Liouville's equation


 * 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.

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 $f(dx2 + dy2)$ on a surface of constant Gaussian curvature $K$:


 * $$\Delta_0\log f = -K f^2,$$

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 $f$ 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:


 * $$\Delta_0 u = - K e^{2u}.$$

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: $$\Delta_0 u = - 2K e^{u}\quad\Longleftrightarrow\quad \frac{\partial^2 u} = - \frac{K}{2} e^{u}.$$

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:


 * $$\Delta_{\mathrm{LB}}\log\; f = -K.$$

Relation to Gauss–Codazzi equations
Liouville's equation is equivalent to the Gauss–Codazzi equations for minimal immersions into the 3-space, when the metric is written in isothermal coordinates $$z$$ such that the Hopf differential is $$\mathrm{d}z^2$$.

General solution of the equation
In a simply connected domain $K = -1/2$, 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 $&Omega;$ is any meromorphic function such that
 * $f (z)$ for every $df⁄dz(z) ≠ 0$.
 * $z ∈ &Omega;$ has at most simple poles in $f (z)$.

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

$$. A surface in the Euclidean 3-space with metric $&Omega;$, and with constant scalar curvature $K$ is locally isometric to:
 * 1) the sphere if $dl = g(z,_ z)dzd_ z$;
 * 2) the Euclidean plane if $K > 0$;
 * 3) the Lobachevskian plane if $K = 0$.