Return to Question

Let $\mathbb D_r=\{z\in\mathbb C:|z|\le r\}$ be the closed unit disk of radius $r$,
let $\mathring {\mathbb D}_r=\{z\in\mathbb C:|z|< r\}$ be its interior,
and let $\mathbb A_r=\mathbb D_r\setminus \mathring{\mathbb D}_1$.

Let $D\subset \mathbb D_1$ be a closed simply connected subset with smooth boundary,
and let let $A_r:=\mathbb D_r \setminus \mathring D$.

I care about the case $\partial D\cap \partial \mathbb D_1\neq\emptyset$.

By the Riemann uniformization theorem, for every $r>1$, there exists an $r'>1$, and a diffeomorphism $$ f_r:A_r \to \mathbb A_{r'} $$ which is holomorphic in the interior.

Question: What can be said about $f_r(\partial \mathbb D_1)$ as $r\to 1$?
Does it converge to $\partial \mathbb D_1$ as a smooth manifold?

In particular, is the domain enclosed by $f_r(\partial \mathbb D_1)$ convex for $r$ sufficiently close to $1$?

Let $\mathbb D_r=\{z\in\mathbb C:|z|\le r\}$ be the closed unit disk of radius $r$,
let $\mathring {\mathbb D}_r=\{z\in\mathbb C:|z|< r\}$ be its interior,
and let $\mathbb A_r=\mathbb D_r\setminus \mathring{\mathbb D}_1$.

Let $D\subset \mathbb D_1$ be a closed simply connected subset with smooth boundary,
and let let $A_r:=\mathbb D_r \setminus \mathring D$, for $r>1$.

I care about the case $\partial D\cap \partial \mathbb D_1\neq\emptyset$.

By the Riemann uniformization theorem, for every $r>1$, there exists an $r'>1$, and a diffeomorphism $$ f_r:A_r \to \mathbb A_{r'} $$ which is holomorphic in the interior.

Question: What can be said about $f_r(\partial \mathbb D_1)$ as $r\to 1$?
Does it converge to $\partial \mathbb D_1$ as a smooth manifold?

In particular, is the domain enclosed by $f_r(\partial \mathbb D_1)$ convex for $r$ sufficiently close to $1$?

Let $\mathbb D_r=\{z\in\mathbb C:|z|\le r\}$ be the closed unit disk of radius $r$,
let $\mathring {\mathbb D}_r=\{z\in\mathbb C:|z|< r\}$ be its interior,
and let $\mathbb A_r=\mathbb D_r\setminus \mathring{\mathbb D}_1$.

Let $D\subset \mathbb D_1$ be closed, simply connected, and with smooth boundary,
and let let $A_r:=\mathbb D_r \setminus \mathring D$.

Assume furthermore that $\partial D\cap \partial \mathbb D_1\neq\emptyset$.

By the Riemann uniformization theorem, for every $r>1$, there exists an $r'>1$, and a diffeomorphism $$ f_r:A_r \to \mathbb A_{r'} $$ which is holomorphic in the interior.

Question: What can be said about $f_r(\partial \mathbb D_1)$ as $r\to 1$?
Does it converge to $\partial \mathbb D_1$ as a smooth manifold?

In particular, is the domain enclosed by $f_r(\partial \mathbb D_1)$ convex for $r$ sufficiently close to $1$?

Let $\mathbb D_r=\{z\in\mathbb C:|z|\le r\}$ be the closed unit disk of radius $r$,
let $\mathring {\mathbb D}_r=\{z\in\mathbb C:|z|< r\}$ be its interior,
and let $\mathbb A_r=\mathbb D_r\setminus \mathring{\mathbb D}_1$.

Let $D\subset \mathbb D_1$ be a closed simply connected subset with smooth boundary,
and let let $A_r:=\mathbb D_r \setminus \mathring D$.

I care about the case $\partial D\cap \partial \mathbb D_1\neq\emptyset$.

By the Riemann uniformization theorem, for every $r>1$, there exists an $r'>1$, and a diffeomorphism $$ f_r:A_r \to \mathbb A_{r'} $$ which is holomorphic in the interior.

Question: What can be said about $f_r(\partial \mathbb D_1)$ as $r\to 1$?
Does it converge to $\partial \mathbb D_1$ as a smooth manifold?

In particular, is the domain enclosed by $f_r(\partial \mathbb D_1)$ convex for $r$ sufficiently close to $1$?

Let $\mathbb D_r=\{z\in\mathbb C:|z|\le r\}$ be the closed unit disk of radius $r$,
let $\mathring {\mathbb D}_r=\{z\in\mathbb C:|z|< r\}$ be its interior,
and let $\mathbb A_r=\mathbb D_r\setminus \mathring{\mathbb D}_1$.

Let $D\subset \mathbb D_1$ be closed, simply connected, and with smooth boundary,
and let let $A_r:=\mathbb D_r \setminus \mathring D$.

By the Riemann uniformization theorem, for every $r>1$, there exists an $r'>1$, and a diffeomorphism $$ f_r:A_r \to \mathbb A_{r'} $$ which is holomorphic in the interior.

Question: What can be said about $f_r(\partial \mathbb D_1)$ as $r\to 1$?
Does it converge to $\partial \mathbb D_1$ as a smooth manifold?

In particular, is the domain enclosed by $f_r(\partial \mathbb D_1)$ convex for $r$ sufficiently close to $1$?

Let $\mathbb D_r=\{z\in\mathbb C:|z|\le r\}$ be the closed unit disk of radius $r$,
let $\mathring {\mathbb D}_r=\{z\in\mathbb C:|z|< r\}$ be its interior,
and let $\mathbb A_r=\mathbb D_r\setminus \mathring{\mathbb D}_1$.

Let $D\subset \mathbb D_1$ be closed, simply connected, and with smooth boundary,
and let let $A_r:=\mathbb D_r \setminus \mathring D$.

Assume furthermore that $\partial D\cap \partial \mathbb D_1\neq\emptyset$.

By the Riemann uniformization theorem, for every $r>1$, there exists an $r'>1$, and a diffeomorphism $$ f_r:A_r \to \mathbb A_{r'} $$ which is holomorphic in the interior.

Question: What can be said about $f_r(\partial \mathbb D_1)$ as $r\to 1$?
Does it converge to $\partial \mathbb D_1$ as a smooth manifold?

In particular, is the domain enclosed by $f_r(\partial \mathbb D_1)$ convex for $r$ sufficiently close to $1$?