Liouville's equation

In differential geometry, Liouville's equation, named after Joseph Liouville, is the nonlinear partial differential equation satisfied by the conformal factor f of a metric \( f^2(\mathrm{d}x^2 + \mathrm{d}y^2) \) on a surface of constant Gaussian curvature K:

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

where \( \Delta_0 is the flat Laplace operator.

\( \Delta_0 = \frac{\partial^2}{\partial x^2} +\frac{\partial^2}{\partial y^2} = 4 \frac{\partial}{\partial z} \frac{\partial}{\partial \bar z} \)

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 \(f^2 \) 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.[1]

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:

\( \Delta_0 u = - K e^{2u}. \)

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]

\( \Delta_0 u = - 2K e^{u}\quad\Longleftrightarrow\quad \frac{\partial^2 u}{{\partial z}{\partial \bar z}} = - \frac{K}{2} e^{u}. \)

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.[1][4]
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

\( \Delta_{\mathrm{LB}} = \frac{1}{f^2} \Delta_0 \)

as follows:

\( \Delta_{\mathrm{LB}}\log\; f = -K. \)

Properties
Relation to Gauss–Codazzi equations

Liouville's equation is a consequence of the Gauss–Codazzi equations when the metric is written in isothermal coordinates.
General solution of the equation

In a simply connected domain \( \Omega \) , the general solution of Liouville's equation can be found by using Wirtinger calculus.[5] Its form is given by

\( u(z,\bar z) = \frac{1}{2} \ln \left( 4 \frac{ \left|{\mathrm{d} f(z)}/{\mathrm{d} z}\right|^2 }{ ( 1+K \left|f(z)\right|^2)^2 } \right) \)

where f(z) is any meromorphic function such that

\( {\mathrm{d} f(z)}/{\mathrm{d} z}\neq 0 for every z \in \Omega \) .[5]
f(z) has at most simple poles in \( \Omeg \) a.[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 \( \mathrm{d}l^2 = g(z,{\bar z}){\mathrm{d}z}{\mathrm{d} \bar z} \) , and with constant scalar curvature K is locally isometric to:

the sphere if K > 0;
the Euclidean plane if K = 0;
the Lobachevskian plane if K < 0.

Notes

See (Hilbert 1900, p. 288): Hilbert does not cite explicitly Joseph Liouville.
See (Dubrovin, Novikov & Fomenko 1992, p. 118) and (Henrici, p. 294).
See (Henrici, pp. 287–294).
Hilbert assumes K = -1/2, therefore the equation appears as the following semilinear elliptic equation:

\( \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} = e^f \)

See (Henrici, p. 294).

See (Dubrovin, Novikov & Fomenko 1992, pp. 118–120).

References

Mathematics Encyclopedia