# Renormalized volume on the Teichm\"uller space of punctured surfaces

Research paper by Colin Guillarmou, Sergiu Moroianu, Frédéric Rochon

Indexed on: 21 Dec '15Published on: 21 Dec '15Published in: Mathematics - Differential Geometry

#### Abstract

We define and study the renormalized volume for geometrically finite hyperbolic \$3\$-manifolds, including with rank-\$1\$ cusps. We prove a variation formula, and show that for certain families of convex co-compact hyperbolic metrics \$g_\eps\$ degenerating to a geometrically finite hyperbolic metric \$g_0\$ with rank-\$1\$ cusps, the renormalized volume converges to the renormalized volume of the limiting metric.