Behaviour of the Schwarzian derivative on long complex projective tubes

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.