Liouville's 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:

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.

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.[1]

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,[2] are obtained by using the slight variant 2 log f ↦ u of the previous change of variables and Wirtinger calculus:[3]

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.[1][4]

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 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.[5] Its form is given by

where f (z) is any meromorphic function such that

  • df/dz(z) ≠ 0 for every z ∈ Ω.[5]
  • f (z) has at most simple poles in Ω.[5]

Application

Liouville's equation can be used to prove the following classification results for surfaces:

Theorem.[6] 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.

See also
  • Liouville field theory, a two-dimensional conformal field theory whose classical equation of motion is a generalization of Liouville's equation

Notes
  1. See (Hilbert 1900, p. 288): Hilbert does not cite explicitly Joseph Liouville.
  2. See (Dubrovin, Novikov & Fomenko 1992, p. 118) and (Henrici, p. 294).
  3. See (Henrici, pp. 287–294).
  4. Hilbert assumes K = -1/2, therefore the equation appears as the following semilinear elliptic equation:
  5. See (Henrici, p. 294).
  6. See (Dubrovin, Novikov & Fomenko 1992, pp. 118–120).

References
  • Dubrovin, B. A.; Novikov, S. P.; Fomenko, A. T. (1992) [1984], Modern Geometry–Methods and Applications. Part I. The Geometry of Surfaces, Transformation Groups, and Fields, Graduate Studies in Mathematics, 93 (2nd ed.), Berlin–Heidelberg–New York: Springer Verlag, pp. xv+468, ISBN 3-540-97663-9, MR 0736837, Zbl 0751.53001
  • Henrici, Peter (1993) [1986], Applied and Computational Complex Analysis Volume 3, Wiley Classics Library (Reprint ed.), New York - Chichester - Brisbane - Toronto - Singapore: John Wiley & Sons, pp. X+637, ISBN 0-471-58986-1, MR 0822470, Zbl 1107.30300.
  • Hilbert, David (1900), "Mathematische Probleme", Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (in German) (3): 253–297, JFM 31.0068.03, translated in English by Mary Frances Winston Newson as Hilbert, David (1902), "Mathematical Problems", Bulletin of the American Mathematical Society, 8 (10): 437–479, doi:10.1090/S0002-9904-1902-00923-3, JFM 33.0976.07, MR 1557926.
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