Liouville equation on compact bordered surfaces

Poincare used the Liouville equation “exponent is half-Laplacian” to solve the problem of uniformization, i.e. restoration of a hyperbolic structure on a surface from its conformal structure. In the case of a bordered surface the equation must be supplemented by the boundary conditions. Different conditions lead to different hyperbolic metrics. Also we propose a conformally invariant action for the Liouville equation.