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Riemann surfaces(Riemannian surfaces) is one dimensional complex manifold. For questions about classical examples in complex analysis, complex geometry, surface topology.

2 votes
1 answer
230 views

Reconstruction of Riemann surface from a germ of holomorphic function

Let $\Sigma$ be a compact Riemann surface of genus $g$, and $f: \Sigma \to \mathbb{C}$ a meromorphic function. Take $U \subset \Sigma$ an open disk in $\Sigma$ biholomorphic to a disk in $\mathbb{C}$, …
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1 vote
1 answer
349 views

Complex structures on topological surfaces

I am interested in the number of complex structures on a surface. More precisely, given a genus $g$ surface (topological manifold of real dimension 2) with $n$ punctures $X_{(g,n)}$, how many complex …
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4 votes
0 answers
106 views

Elliptic integral as quantity associated with Riemann surface?

There are many elliptic integrals, so to show my point let me just pick one of them (complete elliptic integral of the first kind [1]): $$K(k) = \int_{0}^{1} \frac {dx} {\sqrt{(1-x^{2})(1-k^{2}x^{2})} …
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