Hamiltonian structure of 2+1 dimensional gravity

A summary is given of some results and perspectives of the hamiltonian ADM approach to 2 + 1 dimensional gravity. After recalling the classical results for closed universes in the absence of matter, we go over the the case in which matter is present in the form of point spinless particles. Here the maximally slicing gauge proves most effective by relating 2 + 1 dimensional gravity to the Riemann-Hilbert problem. It is possible to solve the gravitational field in terms of the particle degrees of freedom thus reaching a reduced dynamics which involves only the particle positions and momenta. Such a dynamics is proven to be hamiltonian and the hamiltonian is given by the boundary term in the gravitational action. As an illustration, the two body hamiltonian is used to provide the canonical quantization of the two particle system.