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Behaviour of the Schwarzian derivative on long complex projective tubes

Tommaso Cremaschi, Viola Giovannini

TL;DR

This work analyzes how the Schwarzian derivative of developing maps for complex projective structures behaves on long tubes around compressible boundary geodesics. By expressing the real part of the Schwarzian as an Osgood–Stowe tensor and applying a Stokes-type lemma, the authors derive sharp bounds for its pairings with infinitesimal earthquakes and graftings, and relate these to the differential of the renormalized volume V_R for convex cocompact hyperbolic 3-manifolds. A symmetric-tube toy model provides explicit leading terms, which, together with a detailed Fourier analysis on the tube cross-section, yields precise asymptotics and error estimates. These results enable quantitative control of renormalized volume along complex earthquake paths and provide the asymptotic behavior under pinching a compressible curve, highlighting the −π^3/ℓ divergence as curves collapse. Overall, the paper advances understanding of the interplay between Schwarzian geometry, Teichmüller theory, and 3‑manifold renormalized volume in the compressible boundary regime.

Abstract

The Schwarzian derivative parametrizes the fibres of the space of complex projective structures on a surface as vector bundle over its Teichmüller space. We study its behaviour on long complex projective tubes, and get estimates for the pairing of its real part with infinitesimal earthquakes and graftings. As the real part of their Schwarzian coincides with the differential of the renormalized volume we obtain bounds for the variation of renormalized volume under complex earthquake paths, and its asymptotic behaviour under pinching a compressible curve.

Behaviour of the Schwarzian derivative on long complex projective tubes

TL;DR

This work analyzes how the Schwarzian derivative of developing maps for complex projective structures behaves on long tubes around compressible boundary geodesics. By expressing the real part of the Schwarzian as an Osgood–Stowe tensor and applying a Stokes-type lemma, the authors derive sharp bounds for its pairings with infinitesimal earthquakes and graftings, and relate these to the differential of the renormalized volume V_R for convex cocompact hyperbolic 3-manifolds. A symmetric-tube toy model provides explicit leading terms, which, together with a detailed Fourier analysis on the tube cross-section, yields precise asymptotics and error estimates. These results enable quantitative control of renormalized volume along complex earthquake paths and provide the asymptotic behavior under pinching a compressible curve, highlighting the −π^3/ℓ divergence as curves collapse. Overall, the paper advances understanding of the interplay between Schwarzian geometry, Teichmüller theory, and 3‑manifold renormalized volume in the compressible boundary regime.

Abstract

The Schwarzian derivative parametrizes the fibres of the space of complex projective structures on a surface as vector bundle over its Teichmüller space. We study its behaviour on long complex projective tubes, and get estimates for the pairing of its real part with infinitesimal earthquakes and graftings. As the real part of their Schwarzian coincides with the differential of the renormalized volume we obtain bounds for the variation of renormalized volume under complex earthquake paths, and its asymptotic behaviour under pinching a compressible curve.

Paper Structure

This paper contains 26 sections, 31 theorems, 261 equations, 1 figure.

Table of Contents

  1. Introduction and results
  2. Outline of the paper
  3. Notation and background
  4. Riemann, Complex Projective, and Hyperbolic Surfaces
  5. Teichmüller space and Weil-Petersson pairing
  6. Margulis tubes
  7. Complex Earthquakes along simple closed geodesics
  8. A Stokes type lemma
  9. Hyperbolic 3-manifolds.
  10. The Thurston metric
  11. The renormalized volume
  12. Toy model: the symmetric tube case
  13. Application to Renormalized Volume
  14. Renormalized volume bounds under earthquakes on short compressible curves
  15. Asymptotic behaviour of $V_R$ along pinching a compressible curve
  16. ...and 11 more sections

Key Result

Lemma 2.2

Let $X$ be a Riemann surface structure on $S$, with $z$ its complex coordinate. Let $\gamma\colon \mathbb{R}/\ell\mathbb{Z}\rightarrow X$ be a unit length parameterization of a simple closed geodesic with respect to the unique hyperbolic representative in $X$, of length $\ell$. Let also $\mu\in T_{X and

Figures (1)

  • Figure 1: The long tube in the universal covers with the respective metrics.

Theorems & Definitions (75)

  • Definition 2.1
  • Lemma 2.2
  • proof : Proof
  • Definition 2.3
  • Theorem 2.4
  • Remark 3.1
  • Remark 3.2
  • Lemma 3.3
  • proof : Proof
  • Remark 3.4
  • ...and 65 more