Liouville's 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:

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

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

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

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 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:

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

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 dl² = 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.