Liouville's equation

In differential geometry, Liouville's equation, named after Joseph Liouville,[1][2] 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:

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

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

${\displaystyle \Delta _{0}={\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {\partial ^{2}}{\partial y^{2}}}=4{\frac {\partial }{\partial z}}{\frac {\partial }{\partial {\bar {z}}}}.}$

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.[3]