All Questions
Tagged with cv.complex-variables riemann-surfaces
28 questions with no upvoted or accepted answers
5
votes
0
answers
136
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Algebraic dependence of the elliptic functions
Let $\{f_i\}_{i=1}^{n}$ be $n$ elliptic functions in $\mathbb{C}$. We say that $f_1, \dots, f_n$ are algebraic dependent over $\mathbb{C}$ if there exists a polynomial of $n$ variables with constant ...
- 1,706
4
votes
0
answers
112
views
Elliptic integral as quantity associated with Riemann surface?
There are many elliptic integrals, so to show my point let me
just pick one of them (complete elliptic integral of the first
kind [1]):
$$K(k) = \int_{0}^{1} \frac {dx} {\sqrt{(1-x^{2})(1-k^{2}x^{2})}}...
- 5,230
4
votes
0
answers
306
views
Geometric interpretation of Theta functions and the Jacobi inversion problem
A great part of complex geometry and, algebraic geometry has been developed to address the theory of abelian integrals/functions. A very special problem that kept many great mathematicians busy was ...
4
votes
0
answers
229
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Real part of a holomorphic section of a vector bundle
Let $F\to M$ be a holomorphic vector bundle over a complex manifold $M$ and let $s:M\to F$ be a no-zero section. Let $E$ be the complexification of $F$, and suppose that $E$ admits a holomorphic ...
- 163
4
votes
0
answers
157
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Modulus of an annulus with a cut
Let $A_r$ be a complex annulus of modulus $r>0$ obtained from a $1\times r$ rectangle in $\mathbb C$ with vertices $A=0$, $B=r$, $C=r+i$, $D=i$, by identifying isomterically $AB$ with $DC$.
Let us ...
- 14.3k
3
votes
0
answers
195
views
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)},$ ...
- 954
3
votes
0
answers
309
views
Reference request: basics about modular curves
Where can I find a reference (with carefully written proofs) for basic facts about modular curves? Namely:
Congruence subgroups
The open modular curve $Y_\Gamma$ admits the structure of a Riemann ...
- 31
3
votes
0
answers
354
views
Hurwitz Spaces and Rauch Variational Formulas
I have read in some papers about Rauch-type variational formulas on Hurwitz spaces, and I would like to know what exactly is the theory behind them.
A Hurwitz Space $H_g^d$ is the space of coverings ...
2
votes
0
answers
358
views
Triangulating Riemann surfaces by using non-constant meromorphic functions
Let $X$ be a connected Riemann surface, i.e. $X$ is a one dimensional connected complex manifold (Hausdorff and second-countable as a topological space). The following is a classical result:
Theorem (...
- 395
2
votes
0
answers
178
views
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 ...
- 377
2
votes
0
answers
154
views
Algebra of meromorphic functions on a Riemann surface
Let $C$ be compact Riemann surface of high genus. Let $p$ be a general point on $C$ and $z$ a local coordinate around $p$.
Given a meromorphic function on $C$, regular outside $p$, we can look at its ...
- 2,384
2
votes
0
answers
61
views
Criteria for a limit to be a proper function
This question is obviously broad; turning this broadness into something sharp is part of the problem.
Given a sequence of functions defined on a Riemann Surface $R$, valued in $\Bbb C^2$, what ...
- 779
2
votes
0
answers
127
views
Vector bundles over Riemann surfaces of infinite genus
Has any work been done on the description of (finite rank) holomorphic vector bundles over Riemann surfaces of infinite genus?
Is there a theory of moduli spaces of such objects?
This question is ...
- 23.3k
2
votes
0
answers
138
views
Equivariant meromorphic functions
Let $G \subset \rm{PSL}_{2}\mathbb{C}$ be a subgroup of the Mobius group of the 2-sphere $S^2$, and suppose that $G$ also acts on a second surface $M^2$ by automorphisms.
Does there exist a ...
- 21
2
votes
0
answers
441
views
Marten's proof of torelli theorem
I am trying to read the proof of torelli theorem by Henrik H.Martens "A new proof of torelli's theorem" Annals of mathematics vol78 no. 1 .The proof seems to me like using mysterious combination of 3 ...
- 2,106
1
vote
0
answers
126
views
Canonical basis of cycles of Riemann surfaces
Let $\Gamma$ be the compact Riemann surface defiend by the algebraic curve
$$
f(x, y) = y^n + a_1(x)y^{n-1} + a_2(x) y^{n-2} + \dots + a_{n-1}(x)y + a_n(x) = 0,
$$
where $a_1(x), \dots, a_n(x)$ are ...
- 89
1
vote
0
answers
137
views
Existence of meromorphic one-form with a fixed order pole
Let $X$ be a compact Riemann surface of genus $g$. We identify it with $4g$ polygon $\{a_i,b_i, a_i^\prime, b_i^\prime\}_{i=1}^g$. For a meromorphic 1 form $\omega$, we define
$A_i(\omega)= \int_{...
- 541
1
vote
0
answers
77
views
Computing some closed trajectories of meromorphic quadratic differentials
I'm learning about meromorphic (!) quadratic differentials on Riemann surfaces, and would like to determine the closed trajectories [EDIT: I mean closed geodesics, not just closed trajectories; ...
- 121
1
vote
0
answers
47
views
Uniformization of triangulation on a sphere up to Moebius transformations
This is not the most precise question but rather a hope that someone has seen something like this.
I am given a triangulation of the 2-sphere $S^2$ which I only know up to Moebius transformations. I ...
- 41
1
vote
0
answers
119
views
Differentials on tori realised as double of annuli
In this question it was described how to realise a torus as the double of an annulus Explicit construction of mirror surface and complex double for an annulus.
In short, the torus is realised ...
- 1,435
1
vote
0
answers
215
views
Coordinate charts on converging Riemann surfaces
Let $S$ be a $2-$dim manifold and $q \in S$. Furthermore, let $j_{n}$ be a sequence of complex structures on $S$ converging in $C^{\infty}_{\text{loc}}$ to a complex structure $j$ on $S$ as $n\...
- 11
1
vote
0
answers
116
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 ...
- 1,329
1
vote
0
answers
87
views
Are morphisms of intersection graphs of circle packings harmonic?
Let $P$ and $Q$ be circle packings on compact Riemann surfaces (along with some Riemannian metrics) $X$ and $Y$. Let $f\colon X\to Y$ be a conformal map taking each circle in $P$ to a circle in $Q$. ...
- 3,079
0
votes
0
answers
66
views
Uniformization and constructive analytic continuation of Taylor-Maclaurin series
Context. In their paper, "Uniformization and Constructive Analytic Continuation of Taylor Series", Costin and Dunne present a constructive method to greatly increase the accuracy of a ...
0
votes
0
answers
73
views
Conformally embedding a finite Riemann surface of genus g
Let $R$ be a compact Riemann surface of genus $g$ and let $S \subset R$ be a Riemann subsurface. Theorem B in Maskit's paper says that we can embed $S$ into a compact Riemann surface $P$ of genus $g$ ...
- 1,159
0
votes
0
answers
204
views
Expansion around a singular point of a multivalued meromorphic function (due to Riemann/Cauchy)
In Riemann's publication about Abelian functions
'Theorie der Abelschen Functionen' (Here the original paper in german)
at the end of Chapter 4, part 2 is clamed that for every Riemann
surface $T$ and ...
- 5,998
0
votes
0
answers
93
views
Meromorphic functions on a modular curves of genus $0$ that take each value exactly once
Let $\Gamma$ be a congruence subgroup of $\operatorname{SL}_2(\mathbb Z)$, and let $\mathfrak H$ be the upper half-plane. Let $X(\Gamma)$ be the compactification of $\Gamma\backslash\mathfrak H$. Then ...
- 2,375
0
votes
0
answers
268
views
Is the absolute value of the j-invariant bounded from below on an annulus
Let $j:\mathbf{H}\to \mathbf{C}$ be the $j$-invariant. It's a modular function for $\Gamma(1) = \textrm{PSL}_2(\mathbf{Z})$.
For $\epsilon>0$ small, let $B(\epsilon)$ be the image of the strip $$\{...
- 225