[Paper Review] Local convexity of renormalized volume for rank-1 cusped manifolds

Franco Vargas Pallete|arXiv (Cornell University)|May 3, 2015

Geometric and Algebraic Topology7 references3 citations

TL;DR

This paper establishes the local convexity of the corrected renormalized volume for acylindrical hyperbolic 3-manifolds with rank-1 cusps by introducing a modified, gluing-additive definition of renormalized volume. It proves that the geodesic conformal class at infinity is a local minimum for this corrected volume, using Hessian analysis via the skinning map and quasiconformal deformation techniques.

ABSTRACT

We study the critical points of the renormalized volume for acylindrical geometrically finite hyperbolic 3-manifolds that include rank-1 cusps, and show that the renormalized volume is locally convex around these critical points. We give a modified definition of the renormalized volume that is additive under gluing, and study some local properties.

Motivation & Objective

• To define a corrected renormalized volume that is additive under gluing of hyperbolic 3-manifolds.
• To extend the local convexity of renormalized volume from convex co-compact to acylindrical manifolds with rank-1 cusps.
• To show that the geodesic class at infinity is a local minimum for the corrected renormalized volume.
• To establish the Hessian of the corrected volume as positive definite at critical points via the skinning map and quasiconformal deformation.
• To provide a geometric and analytic framework for volume minimization in acylindrical hyperbolic 3-manifolds with cusps.

Proposed method

• Introduce a corrected renormalized volume via subtraction of half the volume of a collar region over the boundary, ensuring additivity under gluing.
• Use the skinning map to relate conformal structures at infinity of complementary components in a decomposition.
• Apply quasiconformal deformation theory to parametrize hyperbolic metrics via Beltrami differentials and quadratic holomorphic differentials.
• Compute the first and second variations of the renormalized volume using the Hessian of the metric variation under quasiconformal deformations.
• Express the Hessian in terms of the differential of the skinning map and the second fundamental form of the boundary at infinity.
• Leverage the fact that all eigenvalues of the skinning map differential lie in [−1, 1] to prove positive definiteness of the Hessian at the geodesic class.

Experimental results

Research questions

• RQ1 Is the renormalized volume locally convex for acylindrical hyperbolic 3-manifolds with rank-1 cusps?
• RQ2 Can a modified definition of renormalized volume be made additive under gluing of manifolds?
• RQ3 Does the corrected renormalized volume attain a local minimum at the geodesic conformal class?
• RQ4 How does the Hessian of the renormalized volume behave under quasiconformal deformations in the presence of rank-1 cusps?
• RQ5 What is the role of the skinning map in determining the convexity properties of the corrected volume?

Key findings

• The corrected renormalized volume is additive under gluing of acylindrical hyperbolic 3-manifolds along incompressible surfaces.
• The Hessian of the corrected renormalized volume at the geodesic class is positive definite, implying local convexity.
• The critical point at the geodesic conformal class is a local minimum for the corrected volume functional.
• The Hessian is expressed as $ 32 \text{Hess}\overline{V}_{R}(v,w) = \langle v + d\sigma(v), w - d\sigma(w) \rangle $, where $ d\sigma $ is the differential of the skinning map.
• All eigenvalues of $ d\sigma $ lie in $[-1, 1]$, ensuring the Hessian is positive definite when $ c $ is the geodesic class.
• The result extends local convexity of renormalized volume to the case of rank-1 cusped manifolds, confirming a conjectured geometric minimality property.