Articles of conformal geometry

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?

Schwarz-Christoffel mapping onto infinite L-shaped region

I’m trying to map the upper half plane onto the infinite L-shaped region $$ \Omega = \{z = x+iy; \ x > 0, \ y > 0, \ \min(x,y) < 1 \} $$ My first try is a Schwarz-Christoffel function $$ F(w) = \int_0^w (\zeta+1)^{-1}\zeta^{-1/2}(\zeta-1)^{-1} \ d\zeta $$ This guy looks promising, because the integrand […]

Conformal map from unit disk to strip

I have the following question: Write down the solution $u(x, y)$ to the Dirichlet problem for the following region and boundary conditions: $U = \{x + iy : 0\le y\le1\}; u(x, 0) = 0, u(x, 1) = 1$. Hence use appropriate conformal maps to find to a solution in the following region and with the […]

Understanding angle-preserving definition

My book (Real and complex analysis, by Rudin) gives the following definition: Let $A(z) = \frac z{|z|}$. Then we say $f$ preserves angles at $z_0$ if $$\lim_{r \to 0}e^{-i\theta} A[f(z_0 + re^{i\theta} )- f(z_0)] \ \ \ \ (r > 0)$$ exists and it is independent of $\theta$. It then adds The requirements is that […]

Can an arbitrary curve locally be made into a geodesic through a conformal change of the metric?

Given a nowhere null curve $\gamma: I \rightarrow M$ in a pseudo-Riemannian manifold $(M,g)$. (I is some open interval) Let $t_0 \in I$ arbitrary. Does a conformally equivalent metric $\hat{g}=e^{2\sigma} g$ and some $\epsilon >0$ exist, such that $\gamma \mid _{(t_0-\epsilon, t_0 + \epsilon)}$ is a geodesic in $(M,\hat{g})$? $\gamma$ being a geodesic with respect […]

Stereographic projection is conformal — from the line element

I’m looking over some fairly basic stuff on complex methods and the book I’m using takes the formula for the stereographic projection: $$z = \cot(\beta/2)e^{i\phi} $$ as well as the line element on the sphere: $$ ds^2 = d\beta^2 + \sin^2 \beta d\phi^2 $$ (where $\beta$ is the polar angle and $\phi$ the azimuthal angle) […]

Conformal mapping circle onto square (and back)

I’m programming an implementation of the Peirce quincuncial map projection. The projection involves a stereographic projection of a hemisphere of the globe onto a circle (I’ve got that part), then mapping points on that circle onto a square with a conformal mapping. Wikipedia describes the relationship between a point $(p, \theta)$ on the circle and […]

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, […]