Essential spectral gap for complex-circle limit sets

Establish whether an essential spectral gap exists for convex cocompact complex hyperbolic manifolds whose limit sets contain complex circles, including the more complicated non-Fuchsian examples with such circles.

Background

The paper proves an essential spectral gap for convex cocompact complex hyperbolic manifolds under the hypothesis that the limit set contains no complex circle. It then discusses known examples where complex circles do occur, including complex Fuchsian groups, higher-dimensional analogues, and more complicated non-Fuchsian examples. Although complex Fuchsian groups can be treated by separation of variables, the existence of an essential spectral gap for the more complicated examples is left unresolved.