Real hypersurfaces in complex hyperbolic two-plane Grassmannians with Reeb vector field

In this paper we give a characterization of real hypersurfaces in the noncompact complex two-plane Grassmannian SU(2,m)/S(U(2)⋅U(m)), m⩾2, with Reeb vector field ξ belonging to the maximal quaternionic subbundle Q. Then we show that such a hypersurface must be a tube over a totally real totally geodesic HHn, m=2n, in the noncompact complex two-plane Grassmannian SU(2,m)/S(U(2)⋅U(m)), a horosphere whose center at the infinity is singular or an exceptional case.