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 f^2(dx^2 + dy^2) on a surface of constant Gaussian curvature K:

${\displaystyle \Delta _{0}\log f=-Kf^{2},}$

where ∆₀ is the flat Laplace operator

${\displaystyle \Delta _{0}={\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {\partial ^{2}}{\partial y^{2}}}=4{\frac {\partial }{\partial z}}{\frac {\partial }{\partial {\bar {z}}}}.}$

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 f^2 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:

${\displaystyle \Delta _{0}u=-Ke^{2u}.}$

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] ${\displaystyle \Delta _{0}u=-2Ke^{u}\quad \Longleftrightarrow \quad {\frac {\partial ^{2}u}{{\partial z}{\partial {\bar {z}}}}}=-{\frac {K}{2}}e^{u}.}$

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

${\displaystyle \Delta _{\mathrm {LB} }={\frac {1}{f^{2}}}\Delta _{0}}$

as follows:

${\displaystyle \Delta _{\mathrm {LB} }\log \;f=-K.}$

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 ${\displaystyle z}$ such that the Hopf differential is ${\displaystyle \mathrm {d} z^{2}}$.

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

${\displaystyle u(z,{\bar {z}})=\ln \left(4{\frac {\left|{\mathrm {d} f(z)}/{\mathrm {d} z}\right|^{2}}{(1+K\left|f(z)\right|^{2})^{2}}}\right)}$

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 dl^2 = 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. Hilbert assumes K = -1/2, therefore the equation appears as the following semilinear elliptic equation

   ${\displaystyle {\frac {\partial ^{2}f}{\partial x^{2}}}+{\frac {\partial ^{2}f}{\partial y^{2}}}=e^{f}}$

Citations

1. Liouville, Joseph (1838). "Sur la Theorie de la Variation des constantes arbitraires" (PDF). Journal de mathématiques pures et appliquées. 3: 342–349.
2. Ehrendorfer, Martin. "The Liouville Equation: Background - Historical Background". The Liouville Equation in Atmospheric Predictability (PDF). pp. 48–49.
3. See (Hilbert 1900, p. 288): Hilbert does not cite explicitly Joseph Liouville.
4. See (Dubrovin, Novikov & Fomenko 1992, p. 118) and (Henrici 1993, p. 294).
5. See (Henrici 1993, pp. 287–294).
6. See (Henrici 1993, p. 294).
7. 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.