Lowener Theory on Analytic Universal Covering Maps (1907.11987v1)

Abstract: Let $\mathcal{H}0(\mathbb{D})$ be the class of all analytic functions $f$ in the unit disc $\mathbb{D} = {z \in \mathbb{C} : |z| < 1 }$ with $f(0)=0 \text{ and } f_t'(0) > 0$. Let $\mathfrak{B} = { \omega \in \mathcal{H}_0(\mathbb{D}) : \omega(\mathbb{D}) \subset \mathbb{D}}$. We say that a one parameter family of analytic functions ${ f_t }{t \in I}$ in $\mathcal{H}0(\mathbb{D})$ on an interval $I \subset [-\infty, \infty]$, is a Loewner chain if $f_s$ is subordinate to $f_t$ whenever $s,t \in I$ with $s<t$, i.e., there exists $\omega{s,t} \in \mathfrak{B}$ with $f_s = f_t \circ \omega_{s,t}$. Notice that we omit the univalence assumption on each $f_t$. We shall show that if $f_t'(0)$ is continuous and strictly increasing in $t$, then $f(z,t) := f_t(z)$ satisfies a partial differential equation which is a generalization of Loewner-Kufarev equation, and ${ f_t }{t \in I}$ can be expressed as $f_t = F \circ g_t $, $t \in I$, where $F$ is an analytic function on a disc $\mathbb{D}(0,r) = { z \in \mathbb{C} : |z| < r }$ with $r = \lim{t \uparrow \sup I}f_t'(0) \in (0, \infty ]$ and $F(0)= F'(0)-1=0$, and ${ g_t }{t \in I}$ is a Loewner chain consists of univalent functions. In the second half we deals with Loewner chains ${ f_t }{t \in I}$ consists of universal covering maps which may be the most geometrically natural generalization of Loewner chains of univalent functions. For each $t \in I$ let $C(f_t(\mathbb{D}))$ be the connectivity of image domain of $f_t(\mathbb{D})$. We shall show that if ${ f_t }_{t\in I}$ is continuous, then the function $C(f_t(\mathbb{D}))$ is nondecreasing and left continuous. Then we develop a Loewner theory on Fuchsian groups.