The riemann-surfaces tag has no usage guidance.

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### Branches of the tetration function

Letting $\eta = e^{1/e}$ where $e$ is Euler's constant, there exists a function $F(z)=\, ^z \eta$ with the following relevant properties. (I won't bother showing the existence of this function, or the ...

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### Field of moduli relative to a different extension

Let $X$ be a Riemann surface of genus $g\geq 2$ defined over $\mathbb{Q}(\sqrt{2})$.
Let us consider the automorphism of $\mathbb{Q}(\sqrt{2})$,
$$
...

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### Zeroes of global sections killed by differential operators

I asked this question some two weeks ago on StackExchange, but received no feedback of any sort ...
Let $X$ be a compact connected Riemann surface and let $\Phi:M\rightarrow N$ be an elliptic ...

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### Deformation of the covariant Laplacian

Let $M$ be a Riemann surface and $P \to M$ a principal $G$-bundle (with compact structure group $G$).
Fix a connection $A$ in $P$ and consider a nearby connection $B$, which is in Coulomb gauge ...

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### Comparison of algebraic and analytic q-expansion

I would like to check that algebraic and analytic q-expansion of a modular form coincide.
I'm thinking about modular forms as global sections of some sheaf on modular curves. If $X$ is a modular ...

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### Gromov-Hausdorff limits of 2-dimensional Riemannian surfaces

Let $\{M_i\}$ be a sequence of 2-dimensional orientable closed surfaces of genus $g$ with smooth Riemannian metrics with the Gauss curvature at least $-1$ and diameter at most $D$. By the Gromov ...

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### Classification of symmetries of tilings in surfaces?

Is there a general study of the symmetries of tilings on surfaces?
Conway, Goodman-Strauss & Burgiel classified them on $\mathbb S^2, \mathbb R^2$ and $\mathbb H^2$, with their 'Magic Theorem'. ...

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### Embed a bordered Riemann surface into punctured Riemann surfaces?

Let $U$ be a bordered Riemann surface of genus $g$ with $n -1$ punctures and one hole (i.e., the border has one connected component). For any punctured Riemann surface $\Sigma$ of genus $g$ with $n$ ...

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### The class of the diagonal in the symmetric product of a smooth curve

Let $C$ be a smooth curve of genus $g$, and let us consider its $d$-th symmetric product $\textrm{Sym}^d(C)$ and its Jacobian $J(C)$. Fixing a point $p_0 \in C,$ there are two maps $$u_d\colon C_d \to ...

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### Lower bound for Hurwitz's automorphisms theorem

Let $\Sigma_g$ a closed Riemann surface of genus $g\geq 2$, then the cardinality of the orientation-preserving conformal automorphism is less than $84(g-1)$, and we have equality for an infinite ...

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### Criterion for deciding the conformal class of a metric on a complete surface

For orientable closed Riemannian surfaces $(S,g)$, there is a constant curvature metric $\overline{g}$ on $S$ that is conformal to $g$ in the sense that $\overline{g} = e^ug$ for some smooth function ...

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### Foliation by Umbilic Surfaces

Suppose $(M,g)$ is a simply connected 3 dimensional Riemannian Manifold which is a foliation by Umbilic surfaces.
Can I make the claim that there exists a coordinate system $(x_1,x_2,x_3)$ in which ...

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### Quotients of curves of genus $4$ by a free $\mathbb{Z}/ 3 \mathbb{Z}$-action

Let $V_2$ and $V_3$ be the two hypersurfaces of $\mathbb P^3$ defined by
\begin{equation*}
V_2:={x_2x_3 + r(x_0, \, x_1)=0}, \quad V_3:={x_2^3+x_3^3+s(x_0, \, x_1)=0},
\end{equation*}
where $r, \, s ...

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### Are there any non-trivial $G$-gerbes over the analytic space $\mathbb C$

Does there exist a finite (abstract) group $G$ and a non-trivial $G$-gerbe $\mathcal X\to \mathbb C$, where we work in the category of analytic stacks.
My guess is that $G$-gerbes for $G$ an abelian ...

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### Given a curve $C$, does there exist a rational function on $C$ totally ramified at two given points?

Let $C$ be a smooth projective irreducible curve over $\mathbb C$. Let $x$ and $y$ be distinct points of $C$.
We say that $f$ is totally ramified at a point $p$ if the ramification index of $p$ ...

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### Moduli of stable bundles - analytic approach

Consider a compact Riemann surface $X$ of genus $\ge 2$, and consider the set $M$ of stable holomorphic vector bundles of rank $n$ and degree $d$ on $X$, up to isomorphism.
At that point, one states ...

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### Complex structure on a punctured torus giving a complex structure on the torus?

Can anyone provide an idea of the proof or a reference of the fact that a complex structure on the once punctured torus extends to one on the torus?
In other words, the Teichmuller space of the ...

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### Construction of algebraic curves using line bundles on graphs

In this paper http://arxiv.org/abs/0707.1309 Matthew Baker and Serguei Norine, construct a analogue of the Riemman Roch formula for Lineal Systems defined on graphs. In the paper ...

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### Is every closed Sasakian 3-manifold a circle bundle on a Riemann surface?

It suffices to say that all circle bundles on compact Riemann surfaces admit the structure of a closed Sasakian 3-manifold. The question is, the converse of this statement and/or what are the ...

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### Classification of open subset of $\mathbb{R}^{3}$ [closed]

There is a theorem which gives a classification of connected open sets of $\mathbb{R}^{2}$. Unfortunately, I don't remember the correct statement, but it looks like this
Theorem ? Let ...

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### Deep/precise relationship between two approaches to FLT for polynomials, $n = 3$

David Speyer commented the following here.
I saw Brian Conrad give an excellent one hour talk to undergraduates where he proved that there do not exist nonconstant, relatively prime, polynomials ...

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### Frucht's type theorem for Riemann surface

Frucht's theorem is a theorem in algebraic graph theory conjectured by Dénes Kőnig in 1936 and proved by Robert Frucht in 1939. It states that every finite group is the group of symmetries of a finite ...

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### Compact surface with arbitrarily large eigenvalue

Consider a compact surface $M$ with genus $\gamma \geq 2$ and fix a positive real number $V$. Is it known whether it is possible to produce a metric $g$ on the surface $M$ such that $(M. g)$ has ...

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### Is there a proof of the uniformization theorem using circle packing?

In this paper: http://www.dm.unipi.it/~benedett/rodin-sullivan.pdf
Rodin and Sullivan show that circle packings converge to the Riemann map. Later, Scharmm and He found another proof of the same ...

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### Explicit constant terms of volumes of moduli spaces

In Simple geodesics and Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces, Mirzakhani gave a recursive formula for WP volumes of moduli spaces $\mathcal{M}_{g,n}(L)$ of bordered ...

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### Is there a complex surface into which every Riemann surface embeds?

This question was previously asked on Math SE.
Every Riemann surface can be embedded in some complex projective space. In fact, every Riemann surface $\Sigma$ admits an embedding $\varphi : \Sigma ...

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### Laplacian Spectra on Nearly Nodal Riemann Surfaces

Consider a family of complex curves ${\mathcal C} \to {\mathbb D}$ such that the central fibre is a nodal Riemann surface while other fibres are smooth Riemann surfaces. We choose a family of ...

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### A question on Ahlfors covering surface

Given a transcendental entire function $f$, and three Jordan domains $D_1$, $D_2$, and $D_3$ such that the closures of the three Jordan domains do not intersect with each other. Then from Ahlfors ...

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### Compact surfaces with arbitrary gaps in spectrum

Consider a sequence of positive numbers $a_n$. My question is, can we select a closed Riemann surface whose spectrum $\lambda_i$ satisfies the condition that $\lambda_{i + 1} - \lambda_i > a_i$? Of ...

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### Mapping class group of a punctured genus 0 surface

Let $T_{0,n}$ be the Teichmuller space of $n$-punctured genus $0$ Riemann surface, and $M_{0,n}$ the Moduli space (assume $n\geq 3$ and the punctures are numbered). What is the correct notion of the ...

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### Finite orbits on an elliptic curve with two generic involutions

Let $C$ be a (very) general genus 1 curve embedded in $\mathbb{CP}^1\times \mathbb{CP}^1$ as a (2,2)-divisor.
Each projection defines $C$ as a double cover of $\mathbb{CP}^1$ and induces an ...

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### What are the possible automorphism groups of Riemann surfaces of low genus?

Skipping the easy cases of genus 0 and 1, what groups can arise as the group of conformal transformations of a Riemann surfaces of genus, say, 2 or 3?
I'm frustrated because there are papers that ...

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### Equivalence of Harmonic Maps and Conformal Maps on Genus-0 closed surfaces

By the uniformization theorem, for every genus-0 closed surface $\mathcal{M}\subset\mathbb{R}^3$, there is a conformal map $f:\mathcal{M}\rightarrow \mathbb{S}^2$. Furthermore consider the Dirichlet ...

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### Is there an algorithm to compute the intersection of tautological classes on the moduli space of genus one curves?

Let $\overline{M}_{1,1}(\mathbb{P}^2, d) $ be the moduli space of degree
$d$ genus one curves on $\mathbb{P}^2$ with one marked point. Let
$L\longrightarrow \overline{M}_{1,1}(\mathbb{P}^2, d) $ ...

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### Rational functions on hyperelliptic Riemann surface

Let $\mathcal R$ be an hyperelliptic Riemann surface of genus $g\geq 1$. Is it true that the only possible rational functions on $\mathcal R$ with $\leq g$ poles are the liftings of rational functions ...

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### Uniqueness of Riemann Constant Vector Solution

Let $X$ be a compact, genus $g$ Riemann surface (given as the desingularization and compactification of a plane algebraic curve), $J(X)$ its Jacobian, and $A : X \to J(X)$ the Abel map
$$A(P) = ...

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### $\eta$-invariants of Riemann Surface

I am curious about a concrete computation of $\eta$-invariants for Riemann surface, e.g. Torus.
Is there any nice review or notes talking about the computation? Or is it possible to express it as ...

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### Conformal welding of annuli

This question is similar to that stated in Conformal Welding Reference:
Let $\Sigma$ be a 1-dim compact connected complex manifold with boundary $\partial \Sigma= \partial\Sigma^+ \cup ...

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### Can we foliate the anti-de sitter space in 3 dimensions by Riemann surfaces?

Can we foliate the anti-de sitter spacetime in 3 dimensions by hyperbolic Riemann surfaces? -- I think this is possible, but got stuck at finding the particular projective mapping that does this. Can ...

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### criterion for a differential of the third kind to be a logarithmic derivative of a function

Let $X$ be a compact Riemann surface of genus $g\geq 1$. If $f$ is a meromorphic function on $X$ then, the meromorphic differential $\omega=\frac{df}{f}$ is a differential
of the third kind with ...

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### How to get a polygon from a translation surface $(X,\omega)$

Let $S_g$ be a compact topological surface of genus $g$. I know there is the correspondence
$\{$Abelian differentials on compact Riemann surfaces of genus g$\}\leftrightarrow\{$ Translation surfaces ...

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### Can every hyperelliptic genus 3 surface be minimally immersed in flat $T^3$

Every minimally immersed genus 3 surface in flat $T^3$ must be hyperelliptic, as the Gauss map gives the degree 2 covering map. How about the converse of this problem?
The only thing I can find is ...

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### Decomposition of a closed surface

I know that I can decompose an hyperbolic closed surface of genus $g>1$ into $2(g−1)$ pants bounded by $3$ geodesics. It seems reasonable to think the same can be done for a closed surface of genus ...

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### Equivalence of Definitions of Quasiconformal Surfaces?

I have been reading John H. Hubbard's book Teichmüller Theory vol. 1 and I am a little bit concerned with his definition of Quasiconformal Surface.
Definition: A Quasiconformal surface $S$ is a ...

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### What are the easiest examples of irreducible, but not big, monodromy representations

Let $\rho: \pi_1(S,s_0) \to GL(V)$ be the monodromy representation associated to a local system of $\mathbb Q$-modules $\mathbb V$ with $\mathbb V_{s_0} = V$.
Let $H$ be the Zariski closure of the ...

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### Maryam Mirzakhani's works

Maryam Mirzakhani has made several contributions to the theory of moduli spaces of Riemann surfaces.
Mirzakhani was awarded the Fields Medal in 2014 for "her outstanding contributions to the dynamics ...

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### Structure of the automorphism group of a Riemann surface

I was wondering if anything is known about the possible structure of $\mathrm{Aut}(S)$ for a Riemann surface $S$. More precisely, are there known obstructions for a finite group $G$ to be such an ...

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### Torsion elements in the mapping class group

Let $S$ be an orientable surface of genus $g$ with $b>0$ boundary components, and let $\mathrm{Mod}(S)$ be its mapping class group, that is, the group of isotopy classes of its homeomorphisms ...

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### Translation surfaces & integer multiples of $\pi$

Richard Schwartz, in Mostly Surfaces (Vol. 60. American Mathematical Soc., 2011),
defines (on p.14) a translation surface as "a Euclidean cone surface, all of whose 'angle errors' are integer ...

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### An explicit description of the Torelli spaces of pointed genus 2 Riemann surfaces

In [N], there is a nice and very explicit description of the Torelli space ${\rm Tor}_{1,n}$ of $n$-pointed elliptic curves, for any $n\geq 1$:
$$
{\rm Tor}_{1,n}=\left\{
\big(\tau, ...