Articles of conformal geometry

Mapping circles using Möbius transformations.

I need some help with the following problem from Ahlfors’ Complex Analysis. Problem: Find a single Möbius transformation $\phi$ (that is, a map of the form $\phi(z) = \dfrac{az + b}{cz + d}$, where $a,b,c,d$ are complex numbers) that maps the circles $|z| = 1$ and $\left|z – \frac{1}{4}\right| = \frac{1}{4}$ to concentric circles. Infinite […]

A holomorphic function is conformal

I am trying to show that if a function $f = u+iv$ is holomorphic with $\partial_z f(z)$ always non zero, then $f$ is a conformal mapping, i.e. it preserves angles between smooth curves. If $f$ is holomorphic, by Cauchy-Riemann $$ \begin{vmatrix} u_x & u_y\\ v_x & v_y \end{vmatrix} = \begin{vmatrix} u_x & -v_x\\ v_x & […]

Find a conformal map from the disc to the first quadrant.

Find a conformal mapping of the disk $x^2+(y-1)^2\lt 1$ onto the first quadrant $x, y \gt 0$ I did something, which may be false or not, I cannot exactly say anything. I used the composition of a conformal map, which is conformal. Firstly, let’s get a conformal map from the disk to the unit disk, […]

I cannot see why Ahlfors' statement is true (Extending a conformal map)

In page 234 in Ahlfors’ complex analysis text, the author talks about extending a conformal map. During the proof he states: We note further that $f'(z) \neq 0$ on $\gamma$· Indeed, $f'(x_0)= 0$ would imply that $f(x_0)$ were a multiple value, in which case the two subarcs of $\gamma$ that meet at $x_0$ would be […]

What happens to small squares in Riemann mapping?

I have a square $S$, and I want to convert it to the unit disc $D$. The Riemann mapping theorem says that I can to it with a conformal bijective map. But, any such mapping will cause some distortion. Specifically, if S contains small sub-squares, they will be mapped into sub-shapes of C that are […]

Moving to a conformal metric

Given a generic 2-dimensional metric $$ ds^2=E(x,y)dx^2+2F(x,y)dxdy+G(x,y)dy^2 $$ what is the change of coordinates that move it into the conformal form $$ ds^2=e^{\phi(\xi,\zeta)}(d\xi^2+d\zeta^2) $$ being $\xi=\xi(x,y)$ and $\zeta=\zeta(x,y)$? Is it generally known? Also a good reference will fit the bill. Thanks beforehand.

Boundary of product manifolds such as $S^2 \times \mathbb R$

Simple question but I am confused. What is the boundary of $S^2\times\mathbb{R}$? Is it just $S^2$? What would be the general way to evaluate the boundary of a product manifold? Thanks for the replies!

Hopf's theorem on CMC surfaces

I got stuck reading the proof of the following theorem: Theorem (Heinz Hopf) Let $X: S^2\to \mathbb R^3$ be a constant mean curvature immersion. Then $X(S^2)$ is a round sphere. Proof: Let $g_{S^3}$ denote the round metric on $S^2$ (such that the area is $4\pi$). By the uniformization theorem there exists a map $\phi: S^2 […]

Conformal map from a disk onto a disk with a slit

Find a conformal mapping from D to $D\backslash [-1/2,1)$, where D is the unit disk. I think that maybe we can map $D\backslash [-1/2,1)$ to upper half plane first and the map upper half plane to $D$ and compute the inverses of the maps. Can we map $D\backslash [-1/2,1)$ to upper half plane? Thank you […]

Ricci SCALAR curvature

Are there any manifolds that have NULL SCALAR curvature but not null ricci curvature tensor? For dim>2, obviously. Edit: Are there, also, any manifolds of null scalar curvature but ricci curvature not proportional to the metric tensor?