: ''For Liouville's equation in dynamical systems, see

Liouville's theorem (Hamiltonian) In physics, Liouville's theorem, named after the French mathematician Joseph Liouville, is a key theorem in classical statistical mechanics, statistical and Hamiltonian mechanics. It asserts that ''the phase space, phase-space distribution functi ...

.'' : ''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 Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, lin ...

, Liouville's equation, named after

Joseph Liouville Joseph Liouville ( ; ; 24 March 1809 – 8 September 1882) was a French mathematician and engineer. Life and work He was born in Saint-Omer in France on 24 March 1809. His parents were Claude-Joseph Liouville (an army officer) and Thérès ...

, is the

nonlinear partial differential equation In mathematics and physics, a nonlinear partial differential equation is a partial differential equation with nonlinear system, nonlinear terms. They describe many different physical systems, ranging from gravitation to fluid dynamics, and have b ...

satisfied by the conformal factor of a metric on a

surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...

of constant

Gaussian curvature In differential geometry, the Gaussian curvature or Gauss curvature of a smooth Surface (topology), surface in three-dimensional space at a point is the product of the principal curvatures, and , at the given point: K = \kappa_1 \kappa_2. For ...

: :

\Delta_0\log f = -K f^2,

where is the flat

Laplace operator In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a Scalar field, scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \ ...

\Delta_0 = \frac +\frac
= 4 \frac \frac.

Liouville's equation appears in the study of

isothermal coordinates In mathematics, specifically in differential geometry, isothermal coordinates on a Riemannian manifold are local coordinates where the metric is conformal to the Euclidean metric. This means that in isothermal coordinates, the Riemannian metric ...

in differential geometry: the

independent variables A variable is considered dependent if it depends on (or is hypothesized to depend on) an independent variable. Dependent variables are studied under the supposition or demand that they depend, by some law or rule (e.g., by a mathematical function ...

are the coordinates, while can be described as the conformal factor with respect to the flat metric. Occasionally it is the square that is referred to as the conformal factor, instead of itself. Liouville's equation was also taken as an example by

David Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician and philosopher of mathematics and one of the most influential mathematicians of his time. Hilbert discovered and developed a broad range of fundamental idea ...

in the formulation of his nineteenth problem.See : Hilbert does not cite explicitly Joseph Liouville.

Other common forms of Liouville's equation

By using the

change of variables In mathematics, a change of variables is a basic technique used to simplify problems in which the original variables are replaced with functions of other variables. The intent is that when expressed in new variables, the problem may become si ...

, another commonly found form of Liouville's equation is obtained: :

\Delta_0 u = - K e^.

Other two forms of the equation, commonly found in the literature, are obtained by using the slight variant of the previous change of variables and Wirtinger calculus:

\Delta_0 u = - 2K e^\quad\Longleftrightarrow\quad \frac = - \frac e^.

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 In differential geometry, the Laplace–Beltrami operator is a generalization of the Laplace operator to functions defined on submanifolds in Euclidean space and, even more generally, on Riemannian and pseudo-Riemannian manifolds. It is named aft ...

\Delta_ = \frac \Delta_0

as follows: :

\Delta_\log\; f = -K.

Properties

Relation to Gauss–Codazzi equations

Liouville's equation is equivalent to the

Gauss–Codazzi equations In Riemannian geometry and pseudo-Riemannian geometry, the Gauss–Codazzi equations (also called the Gauss–Codazzi–Weingarten-Mainardi equations or Gauss–Peterson–Codazzi formulas) are fundamental formulas that link together the induced m ...

for minimal immersions into the 3-space, when the metric is written in

z

such that the Hopf differential is

\mathrmz^2

General solution of the equation

In a

simply connected In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every Path (topology), path between two points can be continuously transformed into any other such path while preserving ...

domain , the general solution of Liouville's equation can be found by using Wirtinger calculus.See . Its form is given by :

u(z,\bar z) = \ln \left(
4 \frac
\right)

where is any

meromorphic function In the mathematical field of complex analysis, a meromorphic function on an open subset ''D'' of the complex plane is a function that is holomorphic on all of ''D'' ''except'' for a set of isolated points, which are ''poles'' of the function. ...

such that * for every . * has at most simple poles in .

Application

Liouville's equation can be used to prove the following classification results for surfaces: .See . A surface in the Euclidean 3-space with metric , and with constant scalar curvature is locally isometric to: # the

sphere A sphere (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, a sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...

if ; # the

Euclidean plane In mathematics, a Euclidean plane is a Euclidean space of Two-dimensional space, dimension two, denoted \textbf^2 or \mathbb^2. It is a geometric space in which two real numbers are required to determine the position (geometry), position of eac ...

if ; # the Lobachevskian plane if .

Notes

Citations

Works cited

*. *. *, translated into English by Mary Frances Winston Newson as . {{refend Differential equations Differential geometry