# 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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### 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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### Bound of Riemann theta function and the determinant of periodic matrix

Let $\Omega$ be the Riemann period matrix for a compact Riemann surface of genus $g$. What do we know about the term below?
$$
f(\Omega)=|\theta(0,\Omega)|^{4}\det (\textrm{Im} \Omega)
$$
I am sure ...

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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 ...