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

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 $g_{\text{eps}}$ 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.