# Tagged Questions

286 views

### searching for an elementary proof a complex analysis result

Given a function $g$ entire on the whole complex plane $C$, it is possible to find an entire function $f$ such that $f(z+1) -f(z)=g(z)$. The proof can be given using riemann ...
643 views

### Spicing up Riemann surfaces course (revised)

I am a master's student planning to write a master's thesis on Riemann surfaces. I plan to study Forster's Lectures on Riemann surfaces. What side topics could one study to spice up the thesis? I am ...
102 views

### Uniform estimate for the Cauchy-Riemann equations on a hyperbolic Riemann surface

I have been trying to find the answer to this question in the literature, but have not succeeded. The question is as follows. Suppose $X$ is a Riemann surface that admits a green's function (i.e. ...
136 views

### Inverse “Riemann mapping” [closed]

The Riemann mapping theorem states, that any simply connected domain $U \subset \mathbb C$ can be conformally mapped to the open unit disk $D$. I.e. there is a Diffeomorphism $\Psi: D \to U$ such that ...
172 views

### Fixed points on Riemann surface

It is well known theorem that for a conformal mapping $\phi$ from a bounded and planar domain $\Omega$ to itself has three fixed points , then it must be identity mapping. However, I cannot find a ...
56 views

### Are morphisms of intersection graphs of circle packings harmonic?

Let $P$ and $Q$ be circle packings on compact Riemann surfaces (along with some Riemannian metrics) $X$ and $Y$. Let $f\colon X\to Y$ be a conformal map taking each circle in $P$ to a circle in $Q$. ...
454 views

### Essential uniqueness of the real-analytic structure on $\mathbb R$

It is well-known that any $C^k$-smooth $1$-manifold homeomorphic to $\mathbb R$ is $C^k$-diffeomorphic to $\mathbb R$. The cases of $k\in{\mathbb N}\cup$ {$\infty$} may all be handled similarly by ...
542 views

### Belyi functions on non-compact surfaces; or: Building Riemann surfaces from equilateral triangles

Some background on (compact) Belyi surfaces $\newcommand{\Ch}{\hat{\mathbb{C}}}$ A compact Riemann surface $X$ is called a Belyi surface if there exists a branched covering map $f:X\to \Ch$ such that ...
416 views

### A “holomorphic” Peano curve?

A Peano curve is a continuous map $[0,1]\to [0,1]^2$ whose image is the whole square. I would like to know if on can obtain "holomorphic" Peano curves. Namely, is it possible to find a continuous ...
139 views

### Hurwitz Spaces and Rauch Variational Formulas

I have read in some papers about Rauch-type variational formulas on Hurwitz spaces, and I would like to know what exactly is the theory behind them. A Hurwitz Space $H_g^d$ is the space of coverings ...
407 views

### The version of Montel's theorem used in the proof of Jenkins-Strebel differential

Hello, I am afraid that my main question might be a bit too elementary, but still I ask : In short, my question is "what is the version of Montel's theorem for a family of holomorphic maps from an ...
938 views

### Reducibility (or not) of algebraic curves [closed]

[ I am a bit clueless about why this question is getting downvotes!? I put it up with a genuine serious interest and I don't seem to be making any egregious error either - apart from those unsure ...
810 views

### Motivation behind defining the Ramification Divisor

I would like to understand what exactly is the motivation for defining the notion of a ramification divisor of a function. As I see the definition, If $f$ is a meromrophic function between two ...
619 views

469 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 ...