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7
votes
0answers
128 views
+50

Deformation of the covariant Laplacian

Let $M$ be a Riemann surface and $P \to M$ a principal $G$-bundle (with compact structure group $G$). Fix a connection $A$ in $P$ and consider a nearby connection $B$, which is in Coulomb gauge ...
0
votes
0answers
33 views

Field of moduli relative to a different extension

Let $X$ be a Riemann surface of genus $g\geq 2$ defined over $\mathbb{Q}(\sqrt{2})$. Let us consider the automorphism of $\mathbb{Q}(\sqrt{2})$, $$ ...
4
votes
1answer
76 views

Zeroes of global sections killed by differential operators

I asked this question some two weeks ago on StackExchange, but received no feedback of any sort ... Let $X$ be a compact connected Riemann surface and let $\Phi:M\rightarrow N$ be an elliptic ...
17
votes
3answers
413 views

Gromov-Hausdorff limits of 2-dimensional Riemannian surfaces

Let $\{M_i\}$ be a sequence of 2-dimensional orientable closed surfaces of genus $g$ with smooth Riemannian metrics with the Gauss curvature at least $-1$ and diameter at most $D$. By the Gromov ...
22
votes
6answers
3k views

Problem in Rick Miranda: finding genus of a Projective curve

I asked the following question in stack exchange (http://math.stackexchange.com/questions/21164/problem-in-rick-miranda-finding-genus-of-a-projective-curve) a few days ago, but didn't get any ...
0
votes
1answer
218 views

Are transversely immersed PL surfaces Riemann surfaces?

I have a piecewise linear (PL) surface transversely immersed in $\mathbb{R}^3$; is this a Riemann surface in the sense that I can describe it with a local coordinate $z\in \mathbb{C}$? My basic ...
7
votes
1answer
290 views

Equivalence of Branched Coverings

For equivalence of unbranched coverings of topological spaces, there is a criteria: Two coverings (unbranched) $p_1\colon Y_1\rightarrow X$ and $p_2\colon Y_2\rightarrow X$ are equivalent iff for ...
2
votes
0answers
139 views

Comparison of algebraic and analytic q-expansion

I would like to check that algebraic and analytic q-expansion of a modular form coincide. I'm thinking about modular forms as global sections of some sheaf on modular curves. If $X$ is a modular ...
5
votes
1answer
235 views

How to get a polygon from a translation surface $(X,\omega)$

Let $S_g$ be a compact topological surface of genus $g$. I know there is the correspondence $\{$Abelian differentials on compact Riemann surfaces of genus g$\}\leftrightarrow\{$ Translation surfaces ...
4
votes
2answers
119 views

Classification of symmetries of tilings in surfaces?

Is there a general study of the symmetries of tilings on surfaces? Conway, Goodman-Strauss & Burgiel classified them on $\mathbb S^2, \mathbb R^2$ and $\mathbb H^2$, with their 'Magic Theorem'. ...
2
votes
1answer
176 views

Torsion elements in the mapping class group

Let $S$ be an orientable surface of genus $g$ with $b>0$ boundary components, and let $\mathrm{Mod}(S)$ be its mapping class group, that is, the group of isotopy classes of its homeomorphisms ...
4
votes
1answer
159 views

Embed a bordered Riemann surface into punctured Riemann surfaces?

Let $U$ be a bordered Riemann surface of genus $g$ with $n -1$ punctures and one hole (i.e., the border has one connected component). For any punctured Riemann surface $\Sigma$ of genus $g$ with $n$ ...
8
votes
2answers
308 views

The class of the diagonal in the symmetric product of a smooth curve

Let $C$ be a smooth curve of genus $g$, and let us consider its $d$-th symmetric product $\textrm{Sym}^d(C)$ and its Jacobian $J(C)$. Fixing a point $p_0 \in C,$ there are two maps $$u_d\colon C_d \to ...
0
votes
0answers
36 views

Lower bound for Hurwitz's automorphisms theorem

Let $\Sigma_g$ a closed Riemann surface of genus $g\geq 2$, then the cardinality of the orientation-preserving conformal automorphism is less than $84(g-1)$, and we have equality for an infinite ...
6
votes
2answers
158 views

Criterion for deciding the conformal class of a metric on a complete surface

For orientable closed Riemannian surfaces $(S,g)$, there is a constant curvature metric $\overline{g}$ on $S$ that is conformal to $g$ in the sense that $\overline{g} = e^ug$ for some smooth function ...
0
votes
0answers
43 views

Foliation by Umbilic Surfaces

Suppose $(M,g)$ is a simply connected 3 dimensional Riemannian Manifold which is a foliation by Umbilic surfaces. Can I make the claim that there exists a coordinate system $(x_1,x_2,x_3)$ in which ...
6
votes
1answer
238 views

Quotients of curves of genus $4$ by a free $\mathbb{Z}/ 3 \mathbb{Z}$-action

Let $V_2$ and $V_3$ be the two hypersurfaces of $\mathbb P^3$ defined by \begin{equation*} V_2:={x_2x_3 + r(x_0, \, x_1)=0}, \quad V_3:={x_2^3+x_3^3+s(x_0, \, x_1)=0}, \end{equation*} where $r, \, s ...
2
votes
1answer
199 views

Given a curve $C$, does there exist a rational function on $C$ totally ramified at two given points?

Let $C$ be a smooth projective irreducible curve over $\mathbb C$. Let $x$ and $y$ be distinct points of $C$. We say that $f$ is totally ramified at a point $p$ if the ramification index of $p$ ...
3
votes
1answer
169 views

Are there any non-trivial $G$-gerbes over the analytic space $\mathbb C$

Does there exist a finite (abstract) group $G$ and a non-trivial $G$-gerbe $\mathcal X\to \mathbb C$, where we work in the category of analytic stacks. My guess is that $G$-gerbes for $G$ an abelian ...
2
votes
1answer
208 views

Moduli of stable bundles - analytic approach

Consider a compact Riemann surface $X$ of genus $\ge 2$, and consider the set $M$ of stable holomorphic vector bundles of rank $n$ and degree $d$ on $X$, up to isomorphism. At that point, one states ...
13
votes
3answers
2k views

Automorphisms of Riemann Surfaces

Izumi Kuribayashi and Akikazu Kuribayashi have classified all groups of automorphisms of compact Riemann surfaces of genus 3,4,5. (J. Pure Applied Alg.65(3)-Sept.1990, and J. Alg.134(1) Oct.1990) ...
1
vote
2answers
181 views

Complex structure on a punctured torus giving a complex structure on the torus?

Can anyone provide an idea of the proof or a reference of the fact that a complex structure on the once punctured torus extends to one on the torus? In other words, the Teichmuller space of the ...
3
votes
0answers
100 views

Construction of algebraic curves using line bundles on graphs

In this paper http://arxiv.org/abs/0707.1309 Matthew Baker and Serguei Norine, construct a analogue of the Riemman Roch formula for Lineal Systems defined on graphs. In the paper ...
2
votes
1answer
72 views

Is every closed Sasakian 3-manifold a circle bundle on a Riemann surface?

It suffices to say that all circle bundles on compact Riemann surfaces admit the structure of a closed Sasakian 3-manifold. The question is, the converse of this statement and/or what are the ...
24
votes
4answers
834 views

Deep/precise relationship between two approaches to FLT for polynomials, $n = 3$

David Speyer commented the following here. I saw Brian Conrad give an excellent one hour talk to undergraduates where he proved that there do not exist nonconstant, relatively prime, polynomials ...
2
votes
3answers
260 views

Classification of open subset of $\mathbb{R}^{3}$ [closed]

There is a theorem which gives a classification of connected open sets of $\mathbb{R}^{2}$. Unfortunately, I don't remember the correct statement, but it looks like this Theorem ? Let ...
1
vote
1answer
589 views

Branch points of a non-constant holomorphic map between compact riemann surfaces

While working on a project for mathematics I came across the following lemma: [Kock] If $X$ is a curve defined over an algebraically closed subfield $N$ of $\mathbb{C}$ and let $t:X\rightarrow ...
4
votes
2answers
181 views

Compact surface with arbitrarily large eigenvalue

Consider a compact surface $M$ with genus $\gamma \geq 2$ and fix a positive real number $V$. Is it known whether it is possible to produce a metric $g$ on the surface $M$ such that $(M. g)$ has ...
7
votes
1answer
205 views

Frucht's type theorem for Riemann surface

Frucht's theorem is a theorem in algebraic graph theory conjectured by Dénes Kőnig in 1936 and proved by Robert Frucht in 1939. It states that every finite group is the group of symmetries of a finite ...
11
votes
1answer
185 views

Is there a proof of the uniformization theorem using circle packing?

In this paper: http://www.dm.unipi.it/~benedett/rodin-sullivan.pdf Rodin and Sullivan show that circle packings converge to the Riemann map. Later, Scharmm and He found another proof of the same ...
58
votes
1answer
2k views

Is there a complex surface into which every Riemann surface embeds?

This question was previously asked on Math SE. Every Riemann surface can be embedded in some complex projective space. In fact, every Riemann surface $\Sigma$ admits an embedding $\varphi : \Sigma ...
4
votes
2answers
130 views

Explicit constant terms of volumes of moduli spaces

In Simple geodesics and Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces, Mirzakhani gave a recursive formula for WP volumes of moduli spaces $\mathcal{M}_{g,n}(L)$ of bordered ...
3
votes
0answers
46 views

Laplacian Spectra on Nearly Nodal Riemann Surfaces

Consider a family of complex curves ${\mathcal C} \to {\mathbb D}$ such that the central fibre is a nodal Riemann surface while other fibres are smooth Riemann surfaces. We choose a family of ...
2
votes
4answers
157 views

A question on Ahlfors covering surface

Given a transcendental entire function $f$, and three Jordan domains $D_1$, $D_2$, and $D_3$ such that the closures of the three Jordan domains do not intersect with each other. Then from Ahlfors ...
2
votes
0answers
61 views

Can we foliate the anti-de sitter space in 3 dimensions by Riemann surfaces?

Can we foliate the anti-de sitter spacetime in 3 dimensions by hyperbolic Riemann surfaces? -- I think this is possible, but got stuck at finding the particular projective mapping that does this. Can ...
3
votes
3answers
199 views

Compact surfaces with arbitrary gaps in spectrum

Consider a sequence of positive numbers $a_n$. My question is, can we select a closed Riemann surface whose spectrum $\lambda_i$ satisfies the condition that $\lambda_{i + 1} - \lambda_i > a_i$? Of ...
4
votes
1answer
133 views

Mapping class group of a punctured genus 0 surface

Let $T_{0,n}$ be the Teichmuller space of $n$-punctured genus $0$ Riemann surface, and $M_{0,n}$ the Moduli space (assume $n\geq 3$ and the punctures are numbered). What is the correct notion of the ...
16
votes
9answers
7k views

Teichmuller Theory introduction

What is a good introduction to Teichmuller theory, mapping class groups etc., and relation to moduli space of curves or Riemann surfaces?
2
votes
2answers
180 views

Finite orbits on an elliptic curve with two generic involutions

Let $C$ be a (very) general genus 1 curve embedded in $\mathbb{CP}^1\times \mathbb{CP}^1$ as a (2,2)-divisor. Each projection defines $C$ as a double cover of $\mathbb{CP}^1$ and induces an ...
15
votes
4answers
806 views

What are the possible automorphism groups of Riemann surfaces of low genus?

Skipping the easy cases of genus 0 and 1, what groups can arise as the group of conformal transformations of a Riemann surfaces of genus, say, 2 or 3? I'm frustrated because there are papers that ...
2
votes
3answers
171 views

Equivalence of Harmonic Maps and Conformal Maps on Genus-0 closed surfaces

By the uniformization theorem, for every genus-0 closed surface $\mathcal{M}\subset\mathbb{R}^3$, there is a conformal map $f:\mathcal{M}\rightarrow \mathbb{S}^2$. Furthermore consider the Dirichlet ...
2
votes
0answers
100 views

Is there an algorithm to compute the intersection of tautological classes on the moduli space of genus one curves?

Let $\overline{M}_{1,1}(\mathbb{P}^2, d) $ be the moduli space of degree $d$ genus one curves on $\mathbb{P}^2$ with one marked point. Let $L\longrightarrow \overline{M}_{1,1}(\mathbb{P}^2, d) $ ...
10
votes
3answers
2k views

Teichmuller theory and moduli of Riemann surfaces

This is a sequel to my earlier question asking for references for Teichmuller theory and moduli spaces of Riemann surfaces. In this connection, I have read Chapter 11 of the book Primer of mapping ...
5
votes
3answers
474 views

Flat regions on surfaces of genus greater than 1

Here is a polygonal disk + gluing scheme model of a surface we are attempting to construct. We want the regions of the surface bounded by the two vertical, dotted lines $\alpha,\beta$ to have zero ...
0
votes
1answer
81 views

Rational functions on hyperelliptic Riemann surface

Let $\mathcal R$ be an hyperelliptic Riemann surface of genus $g\geq 1$. Is it true that the only possible rational functions on $\mathcal R$ with $\leq g$ poles are the liftings of rational functions ...
1
vote
0answers
39 views

Uniqueness of Riemann Constant Vector Solution

Let $X$ be a compact, genus $g$ Riemann surface (given as the desingularization and compactification of a plane algebraic curve), $J(X)$ its Jacobian, and $A : X \to J(X)$ the Abel map $$A(P) = ...
37
votes
3answers
9k views

Properly Discontinuous Action

When looking definition, and theorems related to Properly discontinuous action of a group $G$ on a topological space $X$, it is different in different books (Topology and Geometry-Bredon, Complex ...
2
votes
1answer
189 views

The type of a Riemann surface arising from a polynomial vector field

Consider the planar polynomial vector field $$\begin{cases} \dot x=P(x,y)\\ \dot y=Q(x,y)\end{cases}$$ It defines a singular foliation on $\mathbb{C}P^{2}$. Assume that a complex leaf contains ...
2
votes
0answers
86 views

$\eta$-invariants of Riemann Surface

I am curious about a concrete computation of $\eta$-invariants for Riemann surface, e.g. Torus. Is there any nice review or notes talking about the computation? Or is it possible to express it as ...
0
votes
0answers
93 views

Conformal welding of annuli

This question is similar to that stated in Conformal Welding Reference: Let $\Sigma$ be a 1-dim compact connected complex manifold with boundary $\partial \Sigma= \partial\Sigma^+ \cup ...