Quakebend deformations in complex hyperbolic quasi-Fuchsian space

Abstract

We study quakebend deformations in complex hyperbolic quasi-Fuchsian space ${\mathcal{Q}}_{ℂ}\left(\Sigma \right)$ of a closed surface $\Sigma$ of genus $g>1$, that is the space of discrete, faithful, totally loxodromic and geometrically finite representations of the fundamental group of $\Sigma$ into the group of isometries of complex hyperbolic space. Emanating from an $ℝ$–Fuchsian point $\rho \in {\mathcal{Q}}_{ℂ}\left(\Sigma \right)$, we construct curves associated to complex hyperbolic quakebending of $\rho$ and we prove that we may always find an open neighborhood $U\left(\rho \right)$ of $\rho$ in ${\mathcal{Q}}_{ℂ}\left(\Sigma \right)$ containing pieces of such curves. Moreover, we present generalisations of the well known Wolpert–Kerckhoff formulae for the derivatives of geodesic length function in Teichmüller space.