- For Liouville's equation in dynamical systems, see Liouville's theorem (Hamiltonian).
- For Liouville's equation in quantum mechanics, see Von Neumann 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.
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:
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
General solution of the equation
where f (z) is any meromorphic function such that
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,)dzd, and with constant scalar curvature K is locally isometric to:
- 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.
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