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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 …

Yes. The density of the Poincare metric with respect to the spherical metric is a positive continuous function which tends to infinity at the punctures. Thus it is bounded from below by some positive …

Here it is: MR0015154 Salem, R.; Zygmund, A. Lacunary power series and Peano curves. Duke Math. J. 12, (1945). 569–578.

There are two different things which are called "Riemann surface" in the literature. The modern notion (introduced by Hermann Weyl): complex 1-dimensional manifold. In older literature this is somet …

The answer is no. For $g=0$, the problem is equivalent to the following: is there a univalent function in the unit disk which takes given values at finitely many given points. It is well known that th …

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 ( …

Forster just touches the Riemann-Hilbert problem and fiber bundles. Expansion on this can be interesting I recommend the books of Bolibrukh. Applications of compact Riemann surfaces to solitons ("Ex …

The answer to the first question is "yes". This is called conformal gluing, and the proof is based on the following lemma due to Lavrentiev: Let $\phi$ be an increasing diffeomorphism of $[-1,1]$ onto …

On a basic level: W. Abikoff, The uniformization theorem, Amer. Math. Monthly 88 (1981), no. 8, 574–592. L. Ahlfors, Conformal invariants, last chapter. S. Donaldson, Riemann surfaces, Oxford, 2011. V …

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 …

There are indeed very few pairs (except spheres with 3 or 4 singularities, or tori, and what can be obtained from them by finite coverings, where correspondence 2)-3) is completely explicit. See: H. P …

There are (at least) 2 kinds of proofs: analytic ones (which use the existence of Abelian differentials with certain properties) and algebraic ones. The proofs of the first kind use powerful analytic …

This deep fact is essentially the same as the uniformization theorem. The problem is how to construct at least one holomorphic or meromorphic form with prescribed singularity. All known proofs use som …

Let $L$ be your difference operator: $(Lf)(z)=f(z+1)-f(z)$. Consider these polynomials $$P_n(z)=\frac{1}{n!}z(z-1)\ldots(z-n+1),\quad n=0,1,2,\ldots.$$ Simple computation shows that $LP_n=P_{n-1}$. Po …

The gluing does not necessarily exist, neither it is unique when exists, in such generality as stated. (I am not even speaking of what the "boundary" of a Riemann surface could mean in general). Even …