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3
votes
1answer
327 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
212 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
114 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
votes
2answers
185 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
vote
1answer
109 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
votes
3answers
377 views

Hyperbolic Riemann Surface

Let $X$ be a compact Riemann surface and $x\in X$. Is $X - \overline{D(x,r_x)}$ hyperbolic?
1
vote
0answers
196 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 $L$ ...
1
vote
1answer
291 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
159 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, ...
4
votes
2answers
546 views

The existence of meromorphic functions on Riemann surfaces

In Miranda's book on algebraic curves and Riemann surfaces, Miranda writes: It is a basic and highly nontrivial result that a compact Riemann surface has nonconstant meromorphic functions on ...
17
votes
0answers
358 views

On Determinants of Laplacians on Riemann Surfaces

History of the Formula: In their famous paper "On Determinants of Laplacians on Riemann Surfaces" (1986), D'Hoker and Phong computed the determinant of the Laplacian $\Delta_n^+$ on the space $T^n$ of ...
1
vote
0answers
53 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 $Q$. ...
0
votes
1answer
141 views

Fuchsian groups and their normalizers

Let $\Gamma \leq PSL_2(\mathbb{R})$ be a Fuchsian group. What is the relation between $N(\Gamma) = \{ \alpha \in PSL_2(\mathbb{R}) \mid \alpha \Gamma \alpha^{-1} = \Gamma \}$ and $Aut(\Gamma ...
8
votes
1answer
266 views

Injective morphism from an elliptic curve to $\mathbb CP^2$.

Let $E$ be the elliptic curve $x^3+y^3+z^3=0$. Question. Are there injective morphisms $E\to \mathbb CP^2$ of arbitrary high degree? Comments. 1) There are injective morphisms $E\to \mathbb CP^2$ ...
0
votes
2answers
176 views

Fuchsian groups and automorphisms of Riemann surfaces

Let $\Gamma \subseteq PSL_2(\mathbb{R})$ be a Fuchsian group, possibly containing elliptic elements. Is it true that $N(\Gamma) / \Gamma$, where $N(\Gamma)$ the normalizer of $\Gamma$ in ...
13
votes
3answers
783 views

Injective morphism from curves to $\mathbb CP^2$

Is there a smooth compact complex curve that does not admit an injective holomorphic map to $\mathbb CP^2$ ? Let me stress, that the image of the curve in $\mathbb CP^2$ can have singularities. I ...
0
votes
1answer
190 views

When is a cyclic cover hyperelliptic?

Let us work over the complex numbers for simplicity. Consider a curve $C$ presented as a cyclic cover of some lower genus curve $C'$. When $C'$ has genus $0$, we can write $C$ as the normalization of ...
0
votes
1answer
173 views

Reference for notation $H^0(C, mK)$

I am reading the draft of "Equations of Riemann Surfaces of Genus 4, 5 and 6 wih Large Automorphism groups" and the author starts using the notation $H^0(C, mK)$ on page 3, without explaining it. As ...
1
vote
1answer
168 views

Analytic curve on Riemann surface

Suppose there is a closed analytic curve $C$ on a Riemann surface $S$, that is the image of a map $\gamma$ from the equator $E$ of the Riemann sphere to the surface $S$ which is a restriction of a ...
12
votes
1answer
448 views

Essential uniqueness of the real-analytic structure on $\mathbb R$

It is well-known that any $C^k$-smooth $1$-manifold homeomorphic to $\mathbb R$ is $C^k$-diffeomorphic to $\mathbb R$. The cases of $k\in{\mathbb N}\cup$ {$\infty$} may all be handled similarly by ...
11
votes
2answers
523 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 ...
0
votes
1answer
165 views

representation of teichmuller space Teichmuller space

I want to study representation of teichmuller space of surface of genus g in psl(2,R). can you suggest any good references.
11
votes
1answer
422 views

Complex curves covered by smooth plane curves

Question: Is it true that for every smooth compact complex curve $C$ there exists a smooth curve $C'$ in $\mathbb CP^2$ that admits a non-trivial morphism (i.e. holomorphic map) $C'\to C$? ...
2
votes
1answer
262 views

Automorphisms of higher-genus Riemann surfaces act nontrivially on homology (Reference Request)

The following is a folklore result : Let $X$ be a compact Riemann surface of genus at least $2$ and let $f : X \rightarrow X$ be a biholomorphism. Then $f$ acts nontrivially on $H_1(X;\mathbb{Z})$. ...
1
vote
1answer
149 views

Vortex equations on cylinder

Solutions to the vortex equations for a closed Riemann surface are well known (moduli space is a symmetric power). What do we know about solutions on surfaces with boundary or non compact surfaces? In ...
3
votes
3answers
293 views

Is the set of all smoothed closed simple curves on $\mathbb{R}^2$ a manifold?

In the studies of active contours they describe the set of all simple smooth closed curves on $\mathbb{R}^2$ to be a Riemannian Manifold $M$. The tangent space at a curve $c$, $T_cM$ is a set of ...
11
votes
2answers
394 views

A “holomorphic” Peano curve?

A Peano curve is a continuous map $[0,1]\to [0,1]^2$ whose image is the whole square. I would like to know if on can obtain "holomorphic" Peano curves. Namely, is it possible to find a continuous ...
2
votes
1answer
228 views

growth of energy of eigenfunctions on hyperbolic surface

I am looking to the behavior of eigenfunctions associated to small eigenvalues on a degenerating hyperbolic surface. Let $(\Sigma_n, h_n)$ a sequence of compact surfaces with area equal to $1$ and ...
5
votes
1answer
309 views

How does a moduli interpretation give an analytic object an algebraic structure?

I remember hearing this in other contexts, but I encountered it again when reading Elkies' paper "Shimura Curve Computations", where on page 10, he says that: "We now return to the Shimura curves ...
6
votes
2answers
214 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 ...
0
votes
1answer
108 views

Regular (or complex analytic) functions on M_3

Let $M_3$ be the moduli space of genus three curves over $\mathbb C$. Are there non-constant regular functions of this space? What about complex analytic functions? This question is prompted by the ...
1
vote
2answers
368 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 ...
10
votes
3answers
453 views

Does every smoothly embedded surface $\mathbb{R}^3$ inherit a natural complex structure, and if so, which one?

Smoothly embed a genus g surface in $\mathbb{R}^3$, and pick a normal vector pointing "out" of the surface at each point. Then on each tangent plane, I have a map which rotates the tangent plane 90 ...
1
vote
2answers
258 views

Complete metric on a Riemann surface with punctures

If we have a Riemann surface with punctures of negative Euler characterstisc, how can one define a complete hyperbolic metric? I know that in this case the universal cover is the hyperbolic plane ...
0
votes
0answers
139 views

Limit sets of Fuchsian groups and relation between lifts to $H$ of homotopic maps between hyperbolic Riemann surfaces

Let $f,g : X \to Y$ be homotopic (quasiconformal) maps between hyperbolic Riemann surfaces $X,Y$. Consider their (unique) lifts $\tilde{f},\tilde{g}: H\to H$ , that fix $0,1,\infty $. My question is : ...
6
votes
3answers
504 views

How do you see that higher genus surfaces are not homogeneous?

I am trying to get some intuition about why the torus and the sphere are the only surfaces which can be realised as homogeneous spaces. On the one hand, I know this is true because there is the ...
3
votes
1answer
268 views

Strata of quadratic differentials from rational billiards

Given a quadratic differential $q$ on a surface of genus $g$, we say that $q\in \mathcal Q(k_1,\ldots,k_n)$ if $q$ has $n$ distinct zeroes of order $k_1,\ldots,k_n$ respectively. The set $\mathcal ...
0
votes
0answers
153 views

Quick references/sources for the hyperbolic Riemann Surfaces with boundary

Hello, Here I am asking for a reference for the universal cover of hyperbolic Riemann surfaces with geodesic boundaries. For example, I want to know how the universal cover/fundamental domain of ...
6
votes
1answer
187 views

Is the class of $k$-gonal curves dominant

Before I start, let me make a note on terminology. Curves are always smooth projective connected curves over an algebraically closed field of characteristic zero. Let $\mathcal C$ be a class of ...
7
votes
1answer
1k views

What is the current state of the mathematics of Higgs fields?

Topical. I know there are good mathematical theories in which "Higgs" is used, in a geometrical sense. Would someone care to explain? To clarify, I'd like to know about Higgs bundles on Riemann ...
9
votes
2answers
368 views

Embedding a Riemann surface in the sphere

Assume we have a Riemann surface, the underlying topological surface of which is a sphere with (possibly uncountably many) points removed. Can we always conformally embed this Riemann surface in the ...
5
votes
2answers
816 views

What is / are the softwares to use to draw surfaces of the form of a two or three-holed torus , or torus, or torus with cusps attached to it?

I am trying to draw surfaces with complete hyperbolic structures and surfaces which are topologically tori. The hyperbolic surfaces I need to draw are torus with one or two holes on it, or torus with ...
3
votes
4answers
509 views

Softwares for drawing hyperbolic surfaces , closed, with boundaries or with punctures ?

In a paper I am in the process of writing in LaTeX, I need to draw and incorporate some diagrams of hyperbolic surfaces in my LaTeX document. Is there any software I can use to draw hyperbolic ...
7
votes
4answers
629 views

A question on deformations of Theta divisor in the Jacobian of a complex curve

Suppose $C_g$ is a smooth compact complex curve (of genus $g$), and let $J$ be its Jacobian. Recall that the Jacobian $J$ of a curve $C_g$ is a complex torus that can by obtained by contractions of ...
1
vote
0answers
133 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
1answer
176 views

Constructing rational functions with ramification locus the divisor of some $n$-form

I'm still busy learning the theory of linear systems for compact Riemann surfaces. If the answer to the following question is negative, then there might not be any point in continuing. Let $X$ be a ...
4
votes
1answer
333 views

Characterization of the moduli space of the pair of pants in terms of the modules of the extremal ring domains

Hi, I was thinking about the following question ; I will appreciate it if somebody can give me a full or partial answer or can at least cite any reference(s)/ papers etc : By $ \bar{P} $ , we ...
18
votes
2answers
1k views

Is there anything special about the Riemann surface $y^2 = x(x^{10}+11x^5-1)$?

I stumbled upon the fact that the Bolza surface can be obtained as the locus of the equation, $y^2 = x^5-x$ Its automorphism group has the highest order for genus $2$, namely $48$. I recognized ...
2
votes
1answer
186 views

many-valued function with a given set of branch points in addition to simple poles

The question concerns what information is necessary and sufficient to define uniquely a complex function f(z). To set the stage, there is a theorem that a single-valued function with only a finite ...
2
votes
3answers
391 views

$\partial \bar{\partial}$ on a riemann surface

hallo, i have the following question. let $M$ be a Riemann surface and $R \subset M$ be a compact totally real manifold (1 dimensional real). Furthermore assume there is a holomorphic 2-form ...