Liouville's equation

In differential geometry, Liouville's equation, named after Joseph Liouville,[2][3] is the nonlinear partial differential equation satisfied by the conformal factor $f$ of a metric $f 2 (d x 2 + d y 2)$ on a surface of constant Gaussian curvature $K$ :

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

[1]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

\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.[4]

Other common forms of Liouville's equation

By using the change of variables $log f \mapsto u$ , another commonly found form of Liouville's equation is obtained:

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

Other two forms of the equation, commonly found in the literature,[5] are obtained by using the slight variant $2 log f \mapsto u$ of the previous change of variables and Wirtinger calculus:[6] $\Delta _{0}u=-2Ke^{u}\quad \Longleftrightarrow \quad {\frac {\partial ^{2}u}{{\partial z}{\partial {\bar {z}}}}}=-{\frac {K}{2}}e^{u}.$

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.[4][lower-alpha 1]

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

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

as follows:

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

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.[7] Its form is given by

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

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$f (z)$ has at most simple poles in $Ω$ .[7]

Application

Liouville's equation can be used to prove the following classification results for surfaces:

Theorem.[8] A surface in the Euclidean 3-space with metric $d l 2 = g (z,_z)d z d_z$ , and with constant scalar curvature $K$ is locally isometric to:

the sphere if $K > 0$ ;
the Euclidean plane if $K = 0$ ;
the Lobachevskian plane if $K < 0$ .

Notes

Hilbert assumes $K = -1/2$ , therefore the equation appears as the following semilinear elliptic equation
${\frac {\partial ^{2}f}{\partial x^{2}}}+{\frac {\partial ^{2}f}{\partial y^{2}}}=e^{f}$

Citations

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

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[7] Hilbert assumes $K = -1/2$ , therefore the equation appears as the following semilinear elliptic equation
${\frac {\partial ^{2}f}{\partial x^{2}}}+{\frac {\partial ^{2}f}{\partial y^{2}}}=e^{f}$

[1] Liouville, Joseph (1838). "Sur la Theorie de la Variation des constantes arbitraires". Journal de Mathématiques Pures et Appliquées. 3: 342–349.

[2] Liouville, Joseph. "Sur la Theorie de la Variation des constantes arbitraires" (PDF). Journal de mathématiques pures et appliquées. 3: 342–349.

[3] Ehrendorfer, Martin. "The Liouville Equation: Background - Historical Background". The Liouville Equation in Atmospheric Predictability (PDF). pp. 48–49.

[Hilbertp288-4] See (Hilbert 1900, p. 288): Hilbert does not cite explicitly Joseph Liouville.

[5] See (Dubrovin, Novikov & Fomenko 1992, p. 118) and (Henrici 1993, p. 294).

[6] See (Henrici 1993, pp. 287–294).

[JJSON-8] See (Henrici 1993, p. 294).

[9] See (Dubrovin, Novikov & Fomenko 1992, pp. 118–120).