# Liouville's equation

Liouville's equation

${\displaystyle \Box u=\exp(u)}$

in ${\displaystyle R^{1+1}}$ first arose in the problem of prescribing scalar curvature on a surface. It can be explicitly solved as

${\displaystyle u=\log {\frac {8f'(x+t)g'(x-t)}{(f(x+t)+g(x-t))^{2}}},}$

as was first observed by Liouville.

It is a limiting case of the sinh-gordon equation.

Standard energy methods give GWP in H^1.

Liouville equation turns out to be an equation for a Ricci soliton in ${\displaystyle R^{2}}$. This can be seen by noticing that the Ricci flow in this case take the very simple form

${\displaystyle \partial _{t}\phi =\exp(-2\phi )\triangle \phi .}$

Then, a Ricci soliton is given by

${\displaystyle \triangle u=\Lambda \exp(u)}$

after having set ${\displaystyle u=2\phi }$ and ${\displaystyle \Lambda }$ being a constant. We have used the fact that in dimension two, a set of isothermal coordinates always exists such that the Riemannian metric takes the simple form ${\displaystyle g=\exp(\phi )g_{0}}$ being ${\displaystyle g_{0}}$ the usual Euclidean metric. The equation for the Ricci soliton can be turned back to the original Liouville equation by a ${\displaystyle {\frac {\pi }{2}}}$ rotation of one of the coordinates in the complex plane.

## References

1. J. Liouville, Sur l'equation aux differences partielles ${\displaystyle \partial ^{2}\ln \lambda /\partial u\partial v\pm 2\lambda q^{2}=0}$, J. Math. Pure Appl. 18(1853), 71--74.