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: 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. 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: 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: 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 as follows:

Properties

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.

General solution of the equation

In a simply connected domain Ω , the general solution of Liouville's equation can be found by using Wirtinger calculus. Its form is given by where f (z) is any meromorphic function such that df⁄dz(z) ≠ 0 for every z ∈ Ω . f (z) has at most simple poles in Ω .

Application

Liouville's equation can be used to prove the following classification results for surfaces: . A surface in the Euclidean 3-space with metric , and with constant scalar curvature K is locally isometric to: K > 0 K < 0 .

Citations