(2+1)-dimensional pure gravity for an arbitrary closed initial surface

The (2+1)-dimensional pure Einstein gravity is studied in the ADM formalism. The authors solve the initial value and the time evolution problems with a closed Riemann surface being an initial surface, choosing the time slicing so that the trace of the extrinsic curvature is independent of spatial coordinates. The possible topology of the 2-surface is either a torus or a Riemann surface of genus g>or=2. It is shown that the moduli parameters of the torus follow the geodesic curve in the moduli space.