# Ngô Quốc Anh

## August 22, 2010

### Liouville’s theorem and related problems

Filed under: Giải tích 7 (MA4247), PDEs — Tags: — Ngô Quốc Anh @ 6:35

The following theorem is well-known

Theorem (Liouville). Let $\Omega$ be a simply connected domain in $\mathbb R^2$. Then all real solutions of

$\displaystyle \Delta u +2Ke^u=0$

in $\Omega$ where $K$ a constant, are of the form

$\displaystyle u=\log\frac{|f'|^2}{\left(1+\frac{K}{4}|f|^2\right)^2}$

where $f$ is a locally univalent meromorphic function in $\Omega$.

In geometry, our PDE

$\displaystyle \Delta u +2Ke^u=0$

says that under the case $\Omega=\mathbb R^2$, it holds

$e^u|dz|^2=f^*g_K$

where $g_K$ denotes the standard metric on $\mathbb S^2$ with constant curvature $K$. Thus we have

Corollary. All solutions of the PDE in $\mathbb R^2$ with $K>0$ and

$\displaystyle \int_{\mathbb R^2} e^u<\infty$

are of the form

$\displaystyle u(x)=\log\frac{16\lambda^2}{\left(4+\lambda^2K|x-x_0|^2\right)^2},\quad \lambda>0, \quad x_0 \in \mathbb R^2$.