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Just few remarks to Alex's answer. If all $u_1,u_2,v_1,v_2$ are distinct, the function mapping the Riemann sphere on your surface is a rational function of degree $3$. And conversly, every rational fu …

You cannot do this "without analytic continuation". A multi-valued function is defined on a Riemann surface. The singularities of this function are not lying on the Riemann sphere, so you cannot "spec …

There is no estimate of the area of the image. I suppose you consider non-trivial embeddings (inducing non-trivial homomorphisms of the fundamental group). On your cylinder $C_R$ make a cut $[\delta, …

R. Fricke, F. Klein, Vorlesungen über die theorie der automorphen functionen, 1897-1912.

Classification. You do not specify what classification (what is your equivalence relation?) Topological classification is due to Kerekjarto. A reference is given in the answer of Richard Kent. Comple …

Yes, in principle. If the curve is given by $F(x,y)=0$ and the differential by $D(x,y)dx$ (every curve and differential can be described like this), then we look for a function in the form $R(x,y)$ wh …

There are such proofs. See, for example Goluzin, Geometric theory of functions (Appendix). He uses the following fact. Let $h$ be an analytic diffeomorphism of the circle onto itself. Then there is a …

Besides the position of ramification points $z_j$ you need monodromy of $Q$ to determine $M$. Once $M$ is defined, you need a normalization of your uniformizing function: it is defined up to a confor …

Let $C$ be the Riemann sphere, $p=0$. Then $$\omega(z)=\left(\sum_{-\infty}^\infty c_nz^n\right)dz.$$ Here the part with negative powers converges for $|z|>0$, while the part with positive powers con …

An algorithm exists in principle, at least when the genus is $0$. But it is very difficult unless the degree is small. For example, if the function is supposed to be a polynomial, as in your example, …

In general, Jensen's formula holds with the integral taken over both sheets (and zeros counted on both sheets). See, for example, MR1069755 Lang, S., Cherry, W. Topics in Nevanlinna theory. Lecture …

Let $C$ be a complex line in $C^2$, say $y=0$. Project it on $x$-line, all properties are satisfied:-) If you really want "connected, but ONLY if one goes near the origin", take the set $\{(x,y): y^2 …

Considering the reciprocal function, it is sufficient to construct a holomorphic function with prescribed zeros, and prescribed finite portions of Taylor series at those zeros. For the plane and the u …

First, as you noticed, it is enough to consider the case that the equation has the form $$w^n+a_{n-1}(z)w^{n-1}+\ldots+a_0(z)=0,$$ where the coefficients are entire. Then $w$ is holomorphic on its Ri …

That your $f_t$ are local homeomorphisms away from isolated points is not sufficient for the conclusion you want. Your $f_t$ must be at least topologically holomorphic. (A continuous map is called top …