Moving to a conformal metric

Question

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.

Answer 1

This problem is equivalent to solving the Beltrami equation $f_{\bar z}=\mu f_{z}$ where the coefficient $\mu$ comes from the given metric, as explained on the Wikipedia page linked by @WillieWong. A solution can be sometimes semi-guessed when the coefficient is really simple. You should at least try it. But in general the solution comes as an infinite series involving singular integral operators. This is carefully written out in the book by Astala, Iwaniec, Martin.

Answer 2

Please check Chandrashekhar's Mathematical Theory of Black holes, section 11 of chapter 2.