 Liouville equation

The elliptic Liouville equation in the conventionarises from a planar conformal metric $d{s}^2=e^{2u}(d{x}^2+d{y}^2)$. Its Gaussian curvature is $K=-e^{-2u}\Delta u$, so this equation is precisely the condition $K=-1$. Different curvature normalizations change the constant on the right; writing an exponential without specifying that convention can hide a sign or factor.

For example, on $y>0$, $u(x,y)=-\log y$ has $\Delta u=1/y^2=e^{2u}$. The resulting metric $(d{x}^2+d{y}^2)/y^2$ is the Poincaré half-plane model. The nonlinear partial differential equation therefore constructs a geometry of constant negative curvature, rather than merely adding an arbitrary reaction term to the Laplace equation.