In the complex hyperbolic space CHn 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 CHn 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.
