Liouville's equation

Summary

Liouville's equation is the semilinear elliptic equation $\Delta u + K e^{2u} = 0$ for a function $u$ of two real variables, with $K$ a constant. It says that the metric $e^{2u}(dx^2 + dy^2)$ has constant Gaussian curvature $K$. Joseph Liouville found its general local solution in terms of holomorphic functions. Equations of this exponential type also appear in models of self-gravitating gases and of vortices.