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Riemann combines what is called Riemann-Roch and Riemann-Hurwitz nowadays. He considers the dimension of the space of holomorphic maps of degree $d$ from the Riemann surface of genus $g$ to the sphere …

The answer is negative. For the non-injective case, the reason is that non-constant complex analytic functions are open, discrete maps, while real analytic functions can be neither open nor discrete ( …

For infinite degree, the definition of "branched covering" can be somewhat ambiguous. But $$z\mapsto \cos z: \mathbb{C}\to \mathbb{C}$$ $$\wp: \mathbb{C}\to S$$ are a simple examples of branched cove …

I suppose that Borel is mentioned in the text you refer to by mistake. I cannot prove this (Borel has 335 publications including 85 books according to Zentralblatt) but a large recent book on the unif …

The simplest "non-trivial" example is $$z\mapsto \int_0^ze^{-\zeta^2}d\zeta:\quad C\to C.$$ It is surjective, and not ramified. But it is certainly not a covering because every covering over a simply …

For the case of a plane domain, the first definition is a special case of the second. Assuming that $M$ is a plane domain, take $G=M\backslash K$ in the second definition. Then, if $\omega$ is the har …

There is a simple removability theorem for subharmonic functions: if it is bounded from above in a neighborhood of an isolated singularity then this singularity is removable. Proof. Suppose our singul …

Gauss map is holomorphic (as a map to the Riemann sphere) if the surface is minimal. This is Lemma 8.3 in the book of Osserman, A Survey of Minimal Surfaces. In fact, if you replace "embedded" by "imm …

A good reference is W. Abikoff, The real analytic theory of Teichmuller space, Springer, 1980. (Chap. II section 1). The idea is that you construct the double: it is the result of gluing of your surfa …

An elementary proof is the following: first you prove that every holomorphic map $f:S^2\to S^2$ is a rational function. Indeed, such map must be a meromorphic function, it has finitely many zeros and …

The expression "fixes $\partial\Omega$" is ambiguous. Do you mean that $f(\partial\Omega)=\partial\Omega$ or that $f(z)=z$ for all $z\in\partial\Omega$? For the first, weaker condition, all exceptiona …

These classifications coincide for simply connected Riemann surfaces. In general, an open parabolic surface in the second sense is also parabolic in the first sense, but not vice versa. In fact there …

I suppose that "non-compact complex algebraic curve" means complex affine curve. The following counterexample was proposed by my friend Fedor Pakovich. Let $D=\mathbf{C}\backslash\{-1,1\}$. Consider t …

Take a curvilinear quadrilateral $A',B',C',D'$ satisfying all your conditions, that is $[A',B'],[C',D']$ are disjoint arcs of the unit circle, while the other sides do not belong to the unit circle. Y …

For open surfaces, there are counterexamples. The first one was constructed by P. Myrberg: Ueber die analytische Fortsetzung von beschrankten Funktionen, Ann. Acad. Sci. Fenn., Ser. A. I N:o 58 (1949) …