Articles of riemann surfaces

Is conformal equivalence the same as topological equivalence?

Is it true that if I take two surfaces that are topologically equivalent, I can find a conformal mapping between them?

Calculating germs for complete holomorphic function

I’m trying to find the germs $[z,f]$ for the complete holomorphic function $\sqrt{1+\sqrt{z}}$ . The question indicates that I should find 2 germs for z = 1, but I seem to be able to find 3! Where have I gone wrong? I start by defining $U_k={\frac{k\pi}{2}&lt\theta&lt\pi+\frac{k\pi}{2}}$, then ${(U_k, f_k)}$ for $0\leq k \leq 15$ is […]

Calculus itegral on Riemann surfaces

Suppose that $\Omega$ is a compact region with a smooth boundary in a Riemann surface and that $\Phi$ is a real-valued function which is positive on $\Omega$ and vanishes on the boundary of $\Omega$. Show that $\int_{\partial\Omega} i \space\partial\Phi \ge0$. The problem makes a suggestion: Consider $\Omega ⊂ C$ and see that the integral is, […]

Fixed points of map $h$ in compact riemann surface

I have the following problem: Let $h : Σ → Σ$ be a conformal self-map, different from the identity, of a compact Riemann surface $Σ$ of genus $p$. Show that $h$ has at most $2p+2$ fixed points. The problem has a suggestion. Hint: Consider a meromorphic function $f : Σ → S^2$ with a single […]

Constructing the Riemann Sphere

Problem: In the construction of the Riemann sphere, we begin with the sphere $\mathbb{S}^2$ with two charts: the stereographic projection $\sigma_N : \mathbb{S}^2 \setminus \{N\} \to \mathbb{R}^2 \cong \mathbb{C}$ from the North pole, $N$, given by $$ \sigma_N (x_1, x_2, x_3) := \frac{(x_1, x_2)}{1-x_3}, $$ the stereographic projection $\sigma_S : \mathbb{S}^2 \setminus \{S\} \to \mathbb{R}^2 […]

What is a cusp neighborhood corresponding to a parabolic Möbius transformation in a Riemann surface?

I am referring to this wikipedia entry. So what I understand is that they are defining it using the Fuchsian model. If $\Gamma$ is a Fuchsian group, its parabolic elements correspond to the cusps of $\mathbb{H} / \Gamma$. I do not know why, or what a cusp mathematically is, but I have seen pictures. Then […]

When are two tori biholomorphic?

This question already has an answer here: conformally equivalent flat tori 1 answer

Why is Klein's quartic curve not hyperelliptic

Let $X$ be Klein’s quartic curve given by $x^3y + y^3z+z^3x=0$ in $\mathbf{P}^2$. It is isomorphic to $X(7)$. How do I easily show that $X$ is not hyperelliptic? I can see that $X$ is of genus $3$ and has gonality $\leq 3$ (consider the projection). I’m trying to prove that it has gonality $3$. More […]

How does the topology of the graphs' Riemann surface relate to its knot representation?

Let me give a worked-out example: The following cubic planar non-simple graph $\hskip2.3in$ has the adjacency matrix $A=\pmatrix{0&3\\3&0}$. The graph has three faces, so the rank of $G$ is $\chi(G)=2$. The reciprocal of Ihara’s $\zeta$ function can be evaluated $$ \frac{1}{\zeta_G(u)}={(1-u^2)^{\chi(G)-1}\det(I – Au + 2u^2I)}\\ ={(1-u^2) (4u^4-5u^2+1)} $$ EDIT: Then $\zeta_G(u)=\frac{(1-u^2)^{-1} }{(4u^4-5u^2+1)}=\prod_p (1-u^{L(p)})$ with the […]

Motivation for Mobius Transformation

Let $S$ denote the Riemann Sphere. Recall that a Mobius transformation is a function $f:S \to S$ defines as $z \to \frac {az+b}{cz+d}$ where $a,b,c,d \in \mathbb C$ with $ad-bc=1$. What is the motivation to study Mobius Transformation? Why should one look at the map defined in the above way?