Black-hole thermodynamics and Riemann surfaces

We use the analytic continuation procedure proposed in our earlier works to study the thermodynamics of black holes in 2 + 1 dimensions. A general black hole in 2 + 1 dimensions has g handles hidden behind h horizons. The result of the analytic continuation of a black-hole spacetime is a hyperbolic 3-manifold having the topology of a handlebody. The boundary of this handlebody is a compact Riemann surface of genus G = 2g + h - 1. Conformal moduli of this surface encode in a simple way the physical characteristics of the black hole. The moduli space of black holes of a given type (g, h) is then the Schottky space at genus G. The (logarithm of the) thermodynamic partition function of the hole is the K?hler potential for the Weil-Peterson metric on the Schottky space. The Bekenstein bound on the black-hole entropy leads us to conjecture a new strong bound on this K?hler potential.