# Liouville's equation

In differential geometry, Liouville's equation, named after Joseph Liouville, is the nonlinear partial differential equation satisfied by the conformal factor f of a metric f2(dx2 + dy2) on a surface of constant Gaussian curvature K:

$\Delta _{0}\log f=-Kf^{2},$ For Liouville's equation in dynamical systems, see Liouville's theorem (Hamiltonian).
For Liouville's equation in quantum mechanics, see Von Neumann equation.
For Liouville's equation in Euclidean space, see Liouville–Bratu–Gelfand equation.

where 0 is the flat Laplace operator

$\Delta _{0}={\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {\partial ^{2}}{\partial y^{2}}}=4{\frac {\partial }{\partial z}}{\frac {\partial }{\partial {\bar {z}}}}.$ Liouville's equation appears in the study of isothermal coordinates in differential geometry: the independent variables x,y are the coordinates, while f can be described as the conformal factor with respect to the flat metric. Occasionally it is the square f2 that is referred to as the conformal factor, instead of f itself.

Liouville's equation was also taken as an example by David Hilbert in the formulation of his nineteenth problem.

## Other common forms of Liouville's equation

By using the change of variables log f  u, another commonly found form of Liouville's equation is obtained:

$\Delta _{0}u=-Ke^{2u}.$ Other two forms of the equation, commonly found in the literature, are obtained by using the slight variant 2 log f  u of the previous change of variables and Wirtinger calculus:

$\Delta _{0}u=-2Ke^{u}\quad \Longleftrightarrow \quad {\frac {\partial ^{2}u}{{\partial z}{\partial {\bar {z}}}}}=-{\frac {K}{2}}e^{u}.$ Note that it is exactly in the first one of the preceding two forms that Liouville's equation was cited by David Hilbert in the formulation of his nineteenth problem.

### A formulation using the Laplace–Beltrami operator

In a more invariant fashion, the equation can be written in terms of the intrinsic Laplace–Beltrami operator

$\Delta _{\mathrm {LB} }={\frac {1}{f^{2}}}\Delta _{0}$ as follows:

$\Delta _{\mathrm {LB} }\log \;f=-K.$ ## Properties

### Relation to Gauss–Codazzi equations

Liouville's equation is a consequence of the Gauss–Codazzi equations when the metric is written in isothermal coordinates.

### General solution of the equation

In a simply connected domain Ω, the general solution of Liouville's equation can be found by using Wirtinger calculus. Its form is given by

$u(z,{\bar {z}})=\ln \left(4{\frac {\left|{\mathrm {d} f(z)}/{\mathrm {d} z}\right|^{2}}{(1+K\left|f(z)\right|^{2})^{2}}}\right)$ where f (z) is any meromorphic function such that

## Application

Liouville's equation can be used to prove the following classification results for surfaces:

Theorem. A surface in the Euclidean 3-space with metric dl2 = g(z,)dzd, and with constant scalar curvature K is locally isometric to:

1. the sphere if K > 0;
2. the Euclidean plane if K = 0;
3. the Lobachevskian plane if K < 0.