ON THE RENORMALIZED VOLUME OF HYPERBOLIC 3-MANIFOLDS WITH COMPRESSIBLE BOUNDARY

We investigate the renormalized volume of convex co-compact hyperbolic $3$-manifolds in the setting where the boundary is compressible. We make some steps forward in the study of Maldacena's question, which asks: given a Riemann surface, what is the most efficient way to fill it with a convex co-compact hyperbolic $3$-manifold in order to minimize the renormalized volume? In particular, we prove that every closed Riemann surface of genus at least two, with enough curves of sufficiently short hyperbolic length, is the conformal boundary at infinity of a convex co-compact handlebody of negative renormalized volume. We give an explicit description of the behaviour of the Schwarzian derivative - and thus of the differential of the renormalized volume - on long compressible tubes in complex projective surfaces. Furthermore, we establish bounds for its pairing with infinitesimal earthquakes and infinitesimal graftings. We derive how the renormalized volume changes along earthquake and grafting paths. We define a new version of renormalized volume that adapts to the compressible boundary case, and which, unlike the standard one in this setting, is bounded from below. As a function on Teichm\"uller space, we show that its differential has infinity norm bounded by a constant depending only on the genus of the surface, as well as the Weil-Petersson norm of its Weil-Petersson gradient. Moreover, we define the adapted renormalized volume for surfaces in the compressible strata of the Weil-Petersson completion of Teichm\"uller space, and we prove that the adapted renormalized volume extends continuously to these strata, up to a codimension-one subset.