Liouville's equation

^[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.

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 f²(dx² + dy²) 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 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.^[4]

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

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.^[4][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.^[7] 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.^[7]
• 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 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[]

Citations[]

Works cited[]