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,[1][2] 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.[3]
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,[4] are obtained by using the slight variant 2 log f ↦ u of the previous change of variables and Wirtinger calculus:[5]
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.[3][a]
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.[6] Its form is given by
where f (z) is any meromorphic function such that
- df/dz(z) ≠ 0 for every z ∈ Ω.[6]
- f (z) has at most simple poles in Ω.[6]
Application
Liouville's equation can be used to prove the following classification results for surfaces:
Theorem.[7] 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:
- 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
Notes
- ^ Hilbert assumes K = -1/2, therefore the equation appears as the following semilinear elliptic equation
Citations
- ^ Liouville, Joseph (1838). "Sur la Theorie de la Variation des constantes arbitraires" (PDF). Journal de mathématiques pures et appliquées. 3: 342–349.
- ^ Ehrendorfer, Martin. "The Liouville Equation: Background - Historical Background". The Liouville Equation in Atmospheric Predictability (PDF). pp. 48–49.
- ^ a b See (Hilbert 1900, p. 288): Hilbert does not cite explicitly Joseph Liouville.
- ^ See (Dubrovin, Novikov & Fomenko 1992, p. 118) and (Henrici 1993, p. 294).
- ^ See (Henrici 1993, pp. 287–294).
- ^ a b c See (Henrici 1993, p. 294).
- ^ See (Dubrovin, Novikov & Fomenko 1992, pp. 118–120).
Works cited
- Dubrovin, B. A.; Novikov, S. P.; Fomenko, A. T. (1992) [First published 1984], Modern Geometry–Methods and Applications. Part I. The Geometry of Surfaces, Transformation Groups, and Fields, Graduate Studies in Mathematics, vol. 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) [First published 1986], Applied and Computational Complex Analysis, Wiley Classics Library, vol. 3 (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 into 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.