[Paper Review] Additive continuity of the renormalized volume under geometric limits

Franco Vargas Pallete|arXiv (Cornell University)|Aug 14, 2017

Geometric Analysis and Curvature Flows13 references3 citations

TL;DR

This paper establishes the additive continuity of the renormalized volume for convex-cocompact hyperbolic 3-manifolds under geometric limits, proving that the infimum of the renormalized volume is achieved precisely when the geometric limit consists of Fuchsian or acylindrical components with totally geodesic convex cores. The main result shows that the renormalized volume of a limit manifold equals the sum of the renormalized volumes of its components, generalizing continuity results and characterizing sequences converging to the infimum.

ABSTRACT

We study the infimum of the renormalized volume for convex-cocompact hyperbolic manifolds, as well as describing how a sequence converging to such values behaves. In particular, we show that the renormalized volume is continuous under the appropriate notion of limit. This result generalizes previous work in the subject.

Motivation & Objective

• To characterize the infimum of the renormalized volume for convex-cocompact hyperbolic 3-manifolds with incompressible boundary.
• To describe the behavior of sequences of such manifolds converging to this infimum value.
• To define an appropriate notion of geometric convergence—additive geometric convergence—under which the renormalized volume is continuous.
• To prove that any sequence converging to the infimum limit must converge to a manifold whose convex core is totally geodesic, i.e., a critical point of the renormalized volume.

Proposed method

• Introduces the notion of additive geometric convergence to describe limits of sequences of convex-cocompact hyperbolic manifolds with bounded convex core volume.
• Uses Dirichlet fundamental polyhedra and their genericity with respect to the sphere to analyze geometric limits.
• Applies Dehn-filling arguments and Epstein surface constructions to relate renormalized volume to the Gromov norm via the formula $ V_R(N_i) = \frac{v_3}{2}\|DN_i\| $.
• Employs Klein-Maskit combinations and generalized hyperbolic Dehn fillings to construct sequences converging to the limit components.
• Uses the classical continuity argument of [GMR] adapted to the new convergence notion to prove that the limit of renormalized volumes equals the sum of the volumes of the components.
• Combines all components to show that the infimum is achieved only when the limit manifold has a totally geodesic convex core, which characterizes the critical points of the renormalized volume.

Experimental results

Research questions

• RQ1 What is the infimum value of the renormalized volume for a convex-cocompact hyperbolic 3-manifold with incompressible boundary?
• RQ2 How do sequences of such manifolds behave when their renormalized volume approaches the infimum?
• RQ3 What is the appropriate notion of geometric convergence under which the renormalized volume is continuous?
• RQ4 Under what conditions is the limit of a minimizing sequence a critical point of the renormalized volume?
• RQ5 Can the renormalized volume of a limit manifold be expressed additively as the sum of the volumes of its components?

Key findings

• The infimum of the renormalized volume for a convex-cocompact hyperbolic 3-manifold is achieved if and only if the geometric limit has a totally geodesic convex core.
• Any sequence of convex-cocompact hyperbolic manifolds converging to the infimum must converge geometrically to a manifold whose convex core is totally geodesic.
• The renormalized volume is additive under the new notion of additive geometric convergence: the limit volume equals the sum of the volumes of the component manifolds.
• For quasifuchsian manifolds, the renormalized volume is non-negative, and it vanishes if and only if the manifold is Fuchsian.
• In the acylindrical case, any minimizing sequence converges geometrically to a unique Fuchsian metric with totally geodesic convex core.
• The limit components are either finite-volume hyperbolic manifolds (from acylindrical parts) or Seifert fibered manifolds (from Fuchsian parts), with volume contributions related to the Gromov norm via $ V_R(N_i) = \frac{v_3}{2}\|DN_i\| $.