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For a compact set $K$ in the complex plane, define the analytic capacity of $K$ by $$\gamma(K) := \sup |f'(\infty)|$$ where the supremum is taken over all functions $f$ holomorphic and bounded by $1$ …

Hi, I have a question regarding the universality property of the Riemann zeta-function. I am no expert on this, so I'd be glad for any relevant reference. First, recall Voronin's remarkable theorem …

I would like some good references to learn about the Schramm-Loewner evolution (SLE), for a complex analyst with no background in probability. A quick google search gave a lot of references on SLE th …

As mentioned in the comments, this is true if $K$ is compact and the complement of $K$ in the Riemann sphere is connected : it is the content of Mergelyan's Theorem on uniform polynomial approximation …

Let $X$ be a domain in the Riemann sphere $\widehat{\mathbb{C}}$. We say that $X$ is a circle domain if every connected component of its boundary is either a circle or a point. It was conjectured by …

The following is too long for a comment. I will suppose here that your set of measure zero is compact. In this case, Your question is closely related to so-called continuous analytic capacity. Let $K …

Yes, there is a classical proof of the Riemann mapping theorem using harmonic maps and the Dirichlet problem. Riemann's original assumption of boundary smoothness can be removed using Perron's method …

We say that a compact subset $E$ of the Riemann sphere $\mathbb{C}_\infty$ is (conformally) removable if every homeomorphism of $\mathbb{C}_\infty$ conformal outside $E$ is actually conformal everywhe …

This is the so-called conformal welding problem. One can ask the same question for any Jordan curve $\gamma$ (non necessarily analytic). With this domain of definition, your map $\Gamma$ is well-known …

Let $\Omega$ be a domain of the complex plane. The Hardy space $H^p(\Omega)$ is defined, for $1 \leq p<\infty$, as the class of functions $f$ that are holomorphic on $\Omega$ such that $|f|^p$ has a h …

Let $X \subsetneq \mathbb{C}$ be a simply connected domain. The Riemann mapping theorem states that there exists a biholomorphism of $X$ onto the unit disk $\mathbb{D}$. A simple and elegant way to ob …

Dear Mathoverflow Community, Suppose that $\Omega$ is a domain in the Riemann Sphere $\widehat{\mathbb{C}}$ with $\infty \in \Omega$, and assume that every connected component of $\partial \Omega$ is …

This theorem is an immediate consequence of a result by B. Maskit, which states that one may associate with $\Omega$ another domain $\Omega'$, conformally equivalent to $\Omega$, such that all conform …

I really recommend the book Potential Theory in the complex plane" by Thomas Ransford. It's a very nice book with exercises and it covers each of the 5 points you mentioned.

The main reference on this topic is the book "Boundary behavior of conformal maps" by Pommerenke. If the curve is $C^\infty$, then the biholomorphic mapping extends to a smooth map on the closure of t …