in first arose in the problem of prescribing scalar curvature on a surface. It can be explicitly solved as
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 . This can be seen by noticing that the Ricci flow in this case take the very simple form
Then, a Ricci soliton is given by
after having set and 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 being the usual Euclidean metric. The equation for the Ricci soliton can be turned back to the original Liouville equation by a rotation of one of the coordinates in the complex plane.
- The wikipedia entry for this equation
- A blog post on this equation by Terence Tao
- A blog post on 2D Ricci solitons and Liouville equation
- J. Liouville, Sur l'equation aux differences partielles , J. Math. Pure Appl. 18(1853), 71--74.