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Let $E=\mathbb C/\Lambda$ be an elliptic curve, and let $D\subset E$ be a very small disc. ($D$ is round for the usual flat metric on $E$) By the main result of [1], there exists a holomorphic immers …

Does every connected, compact Riemann surface $\Sigma$ with boundary, $\partial \Sigma\not =\emptyset$, admit a holomorphic function (smooth on the boundary) $f:\Sigma\to\mathbb C$ whose derivative is …

I know that this is supposed to be standard, but I don't know how to search for it... hence the question: Let $J$ be an almost complex structure on $M:=\mathbb R^2$, i.e., a $C^\infty$ section of $\m …

Let $\Sigma$ be a compact Riemann surface equipped with a spin structure (a square root of $\Omega^1_\Sigma$, denoted $\Omega^{1/2}_\Sigma$). Let $\Gamma(\Omega^1_\Sigma)$ be the space of holomorphic …

Let $\mathbb D_r=\{z\in\mathbb C:|z|\le r\}$ be the closed unit disk of radius $r$, let $\mathring {\mathbb D}_r=\{z\in\mathbb C:|z|< r\}$ be its interior, and let $\mathbb A_r=\mathbb D_r\setminus \m …

Every compact surface is diffeomorphic to $S^2$, $\underbrace{T^2\#\ldots \#T^2}_n$, or $\underbrace{RP^2\#\ldots \#RP^2}_n$ for some $n\ge 1$. Is there a conceptual proof of this classification t …

For $G$ a finite group, it is easy to construct a (connected, orientable) surface with a faithful action of $G$. E.g.: take a disjoint union of $G$ many spheres, and add a 1-handle for every edge in t …