the whole domain. To explore this, we make use of a deep result concerning the classification of domains with certain boundary conditions. Since these domains are well understood, one derives that they cannot have constant squeezing function unless they are the unit ball. In chapter 5, we mainly explore some results about the random iteration in high dimensional space. Given a space Ω and a family F of maps taking Ω into itself. One may consider the map f1 ◦ · · · ◦ fn : Ω → Ω, where fi ∈ F, i = 1, . . . , n. If fis are chosen from certain family following some probability distributions, then we call f1 ◦ · · · ◦ fn a random iteration. In 2004, Beardon, Minda and Ng studied the problem of convergence of random iteration on planar domains. They introduced the notion of Bloch subdomains and Lipschitz subdomains and concluded that the random iteration would converge locally uniformly in Ω if the images of the fis lie inside a Lipschitz subdomain of Ω. In chapter 5, we will try to generalize the results in dimension 1 to higher dimensional spaces. To start with, we first introduce the notions of Bloch radius and hyperbolic Lipschitz constant in high dimensional space. Then we showed that the Lipschitz subdomain and Bloch subdomain of polydisc Dn are still same. The proof of the result also explains the existence of a counter example which is Bloch but not Lipschitz.