# Liouville's 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 *f*^{2}(d*x*^{2} + d*y*^{2}) on a surface of constant Gaussian curvature K:

*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

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.[1]

## 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:

Other two forms of the equation, commonly found in the literature,[2] are obtained by using the slight variant 2 log *f* ↦ *u* of the previous change of variables and Wirtinger calculus:[3]

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.[1][4]

### 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

as follows:

## Properties

### Relation to Gauss–Codazzi equations

Liouville's equation is a consequence of the Gauss–Codazzi equations when the metric is written in isothermal coordinates.

### 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.[5] Its form is given by

where *f* (*z*) is any meromorphic function such that

- d
*f*/d*z*(*z*) ≠ 0 for every*z*∈ Ω.[5] *f*(*z*) has at most simple poles in Ω.[5]

## Application

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

Theorem.[6] A surface in the Euclidean 3-space with metric d*l*^{2} = *g*(*z*,)d*z*d, 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.

## See also

- Liouville field theory, a two-dimensional conformal field theory whose classical equation of motion is a generalization of Liouville's equation

## Notes

- 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).

## References

- Dubrovin, B. A.; Novikov, S. P.; Fomenko, A. T. (1992) [1984],
*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) [1986],
*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.