Questions tagged [riemann-surfaces]
Riemann surfaces(Riemannian surfaces) is one dimensional complex manifold. For questions about classical examples in complex analysis, complex geometry, surface topology.
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How the hyperbolic metric changes when we add a puncture?
Suppose we have a surface $S$ of a finite genus, without boundary with a finite number of punctures. Suppose that this surface comes equipped with a hyperbolic metric of curvature $-1$.
Question 1: If ...
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Kinds of differentials and algebraic groups
This Wikipedia article mentions that the analogues of differentials of the first/second/third kind for algebraic groups are abelian varieties/algebraic tori/linear algebraic groups. I guess ...
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Automorphism groups of projective bundles on Riemann surfaces
Let $X$ be a compact Riemann surface, and let $E \to X$ be a stable vector bundle of rank $r+1$. Then we know that $P = \mathbb{P}(E)$ comes from an irreducible representation of $\pi_1(X)$ into the ...
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Existence of holomorphic coverings having small degree
Let $\Sigma$ be a closed Riemann surface of genus $g$. In the book of Farkas and Kra, they prove that there exists a holomorphic covering map $F : \Sigma \to \mathbb{S}^2$ of degree less than or equal ...
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Why 2-tori with Gauss curvature $\geq -1$ cannot collapse to segment?
Let $\{(\mathbb{T}^2,g_i)\}_{i=1}^\infty$ be a sequence of 2-dimensional tori with smooth Riemannian metrics with Gauss curvature at least $-1$. It was explained in the final answer to the post Gromov-...
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Riemannian metrics on 2-sphere invariant under antipodal involution
EDIT: Let $g$ be a smooth Riemannian metric on $S^2$ which is invariant under the antipodal involution $x\mapsto -x$. Assume it has Gauss curvature at least $-1$ and diameter $D$.
Does there exist ...
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Isomorphism between 2nd symmetric product and Jacobian
Let $X=X_0(N)$ be hyperelliptic with $g(X)\geq 2$ with $\infty$ as a cusp and $\iota$ as the hyperelliptic involution. Then the map
$$X^{(2)} \longrightarrow Jac(X)$$ $$D \longrightarrow [D-\infty -\...
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What finite simple groups appear as factors of surface fundamental groups?
Let $\Sigma_g$ be the a closed orientable surface of genus $g$.
My somewhat naive question: what is known about simple finite factors of $\pi_1(\Sigma_g)?$ In particular, I know that the composition ...
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Disk with punctures and convex geodesical hull of the punctures isomorphic?
Consider a unit disk with marked points $z_i$, $i=1, \dots , n$ on its boundary.
Let us call this surface $X$.
As it is well known, the disk can be equipped with an hyperbolic metric and is then ...
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On a map between Riemann surfaces of genus $1$
Let $C$ be a compact Riemann surface of genus $1$, and $p\in C$, and $w$ be a local holomorphic coordinate on $C$ near $p$ with $w=0$ at $p$.
As usual, for a divisor $D$ denote by $L(D)$ the vector ...
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multivalued holomorphic function on Riemann surfaces
Let $M$ be an open Riemann surface and $f$ a multivalued holomorphic function from $M$ to $\mathbb{H}$, where $\mathbb{H}$ is the upper half plane. Suppose that the monodromy of $f$ lies in the two-...
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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 ...
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Is there an algorithm to compute a Belyi map for the Riemann surface?
Let $y^2=x^5-x-1$ be an affine model of a projective complex curve, is there an algorithm to compute the Belyi map (preferably of small degree), i.e., map to the projective line ramified only at $\{0,...
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Existence of a connection $A$ on a holomorphic line bundle $L$, s.t $F(A)=(\deg L)\omega$
I'm reading this paper and at page 67, he states that for any line bundle $L$ over a Rieman surface there is a connection $A$ whose curvature is
$$
F(A)=(\deg L)\omega,
$$
where $\omega$ is a positive ...
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On a corollary of a paper by Colin and Honda
The question is about the last sentence of the last corollary of Stabilizing the monodromy of an open book decomposition by Vicent Colin and Ko Honda. This question is also related to this other ...
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Space of biholomorphic maps into a Riemann surface
Let $F$ be a Riemann surface and $Q\in F$. Consider $U:=(\mathbb{C}\cup\infty)\setminus [-1,1]^2$. I am interested in the space
$$X:=\{f:U\to F;\,\text{$f:U\to f(U)$ biholomorphic and $f(\infty)=Q$}\},...
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Ramification concept in complex analysis and algebraic number theory
I have a question about the connection between the concept of ramification/branching out for Riemann surfaces and algebraic number theory:
In the theory of Riemann surfaces we have following ...
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Almost complex manifold of dimension 2... locally isomorphic to ℂ?
I know that this is supposed to be standard, but I don't know how to search for it... hence the question:
Let $J$ be an almost complex structure on $M:=\mathbb R^2$, i.e., a $C^\infty$ section of $\...
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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 ...
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Stabilizing an open book with Anosov piece
It was proven by Colin and Honda in Stabilizing the monodromy of an open book decomposition that any diffeomorphism can be made pseudo-Anosov and right-veering after a series of positive ...
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Genus of non-reduced curves
Let $X$ be a smooth projective variety of dimension $3$, and $L$ an ample line bundle with $h^0(X,L)\geq 2$. Let $s$ and $t$ be two generic linearly independent sections of $L$, and $C$ the curve that ...
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Notational question about quadratic differentials in Strebel's book "Quadratic differentials"
In Kurt Strebel's book "Quadratic Differentials", in Chapter 2, $\S4$, he begins by saying:
"Every analytic function $\varphi$ is a domain $G$ of the $z$-plane defines, in a natural way, a field of ...
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Any holomorphic vector bundle over a compact Riemann surface can be defined by only one transition function?
It is known that any holomorphic bundle of any rank over a noncompact Riemann surface is trivial. A proof can be found in Forster's "Lectures on Riemann surfaces", section 30.
Let $E$ be a ...
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Reference request: uniformization theorem
I would appreciate if someone could point me to some introductory literature/resources where I can learn about Poincaré's uniformization theorem at a basic level.
Any good powerpoint notes, short ...
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Conjugate points
Suppose $(M,g)$ is a two dimensional Riemannian manifold. Let $\gamma:(-\delta,\delta)\to M$ be a geodesic segment and suppose that $\gamma(0)$ is not conjugate to any other point in $(-\delta,\delta)$...
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Riemann Surfaces of Infinite Genus and Transcendental Curves
I'm a graduate student struggling to find a topic for his doctoral dissertation which hasn't already been explored. At present, my hope was to see if classical results about Jacobian Varieties, ...
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Isometric embedding of a genus g surface
Can a genus $g$ surface with constant negative curvature and $g>1$ be isometrically embedded in $\mathbb{R}^4?$
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Integration over a Surface without using Partition of Unity
Suppose we are given a compact Riemann surface $M$, an open cover $\mathscr{U}=\{U_1,U_2,\dots\}$ of $M$, charts $\{(U_1,\phi_1),(U_2,\phi_2),\dots\}$, holomorphic coordinates, $\phi_m:p\in U_m\mapsto ...
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Existence transverse sections in $\mathbb{CP}^1$-bundles over compact Riemann surfaces
We have that every holomorphic $\mathbb{CP}^1$-bundle on a compact Riemann surface admits a holomorphic section, due Tsen and as found in Compact Compact Surfaces of Barth, Peters and Van de Ven, for ...
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How to visualize the Riemann-Roch theorem from complex analysis or geometric topology considerations?
As the question title asks for, how do others visualize the Riemann-Roch theorem with complex analysis or geometric topology considerations? That is all Riemann would have had back in the day, and he ...
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Geometric description of map from degree $3$ curve to $\mathbb{P}^2$ of degree $2$
Let $\overline{C} \subset \mathbb{P}^2$ be a smooth curve of degree $3$. Let $x, y \in \overline{C}$ be two points and $z = x + y$. Now, it's evident $L(C)$ determines a morphism $f: \overline{C} \to \...
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Precise relationship between two proofs of the Hurwitz theorem?
Let $X$ and $Y$ be connected smooth projective algebraic curves over an algebraically closed field $k$, $f: X \to Y$ a finite morphism, $g_X$ the genus of $X$. What is the precise relationship between ...
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Homogeneous Riemann Surfaces
A Riemann surface $X$ is a connected complex manifold of complex dimension one. A homogeneous space is a manifold with a transitive smooth action of a Lie group. I guess there must be a classification ...
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Does uniformization require choice?
On Twitter, Ricardo Pérez Marco recently asserted that all known proofs of the Poincaré-Koebe uniformization theorem rely on the Axiom of Choice. Is this true?
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Counting the number of poles for rational functions in a coordinate ring of a curve
I'm working in an algebra of rational functions in compact Riemann surfaces with arbitrary genus. The idea I'm struggling is how to count the number $n$ of poles for the rational functions defined in ...
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Extended Abel-Jacobi theorem
Let $X$ be a compact Riemann surface, $u$ a meromorphic function on $X$ with divisor supported on a set of points $\{P_1, ..., P_n\}$, and $f$ a meromorphic function on $X$ such that $f$ has no pole ...
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On finite extensions of the field of meromorphic functions
Let $\mathcal{M}$ be the field of meromorphic functions of one (complex) variable and $w = w(z)$ an analytic function satisfying a polynomial equation
$P(w; z) := w^n + a_{n-1}(z) w^{n-1} + \cdots + ...
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Bounded holomorphic functions on a Riemann surface separating points
Let $R$ be a Riemann surface that admits a non-constant bounded holomorphic function. Then is it true that any two points of $R$ can be separated by a bounded holomorphic function? This is easy to see ...
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fundamental domains in H^2 containing large balls
I would like to construct a genus $g$ surface regularly tiled by triangles (for example by 238 triangles). Edmunds-Ewing-Kulkarni prove that the only obstruction to doing this is Euler characteristic ...
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Poincaré metric on the Riemann sphere minus more than two points
If we omit more than two points from the Riemann sphere, we will obtain a hyperbolic Riemann surface endowed with a canonical metric descending from its universal cover which is the Poincaré disk. Let ...
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Complex structures on topological surfaces
I am interested in the number of complex structures on a surface. More precisely, given a genus $g$ surface (topological manifold of real dimension 2) with $n$ punctures $X_{(g,n)}$, how many complex ...
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Stable extensions by line bundles on Riemann surfaces
Is there a compact Riemann surface $X$ and a line bundle $L$ of negative degree on $X$, such that for any nontrivial extension
$$ 0 \rightarrow L \rightarrow E \rightarrow L^{-1} \rightarrow 0, $$
$E$ ...
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Generalized Jacobians and modular units
Let $X$ be a proper algebraic smooth curve over a characteristic zero field $k$ and let $J$ be the Jacobian variety of $X$. Let $K$ be the function field of $X$. Assume that we are given $n$ distinct ...
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The Deligne-Mumford Compactification for Closed Surfaces
I am reading this note on super-Riemann surfaces. In the second paragraph of section 7.4.1 (page 87), there is a statement that I am trying to understand:
The compactified moduli space of closed ...
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English literature close to "Algébre et Théories Galoisiennes" by Régine and Adrien Douady
I'm currently working on my undergraduate dissertation. I'm working on covering sapces of Riemann surfaces so my supervisor asked me to read the book I mention in the title: "Algébre et Théories ...
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Embedding open connected Riemann surfaces in $\mathbb{C}^2$
This question arises in the context of a question asked on MSE: Are concrete Riemann surfaces Riemann domains over $\mathbb{C}$. Part of the answer to that question is the question above which is ...
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Is every element of $Mod(S_{g,1})$ a composition of right handed Dehn twists?
Let $S_{g,1}$ be the surface of genus $g \geq 1$ and $1$ boundary component. Let $Mod(S_{g,1})$ be the mapping class group in which we allow isotopies to rotate the action on the boundary (...
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Towards recognizing St. Venant geometrical invariant
Using partial derivative notation we can express Gauss curvature $K$ in cartesian coordinates:
$$\quad p= \partial w/ \partial x, q= \partial w/ \partial y; r=\frac{\partial ^2w}{{\partial {x} ^2} },...
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References for Riemann surfaces
I know this question has been asked before on MO and MSE (here, here, here, here) but the answers that were given were only partially helpful to me, and I suspect that I am not the only one.
I am ...
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Maps between moduli of curves
Let $M_{g,n}$ be the moduli space of $n$-pointed curves, and $M_g[m]$ the moduli space of (unpointed) curves with $m$-level structure.
Fix $m>0$. Is it true that for $n$ large enough, there is a ...