- For Liouville's equation in dynamical systems, see Liouville's theorem (Hamiltonian).
- For Liouville's equation in quantum mechanics, see Density operator#Von Neumann equation.
In differential geometry, Liouville's equation, named after Joseph Liouville, is the nonlinear ordinary differential equation satisfied by the conformal factor f of a metric on a surface of constant Gaussian curvature K:
where is the flat Laplace operator.
Liouville's equation typically appears in differential geometry books under the heading isothermal coordinates. This term refers to the coordinates x,y, while f can be described as the conformal factor with respect to the flat metric (sometimes the square is referred to as the conformal factor, instead of f itself).
Replacing f using , we obtain another commonly found form of the same equation:
In a more invariant fashion, the equation can be written in terms of the intrinsic Laplace-Beltrami operator
Relation to Gauss–Codazzi equations
In a simply connected domain the general solution is given by
where is a locally univalent meromorphic function and when .
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