# Questions tagged [riemann-surfaces]

For questions about Riemann surfaces, that is complex manifolds of (complex) dimension 1, and related topics.

1,276 questions
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### Help with gluing together surfaces of infinite genus

Let $S=\bigwedge_{m\in\mathbb{Z}}\mathbb{S}^{1}$ denote the wedge sum of countably infinitely many circles—that is, the topological space obtained by removing countably infinitely many distinct points ...
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### Checking Holomorphicity/Meromorphicity of a Differential 1-Form On A Curve

Let $f\left(x,y\right)$ be an elementary function of $x$ and $y$. Examples: $$4x^{3}-ax-b-y^{2}$$ $$a^{x}-y^{2}+x^{3}y-1$$ $$x^{3}-3xy+y^{3}$$ $$\frac{\sin^{2}x}{x^{2}+\sin^{2}y}-\frac{1}{3}$$ Let ...
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### Reference request: introductory level book for Riemann surfaces

I have been told by different people that probably no subject can unify algebra, analysis and geometry better than Riemann surfaces. Regardless of how true it is, I'm looking for a textbook that ...
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### Is there a term for Gaussian curvature based on geodesic curvature?

Gaussian curvature is usually defined as the product of maximal and minimal normal curvatures of curves through a point on a surface. What if we use geodesic curvatures of the ambient space instead? ...
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### Which smoothness properties are preserved under ramified covering maps?

Setting. Let $M$ be a Riemann surface and $\Gamma$ a discrete group that acts properly discontinuously on $M$ by holomorphic maps. It is well known that each $x \in M$ has a finite stabilizer, that ...
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### Is the genus of biholomorphic Riemann surfaces the same?

Is the statement above true for $X \cong_{bihol} Y$? I would say yes, since I can transform any holomorphic function on a open set in $X$ to one in $Y$ and vice versa.
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### Weak solutions for divisors

I have a question on the following definition in the Forster: I don't get the part where it says "Clearly a weak wolution $f$ is a proper, i.e., meromorphic function, solution precisely if $f$ is ...
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### Branch points on Riemann surfaces

I have a question on the exact definition of branch points (on Riemann surfaces) I have the following definition:: Let $f:X \rightarrow Y$ be a holomorphic function between Riemann surfaces. $x \in X$...
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### Verify Herwitz's formula for $z^3/(1-z^2)$

This is an exercise from Miranda's book "Algebraic curves and Riemann surfaces". Consider $f(z)=z^3/(1-z^2)$ as a holomorphic map from the Riemann sphere $\mathbb{C}_\infty$ to itself. Verify ...
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### Ricci Tensor in an Einstein Manifold

I must prove that an hypersurface $M$ on $\mathbb{R}^{n+1}$ that is Einstein and compact can be only the $n-$dimensional sphere when $n>2$ The Einstein condition we permits to say that scalar ...
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### Unique complex structure on the modular curve $\mathbb{H}/\operatorname{PSL}(2,\mathbb{Z})$

Is the complex structure on the modular curve coming from the quotient $\mathbb{H}/\operatorname{PSL}(2,\mathbb{Z})$ unique? (Here $\mathbb{H}$ is the upper half plane in $\mathbb{C}$) According to ...
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### Algebraic dependence of meromorphic functions on a compact Riemann surface

I am trying to show given two meromorphic functions $f,g$ on a compact Riemann surface $X$ that there exists a non-zero polynomial $P \in \mathbb{C}[X,Y]$ such that $P(f,g) = 0$. I have seen this ...
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### Word length vs hyperbolic length of curves on a hyperbolic surface

Suppose S is a surface which admits a hyperbolic metric - by this I mean a complete Riemannian metric of constant negative curvature, with totally geodesic (possibly empty) boundary. Fix some ...
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### 3D-plots of complex functions

Commonly it's believed that one cannot fully visualize a complex function $f:\mathbb{C}\rightarrow \mathbb{C}$ because the full plot would have to be 4-dimensional. But is this true? And why would the ...
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### Riemann bilinear relations and meromorphic abelian differentials

I am getting quite confused with Riemann bilinear relations. Let $\Sigma$ be a compact Riemann surface of genus $g$, with a canonical homology basis $a_1,b_1,\dots,a_g,b_g$, with associated ...
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### Space of univalent mappings $f: \mathbb{D} \to \mathbb{C}$ has no nesting

Let $S$ be the space of univalent (i.e. injective) mappings from the disk $\mathbb{D}$ to the plane $\mathbb{C}$ normalized so that $f(0) =0$ and $f'(0)=1$. So $$f(z) = z+a_2z^2+a_3z^3+\cdots.$$...
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### A homotopy where all intermediate maps have holomorphic antiderivatives

I will denote by $\mathbb{C}^*$ the punctured complex plane, $\mathbb{C} \setminus \{0\}$. Let's say I have a holomorphic map on the punctured plane, $w: \mathbb{C}^* \to \mathbb{C}$, such that the ...
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### Question about total branching number

Given a map $f:\mathbb{CP^1} \leftarrow \mathbb{CP^1}$ by $f(z)=\frac{4z^2(z-1)^2}{(2z-1)^2}$ Find all branching points and their degrees. If my calculation is correct I got 4 branching points and in ...
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### Jordan curve and Conformal maps

Let $\mathbb D$ be the unitary open disk, $D$ a bounded domain(open and connected) with boundary a jordan curve and $f$ a conformal map from $\mathbb D$ to $D$, is it true that we can extend $f$ as an ...
I'll denote by $\mathbb{C}^*$ the punctured complex plane $\mathbb{C} \setminus \{0\}$. Say that I've got some open cover $\{V_j\}_{j \in J}$ of the closed unit interval $[0,1]$, and continuous ...