Volume functions and boundary data of 3-dimensional hyperbolic manifolds

2025/10/07 by Schlenker, Jean-Marc
#Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Geometric Topology (math.GT) #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.2510.05627

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).

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Related

• Renormalized volume and the volume of the convex core
• Adapted Renormalized Volume for Hyperbolic 3-Manifolds with Compressible Boundary
• Convexity of the renormalized volume of hyperbolic 3-manifolds
• Convex co-compact hyperbolic manifolds are determined by their pleating lamination
• Isoperimetric interpretation for the renormalized volume of convex co-compact hyperbolic 3-manifolds