Liouville's equation

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For Liouville's equation in dynamical systems, see Liouville's theorem (Hamiltonian).

In differential geometry, Liouville's equation, named after Joseph Liouville, is the equation satisfied by the conformal factor f of a metric f²(dx² + dy²) on a surface of constant Gaussian curvature K:

where Δ₀ 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 f² is referred to as the conformal factor, instead of f itself).

Replacing f by , we obtain another commonly found form of the same equation:

Δ₀u = − Ke^2u.

[edit] Laplace-Beltrami operator

In a more invariant fashion, the equation can be written in terms of the intrinsic Laplace-Beltrami operator

as follows: