The riemann-surfaces tag has no wiki summary.
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1answer
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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 ...
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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 ...
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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 ...
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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. ...
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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 ...
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2answers
221 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 ...
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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 ...
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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 ...
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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
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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 ...
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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 ...
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1answer
97 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 ...
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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 ...
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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 ...
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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,
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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 ...
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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 ...
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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
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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 ...
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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 ...
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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 ...
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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\;$ ...
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2answers
384 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 ...
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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
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1answer
242 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 ...
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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
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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
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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...) ?
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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).
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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
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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
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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 ...
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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 ...
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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 ...
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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
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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 ...
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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 ...
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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 ...
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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 ...
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68 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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0answers
92 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 ...
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2answers
457 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 ...
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6answers
682 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
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1answer
213 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
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1answer
135 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
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1answer
195 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 ...
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1answer
547 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
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1answer
325 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 ...
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0answers
211 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 ...
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1answer
112 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 ...
