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.
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
Relation to Gauss–Codazzi equations
Liouville's equation can be used to prove the following classification results for surfaces:
- Liouville field theory, a two-dimensional conformal field theory whose classical equation of motion is a generalization of Liouville's equation
- See (Hilbert 1900, p. 288): Hilbert does not cite explicitly Joseph Liouville.
- See (Dubrovin, Novikov & Fomenko 1992, p. 118) and (Henrici, p. 294).
- See (Henrici, pp. 287–294).
- Hilbert assumes K = -1/2, therefore the equation appears as the following semilinear elliptic equation:
- See (Henrici, p. 294).
- See (Dubrovin, Novikov & Fomenko 1992, pp. 118–120).
- Dubrovin, B. A.; Novikov, S. P.; Fomenko, A. T. (1992) , 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) , 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.