Liouville's equation

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.

${\displaystyle {\mathrm{\Delta }}_0\log f=-K{f}^2,}$

where ∆₀ is the flat Laplace operator

${\displaystyle {\mathrm{\Delta }}_0=\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}^2}+\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{y}^2}=4\frac{\mathrm{\partial }}{\mathrm{\partial }z}\frac{\mathrm{\partial }}{\mathrm{\partial }\overline{z}}.}$

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.

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:

${\displaystyle {\mathrm{\Delta }}_{0}u=-K{e}^{2u}.}$

Other two forms of the equation, commonly found in the literature, are obtained by using the slight variant 2 log f ↦ u of the previous change of variables and Wirtinger calculus: ${\displaystyle {\mathrm{\Delta }}_{0}u=-2K{e}^u⟺\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }z\mathrm{\partial }\overline{z}}=-\frac{K}{2}{e}^u.}$

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.

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

${\displaystyle {\mathrm{\Delta }}_{\mathrm{L}\mathrm{B}}=\frac{1}{f^2}{\mathrm{\Delta }}_0}$

as follows:

${\displaystyle {\mathrm{\Delta }}_{\mathrm{L}\mathrm{B}}\log f=-K.}$