Linear volume bounds for drilling and filling

TL;DR

This work proves uniform linear bounds on volume changes under drilling and Dehn filling in finite-volume hyperbolic 3-manifolds. The authors develop a robust interpolation framework between hyperbolic tubes and cusps, controlling scalar curvature via carefully constructed metric data, and leverage the sigma invariant to compare volumes across modified geometries, avoiding multiplicative volume factors. For drilling, they obtain vol$(M-\gamma)\le$ vol$(M)+c\ell/R$ when the drilled geodesic has length $\ell$ and a tubular radius $R$, and for the shortest geodesic vol$(M-\gamma)\le$ vol$(M)+c\ell$; for cusp filling, they prove vol$(M_\mu)+\frac{A}{2}\frac{\pi^2}{\ell^2}(1-c\frac{\pi^2}{\ell^2})\le$ vol$(M)$, with $A$ the cusp area and $\ell$ the slope length, providing a linear bound on volume decrease. The results extend the linear-volume philosophy behind the $2\pi$-Theorem into a precise quantitative regime, using Dehn-filling limits and sigma-invariant comparisons to achieve additive, rather than multiplicative, volume errors.

Abstract

We prove uniform linear bounds on the volume variation under drilling and filling operations on finite volume hyperbolic 3-manifolds.

Theorems & Definitions (14)

• Theorem 1
• Corollary 2
• Theorem 3
• Definition 2.1: Sigma Invariant
• Proposition 2.2
• proof
• Theorem 2.3: see KL
• Lemma 2.4: see A or BH
• Lemma 3.1
• proof