The riemann-surfaces tag has no wiki summary.

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### Examples of compact hyperbolic surfaces/manifolds with very small or very large eigenvalues

Hello,
Is there any general ways to construct compact hyperbolic 2-manifolds with very small or very large eigenvalues ? Also, as a special case, can we construct a sequence of compact hyperbolic ...

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### Does there exist a Riemann surface corresponding to every field extension? Any other hypothesis needed?

My question is a doubt I had in the last point to the first answer to this MO question - "Algebraic" topologies like the Zariski topology?
Can one associate a Riemann surface to any ...

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### Riemann surfaces

In Hatcher's book on Algebraic Topology, a lot of coverings of $S^1 \vee S^1$ has given. From these coverings, can we get different coverings of double torus (genus 2, compact surface)?

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### What is the definition of ideal boundary?

In many papers about dynamical system, I found the word " ideal boundary". T don't know what is the definition of ideal boundary.

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### How to compute the (co)homology of a compact Riemann surface?

The situation is the following.
A finite-index subgroup $\Gamma$ of $SL_2(\mathbb Z)$ acts on the upper-half plane $\mathcal H$. It has a fundamental domain, obtained by a union of translates of the ...

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### The Riemann correspondence for riemann surfaces made explicit and its generalizations

Riemann showed (not proved rigorously) that there is a correspondence between compact riemann surfaces and algebraic function fields in one variable (does anyone know the year?).
To construct the ...

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### Covers of Riemann surfaces which become arbitrary close in Teichmuller space

Suppose $S$ and $S'$ are two compact Riemann surfaces of genus $g$. Does there exist a sequence of genera $g_i \to \infty$ and covers $S_i, S_{i}'$ of $S,S'$, both of genus $g_i$, such that ...

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### Divisors of solutions of elliptic problems

I learned very recently that in the early-mid 90s Gromov and Shubin proved a generalization of the Riemann-Roch theorem. Let $X$ be a compact closed ${\cal C}^\infty$-variety of dimension $n\geq2$. ...

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### Schwarz Lemma in terms of conformal surfaces or holomorphic curves?

Scharwz Lemma in its general form says that any holomorphic map between hyperbolic surfaces is contracting.
Noting that Riemann surfaces admit a unique metric of constant curvature -1, I wonder if we ...

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### Approximating holomorphic maps by holomorphic embeddings

Let $\mathrm{Hol}^d(\Sigma, \mathbb{C} \mathbb{P}^n)$ denote the space of holomorphic maps of degree $d$ from a Riemann surface $\Sigma$ to complex projective space of dimension $n$. Let ...

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### What should be taught in a 1st course on Riemann Surfaces?

I am teaching a topics course on Riemann Surfaces/Algebraic Curves next term. The course is aimed at 1st and 2nd year US graduate students who have have taken basic coursework in algebra and manifold ...

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### Branched coverings of Riemann surfaces with specified branch points.

Today I showed, using some ad hoc algebraic topology, that if $\Sigma$ is a Riemann surface and $\mathfrak{p} \subset \Sigma$ is a finite set of points, then there is another Riemann surface $S$ and a ...

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### How do you recover the structure of the upper half plane from its description as a coset space?

This is maybe a dumb question. $SL_2(\mathbb{R})$ has a natural action on the upper half plane $\mathbb{H}$ which is transitive with stabilizer isomorphic to $SO_2(\mathbb{R})$. For this reason, ...

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### “Physical” construction of nonconstant meromorphic functions on compact Riemann surfaces?

Miranda's book on Riemann surfaces ignores the analytical details of proving that compact Riemann surfaces admit nonconstant meromorphic functions, preferring instead to work out the algebraic ...

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### Riemann surface disconnected at infinity

This question may be trivial, I did not think hard about it.
A friend of mine is looking for an irreducible (reduced) analytic subspace $C \subset \mathbb{C}^2$ with the following property. Let $f ...

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### Finding Constant Curvature Metrics on Surfaces without full power of Uniformization

(I rewrote this question, hopefully it's more clear now. It's still the same question, but I reordered its parts.)
Let S be a surface (possibly non-compact, but no boundary). It seems that there are ...

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### Hyperelliptic loci in Teichmueller spaces

Let ${\cal M}_g$ be the moduli space of smooth complex genus $g$ curves, let ${\cal H}_g\subset {\cal M}_g$ be the hyperelliptic locus and set ${{\cal H}}'_g$ to be the preimage of ${\cal H}_g$ in the ...

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### Weil's theorem about maps from a discrete group to a Lie group.

Let K be a group (with discrete topology), G be a Lie group. Let $\operatorname{Hom}(K,G)$ be the space of all group homomorphisms from K to G. This is a closed subvariety of the space of all the maps ...

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### Teichmuller theory and moduli of Riemann surfaces

This is a sequel to my earlier question asking for references for Teichmuller theory and moduli spaces of Riemann surfaces.
In this connection, I have read Chapter 11 of the book Primer of mapping ...

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### Covering maps of Riemann surfaces vs covering maps of $k$-algebraic curves

In going from Riemann surface theory to the theory of algebraic curves over fields $k$ that are not necessarily $\mathbb{C}$, I would like to understand more about how the notion of a covering map ...

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### Uniformization theorem for Riemann surfaces

How does one prove that every simply connected Riemann surface is conformally equivalent to the open unit disk, the complex plane, or the Riemann sphere, and these are not conformally equivalent to ...

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### Teichmuller Theory introduction

What is a good introduction to Teichmuller theory, mapping class groups etc., and relation to moduli space of curves or Riemann surfaces?

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### Representations of surface groups via holomorphic connections

EDIT: Tony Pantev has pointed out that the answer to this question will appear in forthcoming work of Bogomolov-Soloviev-Yotov. I look forward to reading it!
Background
Let $E \to X$ be a ...

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### Riemann surfaces: explicit algebraic equations

Suppose $\Gamma$ is a nice discrete subgroup of $SL(2,\mathbb{R})$ such that the genus of the Riemann surface $\mathbb{H}/\Gamma$ is larger than 1. We know that this Riemann surface is also an ...

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### Intuition behind moduli space of curves

For a genus g compact smooth surface $M$, an algebraic structure is the same as a complex structure is the same as a conformal structure. So the moduli space of smooth curves should be the same as the ...

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### Dolbeault cohomology

Hello
I am trying to get a good book that explains the Dolbeault Cohomology, does anyone know of a good one?

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### If Spec Z is like a Riemann surface, what's the analogue of integration along a contour?

Rings of functions on a nonsingular algebraic curve (which, over $\mathbb{C}$, are holomorphic functions on a compact Riemann surface) and rings of integers in number fields are both examples of ...

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### Links between Riemann surfaces and algebraic geometry

I'm taking introductory courses in both Riemann surfaces and algebraic geometry this term. I was surprised to hear that any compact Riemann surface is a projective variety. Apparently deeper links ...