Squeezing function, Fridman function and proper maps on multiply connected domains

Abstract

Let $X$ be a bounded domain in $\mathbb{C}^n$. To study the geometric properties of $X$, one may introduce on $X$ the \textit{squeezing function} $S_X(z)$ and the \textit{Fridman function} $H^d_X(z)$ (where $d=K$ or $C$ refers to the Kobayashi metric or Carath\'{e}dory metric respectively). These two functions are biholomorphic invariants of $X$, i.e., $S_{g(X)}(g(z))=S_X(z)$ and for $H ^d _{ g(X) } ( g(z) )=H ^d _X (z)$ any biholomorphism $g$. They are useful for estimating some of intrinsic metrics of $X$ as well as proving pseudoconvexity of $X$. Some estimates of these functions near the boundary of $X$ are known. However, very few explicit $S_X(z)$ and $H^d_X(z)$ are known. In Chapter 3, we will prove the explicit form for the squeezing function $S_X (z) $ when $X=A_r$ is an annulus and hence determine the explicit form for the squeezing function of all non-degenerated bounded doubly connected planar domains up to biholomorphisms. This serves as the first non-trivial formula of squeezing functions for planar domains except $\mathbb{D}$ and $\mathbb{D} \setminus \{ 0 \}$ as well as their biholomorphic analogue. This also answers a question asked by E.F. Wold in 2017 about the precise form for $S_{A_r}(z)$. The main tools applied in Chapter 3 are explicit formulae of conformal maps from annuli to circularly slit disk in terms of the Schottky-Klein prime functions (introduced in Chapter 2) and slit map approximations using Loewner-type differential equation introduced in Chapter 3. This is a joint work of the candidate with Prof. Tuen-Wai Ng and Dr. Jonathan Tsai.

Squeezing functions were introduced by F. Deng, Q. Guan and L. Zhang in 2012, and it is currently an active research area in several complex variables and complex geometry. On the other hand, the Fridman function has only been recently identified as a dual of the squeezing function by Mahajan and Verma in 2019. In line with the explicit form of the squeezing function we found in Chapter 3, we will also explore in Chapter 4 the explicit form for the Fridman function $H ^d_X(z)$ on some complex manifold $X$. We will prove that the Fridman function $H ^d_X(z)$ over a class of $d$-hyperbolic complex manifolds is bounded above by the injectivity radius function $\iota^d_X (z)$ (introduced in Chapter 4) for $d=K$. This result also suggests us to use the Fridman function to extend the definition of uniform thickness to higher-dimensional $d$-hyperbolic complex manifolds. We will also find an explicit formula for $H^K_X(z)$ when $X= D \setminus \Gamma$ where $D= \mathbb{D}$ or equivalently $\mathbb{H}$ and $\Gamma$ is a torsion-free discrete group of isometries of $D$. In particular, explicit formulae for $H^K_{A_r} (z)$ and $H^C_{A_r} (z)$ will be given in Chapter \ref{Ch:fri}, for comparison with the explicit form for $S_{A_r}(z)$.

Our proofs in Chapter 3 and 4 involve finding the extremal functions for some optimization problems. Often, the extremal functions of the optimization problems considered in function theory are proper holomorphic maps. This simulates us to study formulae of proper holomorphic maps from multiply connected circular domains in $\mathbb{C}$ onto the standard unit disk $\mathbb{D}$. The explicit form of proper holomorphic maps from annulus onto $\mathbb{D}$ has only been recently obtained by Wang and Yin in 2017, and independently, by Bogatyrev in 2019. In Chapter 5, we will establish the explicit form of proper holomorphic maps from the multiply connected circular domain onto the unit disk using again the Schottky-Klein prime functions. This answers a question of Schmieder in 2005, which is about constructing proper holomorhpic mappings of degree $n$ from multiply connected domains to the unit disk $\mathbb{D}$.