# Bisectors and foliations in complex hyperbolic space

## Uniwersytet Łódzki

In the complex hyperbolic space \(\mathbb {C}\mathbb{H}^n\) there are no hypersurfaces (of real dimension \(2n-1\)) which are totally geodesic. The hypersurfaces imitating this condition as well as possible are bisectors i.e. equidistants from pair of points. Every bisector is uniquely described by their poles i.e. two distinct points on the ideal boundary. A spane (rep. complex spine) of the bisector is the geodesic (resp. complex geodesic) joining poles. In my talk I shall formulate a local condition for a family of bisector to form a foliation of \(\mathbb{ C}\mathbb{H}^n\) and observe these foliations on the ideal boundary which has a structure of Heisenberg group. Moreover, we shall give examples of cospinal foliations and compare the situation with totally geodesic foliations of real hyperbolic space.