Liouville equation
= Liouville equation
{title2=$\Delta u=e^{2u}$}
The elliptic <Liouville equation> in the convention
$$
\Delta u=e^{2u}
$$
arises from a planar <conformal metric> $ds^2=e^{2u}(dx^2+dy^2)$. 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 $(dx^2+dy^2)/y^2$ 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>.