# Questions tagged [riemann-surfaces]

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

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### Opers and global differential operators

This is a follow up question to a previous question of mine and my thought of answer to it.
Given a (compact) Riemann surface $\Sigma$, a $SL(n,\mathbb{C})$-oper is a rank $n$ holomorphic vector ...

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### Counting simple closed curves

I'm currently trying to understand how to count simple closed curves. I've been reading Alex Wright's survey (https://arxiv.org/pdf/1905.01753.pdf). However, I don't feel like I'm getting the big ...

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### Small pants in arithmetic hyperbolic surfaces of high degree

Does the following statement hold:
Statement: For any $\epsilon > 0$, there exist a number field $k$ of degree $d_{\epsilon}$ over $\mathbb{Q}$ and an arithmetic hyperbolic surface $\Gamma$ ...

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### Difference of two functions with constant mean curvature

Define the set $\Omega := (-\epsilon,\epsilon) \times (-1,1)^{n-1}$, and define
$\Gamma := \{-\epsilon,\epsilon\} \times (-1,1)^{n-1} \subset \partial \Omega$.
Suppose I have two functions $u,v \in C^...

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### Cutting up the Bring surface into six pairs of pants

The Bring sextic, with 120 automorphisms, is the numerically most symmetric compact Riemann surface of genus 4. To cut it up into six pairs of pants, we need to cut along nine disjoint geodesic loops....

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### Purely analytic proof of the Nielsen-Thurston classification theorem

I hope this question is appropriate for the site. I've been looking at the expositions of Bers' proof of the Nielsen-Thurston classification given in Hubbard's Teichmüller Theory
and Applications to ...

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109 views

### singularities of the theta divisor $\Theta$

By $\Theta_{sing}$ we denote the singularities of the theta
divisor $\Theta$ of the Jacobian variety $J(R)$ of a compact Riemann surface
R of genus $g\ge 4$. Then
$$
\text{dim} \Theta_{sing}=
\{\...

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### Can learning Riemann surfaces be more beneficial than numerical analysis for an analyst?

I am in master program of mathematics, specialized in PDE and numerical analysis. Now I am trying to decide which classes to take for next semester. Of course I want to become an expert in my field, ...

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### Complex plane minus Cantor set admits non-constant bounded harmonic function

Let $K\subset [0,1]$ denote the usual 1/3 Cantor set. I know that $\mathbb{C}\backslash K$ has no non-constant bounded analytic function, since the singularity $K$ can be removed. However, a statement ...

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110 views

### A differential geometric proof of the Riemann--Roch theorem for lines [duplicate]

I am looking for a differential geometric version of the proof of the Riemann--Roch theorem
for Riemann surfaces, that is, $1$-dimensional compact complex manifolds. The proofs one usually finds are ...

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56 views

### Coordinates for Laminations: geometric versus shear

Let $S$ be an orientable surface with a triangulation T.
A lamination $\ell$ is a simple closed curve on $S$, up to isotopy. We will assume that $\ell$ is drawn in such a way that it intersects the ...

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### Linear system corresponding to a holomorphic embedding from compact Riemann surface to projective space is complete

Le $X$ be a compact Riemann surface and $\phi$ be a holomorphic embedding of $X$ into projective space $\mathbb{C}\mathrm{P}^n$ which is induced by $(f_0,\dots , f_n)$. Then there is a linear system ...

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77 views

### What is meromorphic differentials like on Riemann Sphere? [closed]

There is a proposition that every meromorphic differential on Riemann Sphere (or $\mathbb{P}^1 = \mathbb{C} \cup \{ \infty \}$) can be written as $f dz$ where $f$ is a meromorphic function on $\mathbb{...

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190 views

### Descent of vector bundle along branched cover of curve

Suppose $\pi:C'\to C$ is a branched cover of compact Riemann surfaces such that the associated extension of function fields is Galois with group $G$ -- so that $\pi$ presents $C$ as the quotient $C'$ ...

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### Riemann mapping theorem with boundaries and corners

I was reading this paper by Hollands and Yazadjiev, where on page 760, they claim that since $\hat{M}$ is an orientable simply connected analytic $2$-manifold with boundaries and corners, we may map ...

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### Proving the Immersion part of an Embedding

Trying to see the proof of embedding the Jacobian of a Compact Riemann Surface $X$ using Theta functions. So, using the Theta divisor we have the corresponding line bundle say $L$, we want to prove ...

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263 views

### Some questions about Hitchin's self-duality paper

I am reading this paper (The self-duality equations on a Riemann surface by N. Hitchin), and I don't understand a few things in page 67. In proof of Theorem 2.1 after Equation 2.4, he gives the ...

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201 views

### Moduli, Teichmüller spaces and mapping class group of a sphere with four punctures

In the complex analytic setting, it is easy to see that the moduli space of a sphere with four punctures is $\mathcal{M}=\mathbb{CP}^1 / { 0,1,\infty }$, since I can use a Moebius transformation to ...

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56 views

### How large can a planar triangulation be that embeds bi-Lipschitz in a ball of $\mathbb{R^3}$?

Let $G$ be a finite plane graph all bounded faces of which are triangles. For example, $G$ could be
the 1-skeleton of a triangulation of a topological disc. Let $f: V(G) \to \mathbb{R}^3$ be a bi-...

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### The space $M_g$ with the complex structure induced from $T_g$ is a coarse moduli space for compact Riemann surfaces of genus $g$

Proposition: The space $M_g$ with the complex structure induced from $T_g$ is a coarse moduli space for compact Riemann surfaces of genus $g$.
In the proof of (1), I wonder why the holomorphic map $\...

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### Conceptual proof of classification of surfaces?

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 ...

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### When is a holomorphic map from a complex surface to a Riemann surface is a holomorphic family of Riemann surfaces?

A holomorphic family of Riemann surfaces of type $(g, n)$ is a triple $(M, \pi,B)$ defined as follows:
$\bullet$ M is a $2$-dimensional complex manifold (topologically, a $4$-manifold);
$\bullet$ $B$...

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203 views

### A generalization of polynomial algebra on a Riemann surface

Let $M$ be a $1$-dimensional complex manifold. Let $A$ be the space of all holomorphic functions $f:M\to \mathbb{C}$ such that either $f$ is a constant function or every level set $f^{-1}(c)$ is a ...

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154 views

### Visualizing hyperbolic metric of punctured sphere

Uniformization of the 3-punctured sphere generates a "pants" configuration with three legs narrowing down to cusps. This is supposed to have a metric of constant negative curvature, and I can see this ...

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### Caratheodory's theorem in any compact Riemann surface

The classical Caratheodory theorem states that if $G$ is a simply connected domain in the plane, whose boundary is a Jordan curve, then the Riemann uniformization extends continuously to a ...

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279 views

### Isomorphic Jacobian Varieties Just Like Abelian Varieties — Torelli's Theorem

Torelli's theorem states:
Let $R$, $R'$ be compact Riemann surfaces of genus $g$, $J(R)$, $J(R')$ their Jacobian varieties, $\Theta$, $\Theta'$ their respective theta divisors. The Riemann surfaces ...

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208 views

### Riemann-Hurwitz for real maps

Let $S$ be a (compact, connected) Riemann surface of genus $g$ and $f: S\to \mathbb CP^1$ be a degree $d$ meromorphic function. Then the Riemann-Hurwitz formula tells us that the number of ...

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156 views

### Closed simple curves in $\mathbb{R}\mathbb{P}^2$

EDIT: The well known Jordan curve theorem says: let $C\subset S^2$ be a closed simple curve on the 2-sphere. Then its complement $S^2\backslash C$ consists of two connected components, both ...

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155 views

### A generalization of Jordan-Schoenflies theorem on simple plane curves

The well known Jordan-Schoenflies theorem says: let $C\subset \mathbb{R}^2$ be a closed simple curve. Then there exists a homeomorphism $f\colon\mathbb{R}^2\to \mathbb{R}^2$ such that $f(C)$ is the ...

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### Embedding of a Compact Riemann Surface into a Projective Space

Trying to fill up details of a proof I learnt some time ago: $X$ a compact Riemann surface. We want to show that for large enough degree of a divisor $D$, the map
$X\rightarrow \mathbb{P}(H^0(X,\...

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76 views

### Generalising definition of Hurwitz number of compactified moduli space of curve

I am asking mostly for reference if such a definition exists in the literature. I am also interested in the count if it appears somewhere.
Let $\mu:=(\mu_1,\ldots , \mu_n)\vdash d$ for positive ...

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### Can a closed horizontal trajectory on a Riemann surface be freely homotopic to $0$?

Let $R$ be a Riemann surface and let $\varphi=\varphi(z)dz^2$ be a nonzero holomorphic quadratic differential on $R$. A differentiable curve $\gamma$ on $R$ is called a horizontal trajectory if along ...

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### Existence of a symplectic form in a given class for the product of Riemann surfaces

Let $a \in H^2(M, \mathbb{R})$ be a cohomology class of a closed manifold $M$ of dimension $2n$. For the cohomology class $a$ to represent a symplectic form on $M$, we must have $a^n \neq 0$. This is ...

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### Is Wronskian a line bundle for Riemann surfaces?

Suppose $f_1,\dots,f_g$ are holomorphic functons on a domain $U\subset\mathbb{C}$. By the Wronskian determinant $f_1,\dots,f_g$ one means the determinant of the matrix of derivatives $f_k^{(m)},$ ...

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### How did Riemann prove that the moduli space of compact Riemann surfaces of genus $g>1$ has dimension $3g-3$?

Consider the moduli space $M_g$ of compact Riemann surfaces (i.e., smooth complete algebraic curves over $\mathbb{C}$) of genus $g$ for some $g>1$. I'm interested in knowing how Riemann proved that ...

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### Area of balls on flat surfaces

Let $S$ be a closed surface of genus $g \geq 2$. Define $\mathrm{Flat}(S)$ to be the set of marked flat metrics on $S$ with cone angles $2\pi+k\pi$ for $k\geq 0$. It is well-known that these all come ...

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51 views

### Conformal embedding between flat cylinders

Consider a cylinder $C_R$ given by the quotient of the strip $\{z\in \mathbb C|0< Re(z)< R\}$ by the $\mathbb Z$ translation action generated $z\mapsto z+i$, which is endowed with the natural ...

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### Number of curves in an admissible system of Jordan curves on a surface

Consider a compact Riemann surface of genus $g\geq2$. An admissible system of Jordan curves is a finite collection of Jordan curves $\{\gamma_1,\cdots,\gamma_n\}$ such that
they are nonintersecting ...

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### Spaces of Bordered (Ordinary, Spin- or Super-) Riemann Surfaces

It is known from the work of Deligne and Mumford that the "space" of punctured/marked Riemann surfaces is a Deligne-Mumford stack. I have few questions regarding similar statement for the spaces of ...

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### Degree of the direct image of a line bundle

Consider a $n$-branched cover $\pi:S\rightarrow M$, where $S$ and $M$ are both algebraic curves. If $\pi_{0}: L\rightarrow S$ is a line bundle over $S$, we can define the bundle $\pi_{*}L$ on $M$ ...

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### Conjugate points for a family of generalized curves

Let $(M,g)$ denote a compact smooth Riemannian manifold with boundary and let $\mathscr F$ denote a family of smooth curves $\gamma$ such that they solve
$$ \nabla^g_{\dot \gamma} \dot\gamma = F(\...

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### Effect of plumbing a surface on the marked length spectrum

First I'll recall the plumbing procedure.
Let $M$ be a noded Riemann surface with nodes $p_1,\dots, p_n$. There is a family of pairwise disjoint neighbourhoods of each node $U_i$ that has coordinates ...

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### Stability of holomorphic vector bundles

I'm reading the book "R. O. Wells Jr. - Differential Analysis on Complex Manifolds" and on page 247 the author claims two things about the stability of holomorphic vector bundles that I'm struggling ...

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### What is a Lipschitz continuous map between Riemann surfaces in Jost's book Compact Riemann Surfaces?

This appears in the section 3.7 of the book Compact Riemann Surfaces by Jurgen Jost, right after Lemma 3.7.3. The exact words are
Now let $v:\Sigma_1\to\Sigma_2$ be a Lipschitz continuous map. ...

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### How the hyperbolic metric changes when we add a puncture?

Suppose we have a surface of a finite genus, without boundary with a finite number of punctures. This surface admits a unique hyperbolic metric of curvature $-1.$
If I add a puncture somewhere, the ...

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### What is the definition of Sobolev maps between surfaces used by Jurgen Jost in his book Compact Riemann Surfaces?

In the book Compact Riemann Surface by Jurgen Jost, the notion of $W^{1,2}(\Sigma_1,\Sigma_2)$, where $\Sigma_1,\Sigma_2$ are Riemann surfaces, is used. But I can't figure out what definition he is ...

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### Kinds of differentials and algebraic groups

This Wikipedia article mentions that the analogues of differentials of the first/second/third kind for algebraic groups are abelian varieties/algebraic tori/linear algebraic groups. I guess ...

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### Automorphism groups of projective bundles on Riemann surfaces

Let $X$ be a compact Riemann surface, and let $E \to X$ be a stable vector bundle of rank $r+1$. Then we know that $P = \mathbb{P}(E)$ comes from an irreducible representation of $\pi_1(X)$ into the ...

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### Existence of holomorphic coverings having small degree

Let $\Sigma$ be a closed Riemann surface of genus $g$. In the book of Farkas and Kra, they prove that there exists a holomorphic covering map $F : \Sigma \to \mathbb{S}^2$ of degree less than or equal ...

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327 views

### Why 2-tori with Gauss curvature $\geq -1$ cannot collapse to segment?

Let $\{(\mathbb{T}^2,g_i)\}_{i=1}^\infty$ be a sequence of 2-dimensional tori with smooth Riemannian metrics with Gauss curvature at least $-1$. It was explained in the final answer to the post Gromov-...