Liouville's equation
- 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.
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 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. 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.
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 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 dl2 = g(z,)dzd, and with constant scalar curvature K is locally isometric to:
- the sphere if K > 0;
- the Euclidean plane if K = 0;
- the Lobachevskian plane if K < 0.
See also
- Liouville field theory, a two-dimensional conformal field theory whose classical equation of motion is a generalization of Liouville's equation