Liouville's equation

Liouville's equation, named after Joseph Liouville,^[1] is a nonlinear partial differential equation that arises in differential geometry when studying surfaces of constant curvature. It plays a central role in the theory of conformal geometry, where one seeks to understand how a curved surface can be represented in flat (Euclidean) coordinates while preserving angles.

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.

Suppose a surface is described in local coordinates ${\displaystyle (x,y)}$, and we wish to assign it a Riemannian metric that has constant Gaussian curvature ${\displaystyle K}$. A particularly useful form of such a metric is ${\displaystyle f(x,y)^{2}(dx^{2}+dy^{2})}$, where the function ${\displaystyle f(x,y)}$ determines the local scaling of lengths. The equation satisfied by this conformal factor ${\displaystyle f}$ is called Liouville's equation:

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

where ${\displaystyle \Delta _{0}={\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {\partial ^{2}}{\partial y^{2}}}}$ is the flat Laplace operator in two dimensions.

This equation governs how to deform flat space to give it constant curvature. For example, when ${\displaystyle K>0}$, it describes spherical geometry; when ${\displaystyle K<0}$, hyperbolic geometry. The function ${\displaystyle f(x,y)}$ tells how much the flat metric must be stretched or shrunk at each point to reflect this curvature.

Liouville's equation is closely tied to the use of isothermal coordinates, in which the metric takes a conformally flat form. It also appears in complex analysis, as the conformal factor can be expressed using Wirtinger derivatives:

${\displaystyle \Delta _{0}=4{\frac {\partial }{\partial z}}{\frac {\partial }{\partial {\bar {z}}}}.}$

A logarithmic change of variable, ${\displaystyle u=\log f}$, gives a more commonly used form of the equation:

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

Liouville's equation also features in mathematical physics (e.g., in models of 2D gravity) and was cited by David Hilbert in his formulation of the nineteenth problem, concerning the smoothness of solutions to certain elliptic PDEs.

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,^[2] are obtained by using the slight variant 2 log f ↦ u of the previous change of variables and Wirtinger calculus:^[3] ${\displaystyle \Delta _{0}u=-2Ke^{u}\quad \Longleftrightarrow \quad {\frac {\partial ^{2}u}{{\partial z}{\partial {\bar {z}}}}}=-{\frac {K}{2}}e^{u}.}$

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.^[5] 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.^[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 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
• Yamabe equation, the higher dimensional analog of Liouville's equation.