- 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 of a metric on a surface of constant Gaussian curvature K:
where 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 that is referred to as the conformal factor, instead of itself.
Other common forms of Liouville's equation
By using the change of variables "", 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 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 , 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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