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Liouville's equation

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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:

Contents

Δ 0 log ⁡ f = − K f 2 ,

where ∆0 is the flat Laplace operator

Δ 0 = ∂ 2 ∂ x 2 + ∂ 2 ∂ y 2 = 4 ∂ ∂ z ∂ ∂ 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.

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:

Δ 0 u = − K e 2 u .

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:

Δ 0 u = − 2 K e u ⟺ ∂ 2 u ∂ z ∂ z ¯ = − 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.

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

Δ L B = 1 f 2 Δ 0

as follows:

Δ L B log f = − K .

Relation to Gauss–Codazzi equations

Liouville's equation is a consequence of the Gauss–Codazzi equations when the metric is written in isothermal coordinates.

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

u ( z , z ¯ ) = 1 2 ln ⁡ ( 4 | d f ( z ) / d z | 2 ( 1 + K | f ( z ) | 2 ) 2 )

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:

    Theorem. 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.

    References

    Liouville's equation Wikipedia
    (Text) CC BY-SA