Liouville's equation

Liouville's equation

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in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle R^{1+1}} first arose in the problem of prescribing scalar curvature on a surface. It can be explicitly solved as

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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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle R^2} . This can be seen by noticing that the Ricci flow in this case take the very simple form

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Then, a Ricci soliton is given by

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after having set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle u=2\phi} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle g=\exp(\phi)g_0} being Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle g_0} the usual Euclidean metric. The equation for the Ricci soliton can be turned back to the original Liouville equation by a Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{\pi}{2}} rotation of one of the coordinates in the complex plane.

See also

• 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

References

1. J. Liouville, Sur l'equation aux differences partielles Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \partial^2 \ln \lambda /\partial u \partial v \pm 2 \lambda q^2=0} , J. Math. Pure Appl. 18(1853), 71--74.