Volume functions and boundary data of 3-dimensional hyperbolic manifolds

TL;DR

The article surveys the geometry of convex co-compact hyperbolic 3-manifolds, focusing on dual boundary data—the induced boundary metric and the bending lamination—and on the renormalized volume as a bridge between bulk geometry and boundary functionals like the Liouville action and the Schwarzian at infinity. It develops variational formulas that reveal a symplectic/Legendre structure linking boundary data to bulk deformations, and it connects these 3D objects to 2D boundary theories via Teichmüller theory and Weil–Petersson geometry, with deep ties to the AdS/CFT correspondence and to Loewner energy. The survey also extends to larger convex subsets, manifolds with compressible boundaries (via adapted renormalized volume), and Lorentzian analogs (GHMC spacetimes, de Sitter and anti-de Sitter geometries, Minkowski/half-space models), highlighting dualities and open questions about universality and boundary data. Together, these results illuminate how boundary conformal data encode bulk hyperbolic geometry and how holographic-type dualities surface in both mathematical and physical contexts, including sharp bounds and reciprocity principles in the quasifuchsian regime.

Abstract

We review recent progress on two closely related sets of questions concerning convex co-compact hyperbolic manifolds, or convex domains in those manifolds, such as their convex core. The first set of questions is to what extent the hyperbolic metric on such a manifold is uniquely determined by either of two possible geometric data on their boundary. The second aspect is the ``volume'' associated to such a manifold, such as the renormalized volume of a convex co-compact hyperbolic manifold. The relation between the two is provided by the first variation of the volume functions, which involves the two kinds of boundary data as ``conjugate'' variables. While progress has recently been made on some questions, others remain open. New connections have recently emerged, with physics (and in particular the AdS/CFT correspondence) as well as with probability theory (the Loewner energy).