(2+1)-AdS GRAVITY ON RIEMANN SURFACES

Abstract

We discuss a formalism for solving (2+1)-AdS gravity on Riemann surfaces. In the torus case the equations of motion are solved by two functions f and g, solutions of two independent O(2,1) sigma models, which are distinct because their first integrals contain a different time dependent phase factor. We then show that with the gauge choice [Formula: see text] the same couple of first integrals indeed solves exactly the Einstein equations for every Riemann surface. The XA= XA(xμ) polydromic mapping which extends the standard immersion of a constant curvature three-dimensional surface in a flat four-dimensional space to the case of external point sources or topology, is calculable with a simple algebraic formula in terms only of the two sigma model solutions f and g. A trivial time translation of this formalism allows us to introduce a new method which is suitable to study the scattering of black holes in (2+1)-AdS gravity.