(2+1)-Gravity on Riemann Surfaces in Conformal Gauge

We derive a first-order formalism for solving $(2+1)$ gravity on Riemann surfaces, analogously to the recently discovered classical solutions for $N$ moving particles. We choose the York time gauge and the conformal gauge for the spatial metric. We show that Moncrief's equations of motion can be generally solved by the solution f of a $O(2,1$) sigma-model. We build out of f a mapping from a regular coordinate system to a Minkowskian multivalued coordinate system. The polydromy is in correspondence with the branch cuts on the complex plane representing the Riemann surface. The Poincar\'e holonomies, which define the coupling of Riemann surfaces to gravity, describe simply the Minkowskian free motion of the branch points. By solving f we can find the dynamics of the branch points in the physical coordinate system. We check this formalism in some cases, i.e. for the torus and for every Riemann surface with $SO(2,1)$ holonomies.