Liouville's equation

For Liouville's equation in dynamical systems, see Liouville's theorem (Hamiltonian).

For Liouville's equation in quantum mechanics, see Density operator#Von Neumann equation.

In differential geometry, Liouville's equation, named after Joseph Liouville, is the nonlinear ordinary differential equation satisfied by the conformal factor f of a metric $f^{2}(dx^{2}+dy^{2})$ on a surface of constant Gaussian curvature K:

$\Delta _{0}\;\log f=-Kf^{2},$

where $\Delta _{0}$ 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 typically appears in differential geometry books under the heading isothermal coordinates. This term refers to the coordinates x,y, while f can be described as the conformal factor with respect to the flat metric (sometimes the square $f^{2}$ is referred to as the conformal factor, instead of f itself).

Replacing f using $\log \,f=u$, we obtain another commonly found form of the same equation:

$\Delta _{0}u=-Ke^{2u}.$

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:

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

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

In a simply connected domain $\Omega$ the general solution is given by

$u(z,{\bar {z}})={\frac {1}{2}}\ln \left(4{\frac {|df(z)/dz|^{2}}{(1+K|f(z)|^{2})^{2}}}\right)$

where $f(z)$ is a locally univalent meromoprhic function and $1+K|f(z)|^{2}$ when $K<0$.