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3
votes
1answer
658 views

Basic Questions about Teichmuller's theorem/quadratic differentials

I have some basic questions about Teichmuller's theorem, since I am a beginner, my questions might be very basic. If you can give some hints/answers or cite some references to study from, I will ...
1
vote
2answers
184 views

The antipodal action on a connected one dimensional manifold

When I am reading one paper, I have met the following statement: It is impossible to define a $Z_{2}\times Z_{2}$ action on a connected closed curve on a compact Riemann surface. The claim is ...
25
votes
3answers
7k 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 ...
11
votes
3answers
546 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 ...
2
votes
2answers
398 views

Action of a group on a Riemann surface

If $G$ is a finite group of order $n$, and acting on a compact Riemann surface $X_{g'}$ ($g'$ is genus), then $X_{g'}/G$ is another compact Riemann surface, $X_g$, of genus $g$. Also then $X_{g'} ...
17
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 didnt get any solution. ...
0
votes
2answers
668 views

A quick question about Farb-Margalit's book on MCG's proof on Teichmuller's existence theorem

Hello, I was studying Farb-Margalit's " A Primer on MCG " for Teichmuller's existence theorem. On P. 347, proposition 11.14, they proved $ \omega : QD_1(X) -> Teich ( S_g) $ is proper, which, ...
3
votes
2answers
902 views

some questions on Riemann surface

There are several puzzling questions on Riemann surface for me: Q.1 Definition of Riemann surface can be given in at least two ways: Def.1) it is a complex one dimensional manifold; Def.2) for each ...
2
votes
1answer
227 views

Can we prove $ Aut(S_g) , g \geq 2 $ is finite in the following way ?

I was trying to prove that $ Aut( S_g $), g$ \geq 2 $ [ orientation preserving isometries ] is finite in the following way : For fixed $M $ ( positive ) there are finitely many , say $ k $ number of ...
1
vote
0answers
147 views

What is the parametrix for the d-bar operator on Sobolev spaces?

I am attempting to review a certain proof of the Riemann-Roch theorem, a key step of which is "elliptic bootstrapping." In particular, let $X$ be a compact Riemann surface, and $E$ be a holomorphic ...
7
votes
1answer
1k views

Is there a manifold structure on a space of conformal maps?

I would be very grateful for any information or pointers for the following: 1) Fix an open subset $U$ of $\mathbb{CP}^1$. a) Does the set of all holomorphic maps from $U$ to $\mathbb{C}$ (with the ...
7
votes
1answer
1k views

Which Riemann surfaces arise from the Riemann existence theorem?

The following was already known to Riemann. Suppose that one is given a connected Riemann surface $X$, a finite set $\Delta \subset X$ and a homomorphism $\phi: \pi_1(X \backslash \Delta) \to S_d$ ...
1
vote
1answer
449 views

Some basic questions about the proof of Teichmuller's uniqueness theorem

Hello , I was studying the proof of Teichmuller's uniqueness theorem from the note/book " A Primer on Mapping Class Groups " by Farb-Margalit and I got struck at a couple of points, mainly because I ...
3
votes
1answer
585 views

Riemann's theorem on theta

Is there a low-tech way to compute the chern class of the theta bundle, i.e. the bundle associated to theta divisor? Here, the theta divisor is the analytic set of holomorphic line-bundles in ...
2
votes
2answers
292 views

Why a non-simple geodesic in a Y-piece is NOT homotopic to a common perpendicular to the geodesic boundaries ?

This is a basic question, still I dare to ask : Let Y be the Y-piece with geodesic boundaries A,B, C and ( if possible ) c the non simple geodesic from A to B intersecting itself at a point p. I want ...
0
votes
1answer
307 views

Nielsen extension of Riemann surface

Let $S$ be a riemann surface. If S has idea boundary curves,then the intrinsic metric on $S$ can be defined by the restriction to $S$ of poincare metric of the double of $S$. Also this metric can be ...
0
votes
1answer
249 views

Figure eight geodesic on a pair of pants/Y-piece

Consider a figure-eight geodesic $\delta $( geodesic with exactly one self-intersection point at p ) on a pair of pants Y with three geodesic boundaries $ \gamma_i$ and three perpendiculars between ...
1
vote
2answers
1k views

Normalisations of singular plane algebraic curves

In what follows assume that the base field is $\mathbb{C}$ Background: Let $C$ be an irreducible plane algebraic curve, $S$ the set of singular points. There exists a Riemann surface $\tilde{C}$ and ...
12
votes
3answers
2k views

Teichmuller modular forms and number theory

Do higher genus Teichmuller modular forms have, or are they expected to have, implications for number theory that generalize the sorts of results that flow from the study of classical modular forms?
6
votes
2answers
750 views

Bounding the modular discriminant of an elliptic curve in the j-invariant

Consider an elliptic curve $X=\mathbf{C}/ (\mathbf{Z}+\tau \mathbf{Z})$, where $\tau$ is an element in the complex upper half plane. We define $$\Vert \Delta\Vert(X) = (\Im \tau)^6 \vert ...
3
votes
2answers
498 views

finite-dimensionality of cohomology groups on compact riemann surfaces

does the finite dimensionlity of the first cohomology group ($ H^1 $) of the sheaf of meromorphic sections of a holomorphic line bundle on a compact riemann surface follow easily from the finite ...
3
votes
2answers
1k views

Translation surfaces

I know that this definitely have some sort of reference out there, but I did not find any wikipidea page for it or any introductory Mathematical article about it . I just want definition and concrete ...
6
votes
3answers
622 views

An analytic proof of the De Franchis theorem

The De Franchis theorem in its simplest form states that given two compact Riemann surfaces $\Sigma_{g_1},\Sigma_{g_2}$ where $g_1,g_2 > 1$, there are only finitely many non-constant holomorphic ...
6
votes
2answers
499 views

Poles of Kloosterman Zeta Function

I am a string theorist who has encountered the following number theory problem in my research. Consider the sums $$Z_+(s) = \sum_{p=1}^\infty p^{-s} S(1,1; p)$$ and $$Z_-(s) = \sum_{p=1}^\infty ...
2
votes
2answers
659 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 ...
3
votes
2answers
321 views

Books about the spectra of non-compact Riemann surfaces

Hello, Thanks for reading my question ! Could anybody give me some references ( books, papers containing elementary results etc ) on the eigen values and eigenspectra of NON-compact Riemann surfaces. ...
2
votes
1answer
786 views

Question related to the moduli space of Riemann surfaces and a fibration

If $M_{g}$ is the moduli space of Riemann surface of genus $g$, $M^1_{g}$ is the moduli space of Riemann surface of genus $g$ with one boundary, how can we show that the natural map: $M^1_{g} ...
1
vote
2answers
493 views

Geodesics on zero-curvature regions of closed surfaces of genus > 1 of non-positive curvature

Let $M$ be a 2-dimensional Riemannian manifold of non-positive curvature everywhere, of genus > 1. Let $\textbf{D} \subset \textbf{C}$ be the open unit disc in the complex plane, the universal cover ...
6
votes
2answers
675 views

Compact cover of a Hausdorff compact space

In his book "Riemannian Geometry" do Carmo cites the Hopf-Rinow theorem in chapter 7. (theorem 2.8). One of the equivalences there deals with the cover of the manifold using nested sequence of compact ...
5
votes
1answer
221 views

Interpolation on real Riemann surfaces

Background: Generalizing the notion of upper half plane to compact Riemann surfaces: Suppose $p(x,y) \in \mathbb{R}[x,y]$ is a polynomial in 2 variables with real coefficients, defining a smooth ...
4
votes
1answer
598 views

Homology dimension of the mapping class group of a surface with boundary

There is a result on the dimension bound for ${M_{g,n}}/S_n$, (the moduli space for Riemann surfaces of genus $g$ with $n$ marked points) that is $H_{i}({M_{g,n}}/S_n)=0$, for $i\ge 6g-7+2n$ except ...
8
votes
0answers
572 views

Moduli space of semistable bundles

It is well-known that the space of S-equivalence classes of rank 2 semistable holomorphic vector bundles with trivial determinant on a genus 2 surface M is $CP^3$ (more concretely ...
0
votes
0answers
156 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 ...
16
votes
1answer
761 views

Irrational Numbers and the Riemann Surface of a Multi-Valued Function

Suppose a meromorphic function $f(z)$ has two poles, with residues $1$ and $\gamma$, respectively. Then the topology of the Riemann surface of the anti-derivative of $f(z)$ depends on whether or not ...
4
votes
3answers
434 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 ...
5
votes
1answer
514 views

How can we show the spaces $M_{g}(n)$ and $M_{g, n}$ are homotopy equivalent?

How can we prove that the moduli space,$M_{g}(n)$, of genus $g$ Riemann surface with $n$ boundary components is homotopy equivalent to $M_{g,n}$, that is ,the moduli space of genus $g$ Riemann surface ...
1
vote
0answers
295 views

Is there a reference showing that the space $\bar{M_{g,n}}$ is a closed oriented orbifold and it is hausdorff

Is there a reference showing that the space $\bar{M_{g,n}}$ is a closed oriented orbifold and it is Hausdorff? Note: here $\bar{M_{g,n}}$ is not the Deligne-Mumford space in the usual algebraic ...
5
votes
4answers
787 views

To differently gluing of two Riemann surfaces with boundary we get different surfaces

If $M,N$ are two Riemann surfaces with boundary, then we can glue them along one of each of their boundary component, which is $S$, to form a new Riemann surface with boundary, but for different ...
5
votes
1answer
570 views

Examples of compact hyperbolic surfaces/manifolds with very small or very large eigenvalues

Hello, Is there any general ways to construct compact hyperbolic 2-manifolds with very small or very large eigenvalues ? Also, as a special case, can we construct a sequence of compact hyperbolic ...
4
votes
3answers
869 views

Does there exist a Riemann surface corresponding to every field extension? Any other hypothesis needed?

My question is a doubt I had in the last point to the first answer to this MO question - "Algebraic" topologies like the Zariski topology? Can one associate a Riemann surface to any ...
1
vote
0answers
328 views

Riemann surfaces

In Hatcher's book on Algebraic Topology, a lot of coverings of $S^1 \vee S^1$ has given. From these coverings, can we get different coverings of double torus (genus 2, compact surface)?
2
votes
2answers
775 views

What is the definition of ideal boundary?

In many papers about dynamical system, I found the word " ideal boundary". T don't know what is the definition of ideal boundary.
4
votes
1answer
886 views

How to compute the (co)homology of a compact Riemann surface?

The situation is the following. A finite-index subgroup $\Gamma$ of $SL_2(\mathbb Z)$ acts on the upper-half plane $\mathcal H$. It has a fundamental domain, obtained by a union of translates of the ...
2
votes
3answers
969 views

The Riemann correspondence for riemann surfaces made explicit and its generalizations

Riemann showed (not proved rigorously) that there is a correspondence between compact riemann surfaces and algebraic function fields in one variable (does anyone know the year?). To construct the ...
7
votes
2answers
733 views

Covers of Riemann surfaces which become arbitrary close in Teichmuller space

Suppose $S$ and $S'$ are two compact Riemann surfaces of genus $g$. Does there exist a sequence of genera $g_i \to \infty$ and covers $S_i, S_{i}'$ of $S,S'$, both of genus $g_i$, such that ...
4
votes
1answer
163 views

Divisors of solutions of elliptic problems

I learned very recently that in the early-mid 90s Gromov and Shubin proved a generalization of the Riemann-Roch theorem. Let $X$ be a compact closed ${\cal C}^\infty$-variety of dimension $n\geq2$. ...
2
votes
2answers
479 views

Schwarz Lemma in terms of conformal surfaces or holomorphic curves?

Scharwz Lemma in its general form says that any holomorphic map between hyperbolic surfaces is contracting. Noting that Riemann surfaces admit a unique metric of constant curvature -1, I wonder if we ...
7
votes
2answers
370 views

Approximating holomorphic maps by holomorphic embeddings

Let $\mathrm{Hol}^d(\Sigma, \mathbb{C} \mathbb{P}^n)$ denote the space of holomorphic maps of degree $d$ from a Riemann surface $\Sigma$ to complex projective space of dimension $n$. Let ...
12
votes
7answers
2k views

What should be taught in a 1st course on Riemann Surfaces?

I am teaching a topics course on Riemann Surfaces/Algebraic Curves next term. The course is aimed at 1st and 2nd year US graduate students who have have taken basic coursework in algebra and manifold ...
2
votes
2answers
1k views

Branched coverings of Riemann surfaces with specified branch points.

Today I showed, using some ad hoc algebraic topology, that if $\Sigma$ is a Riemann surface and $\mathfrak{p} \subset \Sigma$ is a finite set of points, then there is another Riemann surface $S$ and a ...