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0
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1answer
68 views

A question for hyperbolic metric in the proof for Bohr's lemma

Recently I was reading an interesting proof for Bohr's lemma by the tool of hyperbolic metric, however I have a following question: Given a holomorphic map $f$ on $D$, $f(0)=0$, and $|f|<1$ on ...
6
votes
2answers
212 views

Rational curved lying in the boundary of Deligne-Mumford compactification $\bar M_g$

Let $\bar M_g$ be the Deligne-Mumford compactifiction of the moduli space of complex genus $g$ curves $M_g$. Is this correct that through every point of the boundary $\bar M_g\setminus M_g$ passes a ...
3
votes
1answer
243 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 ...
0
votes
0answers
81 views

Branch points lie in $\mathbb P^1\left(\overline{\mathbb Q}\right)$ [migrated]

Preamble: Probably my question will be highly downvoted and soon closed, because it is too simple. However I will make a tentative because I'm working on this for several days without finding any ...
2
votes
2answers
206 views

Etale covers of a hyperelliptic curve

Let $X$ be a hyperelliptic curve of genus at least two. Let $Y\to X$ be a finite etale morphism with $Y$ connected.Then $Y$ is a smooth projective connected curve. Is $Y$ hyperelliptic? More ...
2
votes
0answers
43 views

Uniform estimate for the cauchy-riemann equations on a hyperbolic riemann surface

I have been trying to find the answer to this question in the literature, but have not succeeded. The question is as follows. Suppose $X$ is a Riemann surface that admits a green's function (i.e. ...
0
votes
2answers
184 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 ...
3
votes
0answers
99 views

Criterion for a subgroup of $PSL_2(\mathbb R)$ to be Fuchsian

Let $\Gamma$ be a finitely generated subgroup of $PSL_2(\mathbb R)$. I'm looking for effective criteria (idealy necessary and sufficient but just necessary would be a good start) ensuring that ...
1
vote
2answers
222 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
71 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 ...
3
votes
0answers
62 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
votes
4answers
447 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
votes
1answer
128 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 ...
3
votes
1answer
326 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 ...
6
votes
1answer
108 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
vote
1answer
98 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 ...
11
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3answers
519 views

Conformal Welding Reference

I'm looking for a reference for the following fact: given two Riemann surfaces and an identification of their boundaries, once I topologically glue the surfaces together there exists a unique ...
0
votes
1answer
88 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
vote
1answer
114 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
votes
2answers
63 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
vote
1answer
112 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 ...
12
votes
3answers
516 views

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 ...
4
votes
1answer
149 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 ...
6
votes
3answers
1k views

Is a non-compact Riemann surface an open subset of a compact one ?

Let $X$ be a non-compact holomorphic manifold of dimension $1$. Is there a compact Riemann surface $\bar{X}$ suc that $X$ is biholomorphic to an open subset of $\bar{X}$ ? Edit: To rule out the case ...
1
vote
0answers
59 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
vote
0answers
65 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
votes
1answer
160 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
187 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 ...
6
votes
2answers
385 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 ...
0
votes
1answer
107 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 $\;f\cdot\omega\;$ ...
9
votes
0answers
137 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 ...
11
votes
2answers
520 views

Belyi functions on non-compact surfaces; or: Building Riemann surfaces from equilateral triangles

Some background on (compact) Belyi surfaces $\newcommand{\Ch}{\hat{\mathbb{C}}}$ A compact Riemann surface $X$ is called a Belyi surface if there exists a branched covering map $f:X\to \Ch$ such that ...
1
vote
2answers
366 views

How do you find the genus of a Fuchsian group derived from a quaternion algebra?

Let $G$ be a Fuchsian group with normalizer $N(G)$ inside $PSL(2,13)$ Due to the Hurwitz formula, it suffices to find a presentation of $G$ of the form: $$\langle ...
0
votes
0answers
66 views

Periods of holomorphic $1$-forms on compact surfaces

There is an old theorem from Otto Haupt which gives necessary and sufficient conditions for $\alpha$ to be the periods of a holomorphic $1$-form. (see ...
2
votes
1answer
327 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
120 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
289 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
votes
0answers
153 views

Representation of the group of automorphisms on the holomorphic forms

Let $X$ be a compact Riemann surface and $G = Aut(X)$ be its group of automorphisms (biholomorphisms between $X$ and $X$). It is known that $G$ acts on the space $Harm(X)$ of all harmonic forms and ...
0
votes
1answer
126 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
votes
1answer
102 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
257 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 ...
3
votes
1answer
169 views

number of ribbon structures (or punctured surfaces) on a graph

Suppose $G$ is connected undirected graph. Does the calculation of the number of topologically distinct punctured surfaces that can arise from putting a ribbon structure on $G$ exist in the ...
1
vote
1answer
129 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 ...
3
votes
0answers
366 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
178 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
163 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
171 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 ...
2
votes
2answers
608 views

A question in R.C.Penner's paper about Teichmuller space

In R.C.Penner "Decorated Teichmuller theory of boarded surface", on Page 7 and 8, it says that (without proof) the Teichmuller space of surface with $s$ labelled punctures and $r$ labelled boundary ...
6
votes
3answers
299 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 ...
7
votes
3answers
463 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 ...