Let $U\subset \mathbb R^n$ be an open.

Let $f: S^1 \times U \to \mathbb C$ be a smooth map s.t. for every $x\in U$ the map $f_x:=f|_{S^1\times \{x\}}:S^1\to \mathbb C$ is a smooth embedding (i.e. with everywhere non-zero derivative) that goes around the origin with winding number $+1$.

For every $x\in U$ there's a unique Riemann map $\Phi_x$ : [unit disc] $\to$ [interior of $f_x(S^1)$] that sends $0\mapsto 0$ and whose derivative at zero is in $\mathbb R_{\ge 0}$.

Is $\Phi:=\coprod_{x\in U}\Phi_x:$ [unit disc] $\times U\to\mathbb C$ smooth?

I'm also interested in the same question where the open unit disc is replaced by the closed unit disc (i.e. smoothness all the way to the boundary).

Let $U\subset \mathbb R^n$ be an open.

Let $f: S^1 \times U \to \mathbb C$ be a smooth map (i.e. $C^\infty$) s.t. for every $x\in U$ the map $f_x:=f|_{S^1\times \{x\}}:S^1\to \mathbb C$ is a smooth embedding (i.e. with everywhere non-zero derivative) that goes around the origin with winding number $+1$.

For every $x\in U$ there's a unique Riemann map $\Phi_x$ : [unit disc] $\to$ [interior of $f_x(S^1)$] that sends $0\mapsto 0$ and whose derivative at zero is in $\mathbb R_{\ge 0}$.

Is $\Phi:=\coprod_{x\in U}\Phi_x:$ [unit disc] $\times U\to\mathbb C$ smooth?

I'm also interested in the same question where the open unit disc is replaced by the closed unit disc (i.e. smoothness all the way to the boundary).

Let $U\subset \mathbb R^n$ be an open.

Let $f: S^1 \times U \to \mathbb C$ be a smooth map s.t. for every $x\in U$ the map $f_x:=f|_{S^1\times \{x\}}:S^1\to \mathbb C$ is an embedding that goes around the origin with winding number $+1$.

For every $x\in U$ there's a unique Riemann map $\Phi_x$ : [unit disc] $\to$ [interior of $f_x(S^1)$] that sends $0\mapsto 0$ and whose derivative at zero is in $\mathbb R_{\ge 0}$.

Is $\Phi:=\coprod_{x\in U}\Phi_x:$ [unit disc] $\times U\to\mathbb C$ smooth?

I'm also interested in the same question where the open unit disc is replaced by the closed unit disc (i.e. smoothness all the way to the boundary).

Let $U\subset \mathbb R^n$ be an open.

Let $f: S^1 \times U \to \mathbb C$ be a smooth map s.t. for every $x\in U$ the map $f_x:=f|_{S^1\times \{x\}}:S^1\to \mathbb C$ is a smooth embedding (i.e. with everywhere non-zero derivative) that goes around the origin with winding number $+1$.

For every $x\in U$ there's a unique Riemann map $\Phi_x$ : [unit disc] $\to$ [interior of $f_x(S^1)$] that sends $0\mapsto 0$ and whose derivative at zero is in $\mathbb R_{\ge 0}$.

Is $\Phi:=\coprod_{x\in U}\Phi_x:$ [unit disc] $\times U\to\mathbb C$ smooth?

I'm also interested in the same question where the open unit disc is replaced by the closed unit disc (i.e. smoothness all the way to the boundary).