Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Schwarzian Bounds on Bending in Hyperbolic 3-Manifolds

Martin Bridgeman, Ming Hong Tee

TL;DR

This work builds an explicit quantitative bridge between two classical ways to measure deviation from Möbius maps: the analytic Schwarzian norm $\|Sf\|_\infty$ and Thurston’s geometric bending norm $\|\beta_f\|_L$. Utilizing Epstein surfaces, the authors derive an exact bound $\|\beta_f\|_L \le B_L(\|Sf\|_\infty)$ for univalent maps when $\|Sf\|_\infty$ is small, with $B_L$ given explicitly and exhibiting linear behavior near zero. They further translate this analytic-geometric dictionary into effective bounds for the bending laminations of quasifuchsian manifolds in terms of Teichmüller distance, via a Lipschitz Bers embedding that connects deformation data to Schwarzian data. The results sharpen the analytic-geometric correspondence, showing that small Schwarzian norm both preserves injectivity and controls convex-hull bending, and provide practical, computable bounds in the quasifuchsian setting. Altogether, the paper enhances the toolkit for relating conformal data on the disk to hyperbolic geometry in 3-space through Epstein-surface geometry and Teichmüller theory.

Abstract

The Schwarzian derivative provides a classical analytic measure of how far a holomorphic map of the disk is from being Möbius, with Nehari's bounds giving sharp criteria for univalence. Independently, Thurston introduced a geometric parametrization of locally univalent maps via bending measured laminations on the hyperbolic plane, capturing deviation from roundness in hyperbolic three-space. While both approaches quantify the same phenomenon, their precise relationship has remained only implicit. In this paper we establish explicit quantitative bounds relating the Schwarzian norm $\|Sf\|_\infty$ and the bending norm $\|β_f\|_L$. In particular, for univalent maps with $\|Sf\|_\infty< 1/2$, we show that $\|β_f\|_L$ is controlled by an elementary function $B_L(\|Sf\|_L)$ that we compute explicitly. As an application, we obtain new effective bounds on the bending laminations of quasifuchsian manifolds in terms of the Teichmüller distance between their conformal boundary components. Our results sharpen the analytic-geometric correspondence between the Schwarzian derivative and hyperbolic geometry showing that just as small Schwarzian norm forces injectivity, it also forces controlled bending of convex hull boundaries.

Schwarzian Bounds on Bending in Hyperbolic 3-Manifolds

TL;DR

This work builds an explicit quantitative bridge between two classical ways to measure deviation from Möbius maps: the analytic Schwarzian norm and Thurston’s geometric bending norm . Utilizing Epstein surfaces, the authors derive an exact bound for univalent maps when is small, with given explicitly and exhibiting linear behavior near zero. They further translate this analytic-geometric dictionary into effective bounds for the bending laminations of quasifuchsian manifolds in terms of Teichmüller distance, via a Lipschitz Bers embedding that connects deformation data to Schwarzian data. The results sharpen the analytic-geometric correspondence, showing that small Schwarzian norm both preserves injectivity and controls convex-hull bending, and provide practical, computable bounds in the quasifuchsian setting. Altogether, the paper enhances the toolkit for relating conformal data on the disk to hyperbolic geometry in 3-space through Epstein-surface geometry and Teichmüller theory.

Abstract

The Schwarzian derivative provides a classical analytic measure of how far a holomorphic map of the disk is from being Möbius, with Nehari's bounds giving sharp criteria for univalence. Independently, Thurston introduced a geometric parametrization of locally univalent maps via bending measured laminations on the hyperbolic plane, capturing deviation from roundness in hyperbolic three-space. While both approaches quantify the same phenomenon, their precise relationship has remained only implicit. In this paper we establish explicit quantitative bounds relating the Schwarzian norm and the bending norm . In particular, for univalent maps with , we show that is controlled by an elementary function that we compute explicitly. As an application, we obtain new effective bounds on the bending laminations of quasifuchsian manifolds in terms of the Teichmüller distance between their conformal boundary components. Our results sharpen the analytic-geometric correspondence between the Schwarzian derivative and hyperbolic geometry showing that just as small Schwarzian norm forces injectivity, it also forces controlled bending of convex hull boundaries.

Paper Structure

This paper contains 9 sections, 13 theorems, 109 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Results
  3. Background
  4. Teichmuller space, hyperbolic 3-manifolds, convex hulls
  5. Thurston's parametrization for locally valent maps
  6. Bounding bending by thickness of convex hull
  7. Hyperbolic Trigonometry
  8. Schwarzian bound on Thickness
  9. Teichmüller distance

Key Result

Theorem 2.1

Let $f:\Delta\rightarrow \hat{\mathbb C}$ be locally univalent. If $f$ is univalent then $\|Sf\|_\infty < 3/2$ and if $\|Sf\|_\infty < 1/2$ then $f$ is univalent.

Figures (3)

  • Figure 1: Graph of $B_1$
  • Figure 2: Case 1 configuration of geodesics
  • Figure 3: Case 2 configuration of geodesics

Theorems & Definitions (13)

  • Theorem 2.1: Nehari, nehari
  • Theorem 2.2: Bridgeman-Bromberg, BB_advances
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 4.1
  • Lemma 4.2
  • Theorem 5.1: Epstein, Epstein:surface
  • Corollary 5.2
  • Theorem 6.1: Ahlfors-Weill ahlforsweill
  • Theorem 6.2
  • ...and 3 more