Let π be the fundamental group of a closed surface Σ of genus g > 1. One of the fundamental problems in complex hyperbolic geometry is to find all discrete, faithful, geometrically finite and purely loxodromic representations of π into SU(2, 1), (the triple cover of) the group of holomorphic isometries of H2C. In particular, given a discrete, faithful, geometrically finite and purely loxodromic representation ρ0 of π1, can we find an open neighbourhood of ρ0 comprising representations with these properties. We show that this is indeed the case when ρ0 preserves a totally real Lagrangian plane.
"Open Sets of Maximal Dimension in Complex Hyperbolic Quasi-Fuchsian Space." J. Differential Geom. 73 (2) 319 - 350, June 2006. https://doi.org/10.4310/jdg/1146169913