Behaviour of the Schwarzian derivative on long complex projective tubes

TL;DR

This paper investigates the behavior of the Schwarzian derivative on long complex projective tubes, providing estimates and bounds for its interaction with geometric deformations and the renormalized volume in complex projective structures.

Contribution

It offers new estimates for the Schwarzian derivative's pairing with earthquakes and graftings, and analyzes the asymptotic behavior of the renormalized volume under complex deformations.

Findings

Bounds for the variation of renormalized volume under complex earthquake paths

Asymptotic behavior of the Schwarzian derivative under pinching curves

Estimates for the pairing of the Schwarzian's real part with geometric deformations

Abstract

The Schwarzian derivative parametrizes the fibres of the space of complex projective structures on a surface as vector bundle over its Teichm\"uller 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.