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Suppose $\Gamma$ is an analytic Jordan curve in the complex plane $\mathbb{C}$. What are ways to test whether $\Gamma$ is (contained in) an algebraic curve, i.e., whether there exists a real polynomia …

There is a necessary and sufficient condition due to Yoccoz in terms of continued fraction expansion. The condition is that $\lambda=e^{2\pi i \theta}$ should be such that $\theta$ is not a Brjuno num …

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 …

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 …

The so-called Continuity Method is a simple yet powerful method to show that a given continuous injective map is surjective. Namely, suppose that $f:X \to Y$ is a map between two connected manifolds $ …

The only example I know of a conformal map that "stretches" a point boundary component to a nondegenerate continuum was constructed by Gehring and Martio in Quasiextremal distance domains and extensio …

I also think there is no "explicit formula" for the conformal modulus, even in simple cases. However, as mentioned in Igor Rivin's answer, there are methods for approximating the conformal map and the …

Hi, Let $f$ be a non-identically zero bounded analytic function in the open unit disk $\mathbb{D}$. It is well-known that $f$ has radial limits almost everywhere on the unit circle $\mathbb{T}$. Let …

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 …

Perhaps a more elementary proof is the following : Assume $\mathbb{C} \setminus K$ is disconnected. Let $V$ be a bounded component of $\mathbb{C} \setminus K$, and let $w \in V$. Let us show that the …

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 …

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 …

Yes it can, see the paper A new proof of the Ahlfors five islands theorem by Walter Bergweiler. The proof is based on a result on Nevanlinna concerning perfectly branched values, Zalcman's rescaling l …

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 answer to your question is yes. Indeed, one can replace $\mathbb{D}$ with the right half plane, applying a Mobius transformation. Then the argument is quite simple if you are familiar with the fol …