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]

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

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

Other two forms of the equation, commonly found in the literature,^[4] are obtained by using the slight variant 2 log f ↦ u of the previous change of variables and Wirtinger calculus:^[5] ${\displaystyle \Delta _{0}u=-2Ke^{u}\quad \Longleftrightarrow \quad {\frac {\partial ^{2}u}{{\partial z}{\partial {\bar {z}}}}}=-{\frac {K}{2}}e^{u}.}$

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.^{[3][lower-alpha 1]}

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

Theorem.^[7] A surface in the Euclidean 3-space with metric dl² = g(z,_z)dzd_z, 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.

• Liouville field theory, a two-dimensional conformal field theory whose classical equation of motion is a generalization of Liouville's equation