[Paper Review] Universal bounds for hyperbolic Dehn surgery

Craig D. Hodgson, Steven P. Kerckhoff|ArXiv.org|Apr 30, 2002

Geometric and Algebraic Topology20 references4 citations

TL;DR

This paper establishes universal quantitative bounds for hyperbolic Dehn surgery on cusped hyperbolic 3-manifolds by constructing one-parameter families of hyperbolic cone-manifold structures via infinitesimal harmonic deformations and geometric limit analysis. It proves that if the normalized length of a Dehn filling curve exceeds 7.515, the resulting manifold is hyperbolic, with volume at least 1.701, and provides universal upper bounds of 60 (one cusp) and 114 (multiple cusps) non-hyperbolic surgeries per boundary torus.

ABSTRACT

This paper gives a quantitative version of Thurston's hyperbolic Dehn surgery theorem. Applications include the first universal bounds on the number of non-hyperbolic Dehn fillings on a cusped hyperbolic 3-manifold, and estimates on the changes in volume and core geodesic length during hyperbolic Dehn filling. The proofs involve the construction of a family of hyperbolic cone-manifold structures, using infinitesimal harmonic deformations and analysis of geometric limits.

Motivation & Objective

• To establish universal upper bounds on the number of non-hyperbolic Dehn fillings for any cusped hyperbolic 3-manifold, independent of the initial manifold.
• To provide quantitative estimates on volume change and core geodesic length during hyperbolic Dehn filling.
• To resolve the long-standing open question of whether the number of non-hyperbolic surgeries is universally bounded across all cusped hyperbolic 3-manifolds.
• To develop a general method using cone-manifold structures and geometric limits to analyze the behavior of hyperbolic structures under Dehn filling.
• To demonstrate that normalized length ≥7.515 guarantees hyperbolicity of the filled manifold, with volume at least 1.701.

Proposed method

• Constructs a one-parameter family of hyperbolic cone-manifold structures on the Dehn-filled manifold, deforming from a complete hyperbolic structure to a cone-manifold with cone angle 2π.
• Applies infinitesimal harmonic deformations to analyze the first-order change in geometry during the deformation process.
• Uses Schläfli’s formula to relate changes in volume to changes in cone angle and core geodesic length.
• Employs a rescaling of the boundary torus metric to define normalized length, ensuring universality independent of the original manifold’s geometry.
• Analyzes geometric limits and tube radius behavior to control the deformation path and ensure convergence to a smooth hyperbolic structure.
• Derives differential inequalities for volume change using functions H(z), G(z), and H′(z), integrating over the deformation path to bound volume loss.

Experimental results

Research questions

• RQ1 Is there a universal upper bound on the number of non-hyperbolic Dehn fillings per cusp across all finite-volume, cusped hyperbolic 3-manifolds?
• RQ2 Can the volume of a Dehn-filled manifold be universally bounded below when the filling curve has sufficiently long normalized length?
• RQ3 Does the normalized length of a filling curve—defined as geodesic length divided by the square root of torus area—provide a universal criterion for hyperbolicity of the filled manifold?
• RQ4 How does the core geodesic length change during the deformation from a cone-manifold to a complete hyperbolic structure?
• RQ5 Can the volume change during Dehn filling be universally bounded, independent of the initial manifold?

Key findings

• For any cusped hyperbolic 3-manifold with a single cusp, at most 60 Dehn fillings per boundary torus yield non-hyperbolic manifolds.
• For manifolds with multiple cusps, at most 114 Dehn fillings per torus yield non-hyperbolic manifolds.
• If the normalized length of the filling curve is at least 7.515, the resulting Dehn-filled manifold is hyperbolic and has volume at least 1.701.
• The volume of the filled manifold decreases by at most 0.3287 compared to the original cusped manifold when normalized length ≥7.515.
• The method provides a universal bound on volume loss, independent of the initial manifold, based on geometric deformation and cone-manifold analysis.
• The results imply that all but a universal number of surgeries yield irreducible, atoroidal, and word hyperbolic manifolds, supporting the geometrization conjecture.