Renormalized volume on the Teichmüller space of punctured surfaces

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.

54 pages, minor corrections. Proof of the fact that renormalized volume of punctured surfaces is a Kahler Potential for Weil Petersson metric added in the quasi-Fuchsian setting

References in corpus (4)

• On the renormalized volume of hyperbolic 3-manifolds
• Normalized Entropy versus Volume
• Local convexity of renormalized volume for rank-1 cusped manifolds
• Positivity of the renormalized volume of almost-Fuchsian hyperbolic $3$-manifolds

Cited by in corpus (5)

• Continuity of the renormalized volume under geometric limits
• Additive continuity of the renormalized volume under geometric limits
• Local convexity of renormalized volume for rank-1 cusped manifolds
• Geometrically finite Poincaré-Einstein metrics
• Compactification and distance on Teichmüller space via renormalized volume