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2answers
269 views

The Schottky group and the fundamental group of a compact Riemann surface

I am quoting the following description from a paper, "...every compact Riemann surface can be obtained as the quotient $\mathbb{C}/\Gamma$ where $\Gamma$ is a Schottky group. The Schottky group of a ...
0
votes
2answers
88 views

Uniformization of n-Sheeted surfaces

Consider a 3-sheeted Riemann surface without a Z_3 symmetry. The first and second sheets are sewn together at an interval (u1,v1), and the second and third are sewn at (u2,v2). This is a Riemann ...
4
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0answers
90 views

tangent developable surface in $\mathbb{R}^3$

Let $C$ be a regular curve embedded in $\mathbb{R}^3$ (i.e. a real 1-dimensional manifold embedded in $\mathbb{R}^3$). Let $S$ be the union of its affine tangent lines: $$S=\bigcup\limits_{p\in ...
11
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4answers
562 views

Analogy between the nodal cubic curve $y^2=x^3+x^2$ and the ring $\mathbb{Z}[\sqrt{-3}]$?

I'm trying to motivate a bit of algebraic geometry in an abstract algebra course (while simultaneously trying to learn a bit of algebraic geometry), and I thought that it might be nice to present an ...
2
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1answer
146 views

Reference for homeomorphism between “analytic” compactification of $M_{g,n}$ and Deligne-Mumford compactification

There are several natural ways to endow the compactification of the space of marked Riemann surfaces $M_{g,n}$ ($2g+n\geq 3$), with a topology, which is defined using "differential geometric or ...
7
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1answer
142 views

Riemann surfaces of $w^3 = (z-a)(z-b)(z-c)$

I've been playing around with Riemann surfaces of cubics, and it seems to me that all coverings of the Riemann sphere from equations of the form $w^3 = q(z)$, where $q(z)$ is a cubic with three ...
1
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1answer
119 views

Image of the map induced on homology by a covering

I asked this question on math.se (http://math.stackexchange.com/questions/647930/image-of-the-map-on-homology-induced-by-a-covering), but it have not attracted much of attention. Let $X$ and $Y$ are ...
0
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1answer
118 views

Use of Jensen's inequality on a Riemann surface

Let $f:\mathbb{C}\to \mathbb{C}$ be entire and consider the composite function $g(z):=f(\sqrt{z^2 - 1})$ on $\mathbb{C}\setminus \big ((-\infty , -1]\cup [1,\infty )\big )$ on the branch of the square ...
1
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1answer
140 views

lift of antiholomorphic involution of Riemann surface to its Jacobian's cohomology

Start from a connected closed Riemann surface $\Sigma_g,$ obtained as the (symmetric) covering of an open and/or unoriented surface $\Sigma,$ namely $\Sigma=\Sigma_g/\Omega,$ where $\Omega$ is an ...
0
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2answers
98 views

Subharmonic function on a twice punctured complex plane

is the twice punctured complex plane parabolic or hyperbolic? In this sense: does $\mathbb{C}-\{0,1\}$ admit a nonconstant, negative, subharmonic function? Thanks,
1
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1answer
159 views

SU(2) Lefschetz decomposition for cohomology of Riemann surface Jacobian

Start with a closed Riemann surface with $g$ handles $\Sigma_g.$ I'm interested in the cohomology of its Jacobian $Jac(\Sigma_g)=T^{2g},$ in particular how the $SU(2)$ or $SL(2,\mathbb{R})$ Lefschetz ...
4
votes
1answer
177 views

hyperbolic orbifolds of small area

Is there a list of 2-dimensional hyperbolic orbifolds obtained from reflection groups (such as the double of a hyperbolic triangle with angles $\pi/p$, etc.) of small area, for instance area smaller ...
1
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0answers
65 views

Weierstrass points of discrete Riemann surfaces

Is there a notion of Weierstrass points for discrete Riemann surfaces? Any help or reference would be welcome! Thanks!
1
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0answers
71 views

How to find number of points at infinity of a Riemann surface

Let $X \subset \mathbb C^2$ be a Riemann surface with boundary $\partial X \subset \mathbb C^2$ and without compact components. Let $\bar X = X \cup \{p_1,\ldots,p_N\} \subseteq \mathbb CP^2$ be its ...
2
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1answer
171 views

A continuous version of Teichmuller uniqueness

By the Teichmuller uniqueness theorem, given a homeomorphism $f:X \rightarrow X$ where $X$ is the $n$-punctured sphere, there is a unique quasiconformal homeomorphism $g$ fixing $0$, $1$, and ...
1
vote
3answers
215 views

polynomial branched cover of the sphere with specified monodromy

We know by the Riemann Existence Theorem that any Riemann surface can arise holmorphically as the branched cover of a sphere: Which Riemann surfaces arise from the Riemann existence theorem? Do ...
0
votes
1answer
121 views

Residues and Mittag-Leffler sequence

Let $X$ be a compact Riemann surface, $\omega$ a meromorphic differential on $X$ and $f$ a meromorphic function on $X$ with poles only over the points $P_1,\dots,P_d$. The product $\;f\cdot\omega\;$ ...
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2answers
461 views

Is there an algorithm to compute efficiently the dessin d'enfant from a Belyi pair?

Let $(X,f)$ be a Belyi pair, i.e. a Riemann surface $X$ together with a morphism $f: X \to \mathbb{P}^1$, ramified only in $0,1, \infty$. Grothendieck's dessin d'enfant is the pre-image $G$ of the ...
15
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1answer
296 views

Avoiding integers in the spectrum of the Laplacian of a Riemann surface

Let $\Sigma^2$ be an orientable compact surface of genus $gen(\Sigma)\geq2$, and denote by $\mathcal M(\Sigma)$ the moduli space of hyperbolic metrics on $\Sigma$, i.e., Riemannian metrics of constant ...
3
votes
1answer
340 views

The Fuchsian monodromy problem

I want to understand the argument being made from equation 6.1 to 6.5 in this paper between pages 27-28 6.2, 6.4 and 6.5 are completely out-of-the-blue to me and I have no clue as to from where they ...
3
votes
1answer
441 views

A question about Abel-Jacobi map

Let $X$ be a Riemann Surface with genus $g$, $S^g(X)$ be the symmetric power of $X$ (which is naturally identified with the set of effective divisors of degree $g$). Let $A$ be the Abel-Jacobi map ...
1
vote
1answer
148 views

Quasi-isometric embeddings of the mapping class group into the Teichmuller space

Does there exist a quasi-isometric embedding $$MCG(S) \to (\mathrm{Teich}(S), d)$$ for $d$ any "known" distance on the Teichmuller space (i.e. Teichmuller, Weil-Petersson, Thurston...) ?
7
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0answers
303 views

What is known of the reverse math of Riemann-Roch?

I hope this is not too trivial, but I think this may be well known to someone (not me).
0
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1answer
140 views

Inverse “Riemann mapping” [closed]

The Riemann mapping theorem states, that any simply connected domain $U \subset \mathbb C$ can be conformally mapped to the open unit disk $D$. I.e. there is a Diffeomorphism $\Psi: D \to U$ such that ...
3
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1answer
139 views

Euclidean surfaces with conical singularities and cusped hyperbolic surfaces

Let $S$ be a compact orientable surface endowed with a singular euclidean metric $g$, with $n$ conical singularities $x_1,\ldots,x_n$. Construction 1: it is well-known that the conformal class ...
3
votes
1answer
288 views

Is there a Riemann-Roch like result for meromorphic differentials with all periods vanishing?

The classical Riemann-Roch theorem for Riemann surfaces makes a connection between the dimension of the space of meromorphic functions and the dimension of the space of meromorphic forms. Thus it is ...
1
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1answer
149 views

Reference on Deligne-Mumford compactness for Riemann surfaces

I am working with closed degenerating hyperbolic Riemann surfaces, and I try to understand the compactification of the moduli space. Looking in different books, notably the one of Hummel, I now get a ...
4
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0answers
442 views

Questions about dessin d'enfants, trees and their Shabat polynomials

This will be a series of questions, a few of which have been troubling me for quite a while now. Before I jump right in, let me first introduce a few notions which I will assume. (Note: All of these ...
1
vote
1answer
194 views

Request for some references exploring the connections of Riemann surfaces with medical imaging

I'd like to know some references for a beginner who has basic background in Riemann surfaces and differential geometry, and would like to start learning/working on more applied areas, medical ...
3
votes
1answer
178 views

Fixed points on Riemann surface

It is well known theorem that for a conformal mapping $\phi$ from a bounded and planar domain $\Omega$ to itself has three fixed points , then it must be identity mapping. However, I cannot find a ...
3
votes
3answers
290 views

Reference for hyperelliptic curves

I was reading a paper the other day that said that all automorphisms of a hyperelliptic curve are liftings of automorphisms of $\mathbb{P}^1$ operating on the set of branch points. Can someone point ...
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3answers
316 views

Are non-isomorphic covers of riemann surfaces also generally nonisomorphic as riemann surfaces?

Suppose you've got a Riemann surface $E$, and two topological covers $X,Y\rightarrow E$. Suppose $X,Y$ are nonisomorphic topological covers of $E$, then would you expect $X,Y$ as Riemann surfaces ...
8
votes
3answers
606 views

What prevents a cover to be Galois?

Let $f:X\rightarrow Y$ be a ramified cover of Riemann surfaces or algebraic curves over $\mathbb{C}$. My question is can one in terms of the ramification data of $f$, determine whether the cover is ...
0
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0answers
77 views

Existence of special pants decompositions for non-elementary representations into PSL(2,R)

A Theorem by Gallo, Goldman and Porter states the following: Let $S_g$ be a closed orientable surface of genus $g$ with fundamental group $\Gamma_g$, and fix a non-elementary representation ...
3
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1answer
156 views

Periods of translation surfaces

A translation surface is a Riemann surface equipped with a holomorphic 1-form $\omega$ and a Riemannian metric $g=\omega \bar \omega$ with conical singularities. It is well-known that there exists ...
12
votes
2answers
680 views

The fundamental group of a closed surface without classification of surfaces?

The fundamental group of a closed oriented surface of genus $g$ has the well-known presentation $$ \langle x_1,\ldots, x_g,y_1,\ldots ,y_g\vert \prod_{i=1}^{g} [x_i,y_i]\rangle. $$ The proof I know ...
8
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6answers
958 views

What is a branched Riemann surface with cuts?

Edit: Let me restate the main claim being made in these two papers, Consider the "branched" Riemann surface which has "n" sheets stuck along the intervals, $[z_i, z_{i+1}]$ for $i=1,..,2N$ then it ...
7
votes
1answer
242 views

Cohomology of the genus 2 mapping class group

Is the cohomology of the genus 2 mapping class group (that is, the cohomology of the moduli stack $M_2$ of genus 2 curves) known? I'd be interested in references. The rational cohomology is known to ...
2
votes
1answer
168 views

Approximation Runge's Theorem

Let $X$ be a Riemann Surface and $K$ a compact subset of $X$. Every holomorphic function in $K$ be uniformly approximable on $K$ by holomorphic functions on $X$ if $X-K$ have no connected component ...
2
votes
1answer
251 views

Hyperbolic structures on once punctured tori

I've been working on a problem about billiards in ideal hyperbolic polygons and I was thinking about how the problem for ideal quadrilaterals relates to closed geodesics on once punctured tori. My ...
16
votes
1answer
623 views

Octonions and the Fano plane.

Does the Fano plane mnemonic for octonion multiplication have any deeper meaning? http://upload.wikimedia.org/wikipedia/commons/2/2d/FanoPlane.svg The symmetry group of the Fano plane is PSL(2,7), ...
3
votes
1answer
388 views

A “Riemannian” analogue of Kobayashi metric?

Recall that Kobayashi metric is defined on any complex manifold $M$. This is a pseudo-metric according to which a tangent vector $v$ at $P$ has length at most $1$ if there is holomorphic map from ...
1
vote
0answers
239 views

Non-compact Riemann surfaces and affine algebraic curves

Is every connected non-compact Riemann surface biholomorphically equivalent to an affine algebraic curve in some ${\mathbb C}^n$? I suspect that surfaces of infinite genus probably are not but could ...
0
votes
1answer
136 views

Trivial Line Bundle-Riemann surfaces

What are the Hermitian metrics in a trivial line bundle on a Riemann surface X? I read that a Hermitian metric in the trivial line bundle is equivalent to a $\mathcal{C}^{\infty}$ weight function ...
0
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2answers
252 views

Green's function - Hyperbolic Riemann surface

A Riemann surface is said to be: -Potential-theoretically hyperbolic if it has a non-constant bounded subharmonic function. -Poincaré hyperbolic if it is covered by the unid disk. Are this ...
1
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1answer
139 views

Computing saddle connections in flat structures

Background: A polygonal billiards table $P$ with rational angles gives rise to a flat structure $S(P)$ in a standard way, described here. Curves of constant argument on $S(P)$ which start and end at a ...
2
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3answers
417 views

Hyperbolic Riemann Surface

Let $X$ be a compact Riemann surface and $x\in X$. Is $X - \overline{D(x,r_x)}$ hyperbolic?
1
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0answers
232 views

Complete curves in $M_g$ and Theta Characteristics

Let $g\geq 3$. Following the reference below, the locus of curves in $M_g$ with an effective even theta characteristic has codimension $1$. (Those are the curves $C$ with an effective line bundle $L$ ...
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1answer
373 views

A question from Otto Forster's book on Riemann surfaces

I am reading section 14, A Finiteness Theorem of Otto Forster's book Lectures on Riemann Surfaces, and come across a problem on Theorem 14.15 on page 117. In the proof Forster introduces a function ...
2
votes
1answer
203 views

Finiteness theorem for first-cohomology group of sheaf of holomorphic functions on compact Riemann surfaces

I have been reading Otto Forster's Lectures on Riemann Surfaces recently, and came across a question on section 15, Finiteness Theorem, which asserts that $H^1(X, \mathcal{O})$ is finite dimensional, ...