In differential geometry, Liouville's equation, named after Joseph Liouville, is the nonlinear ordinary differential equation satisfied by the conformal factor f of a metric on a surface of constant Gaussian curvature K:
where is the flat Laplace operator.
Liouville's equation typically appears in differential geometry books under the heading isothermal coordinates. This term refers to the coordinates x,y, while f can be described as the conformal factor with respect to the flat metric (sometimes the square is referred to as the conformal factor, instead of f itself).
Replacing f using , we obtain another commonly found form of the same equation:
In a more invariant fashion, the equation can be written in terms of the intrinsic Laplace-Beltrami operator
as follows:
Liouville's equation is a consequence of the Gauss–Codazzi equations when the metric is written in isothermal coordinates.
In a simply connected domain the general solution is given by
where is a locally univalent meromorphic function and when .