families of Riemann mappings

Question

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).

Answer 1

The conformal map $\phi$ between the unit disk $D$ and the simply connected domain $\Omega$ with a smooth boundary, subject to normalization $\phi(0)=a$ and $\phi'(0)>0$, depends smoothly on the domain $\Omega$. It's proved, for example, in Bell's book "The Cauchy Transform, Potential Theory and Conformal Mapping", see Theorem 28.1.

There it's proved by integral transform method. It can be proved using other approaches to Riemann Mapping problem. For example, it's easy to prove using method based on Green's function as described in section 5.4 in Taylor's "Partial Differential Equations 1. Basic Theory", 1st ed.

I don't know who was the first to establish that the conformal map depends smoothly on the domain. Quite possible it was Riemann himself, because to find how that map changes under small perturbation of the domain one need to solve Riemann-Hilbert problem.

Now, the relation to gluing analytic disks. That's definitely an overkill for the problem in hands, but it may be useful. Let's consider $\mathbb{C}^{n+1}$ with the standard symplectic form. Then the image $f(S^1\times U)\subset \mathbb{C}\times\mathbb{R}^n\subset\mathbb{C}^{n+1}$ is a Lagrangian submanifold. $\Phi$ is a family of analytic disks attached to that manifold. In this setting, there is no natural normalization, so it's not completely equivalent. Nevertheless, appropriate version of smoothness of the gluing map is classical (and easy). The difficult part (in general) is the existence of at least one disk and some version of compactness (existence of limits).