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: where ∆0 is the flat Laplace operator 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.
[3] By using the change of variables log f ↦ u, another commonly found form of Liouville's equation is obtained: 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]
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.
In a simply connected domain Ω, the general solution of Liouville's equation can be found by using Wirtinger calculus.
[6] Its form is given by where f (z) is any meromorphic function such that 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: