Existence and uniqueness of inhomogeneous ruled hypersurfaces with shape operator of constant norm in the complex hyperbolic space

We complete the classification of ruled real hypersurfaces with shape operator of constant norm in nonflat complex space forms by showing the existence of a unique inhomogeneous example in the complex hyperbolic space.

Real hypersurfaces in the complex hyperbolic quadrics with isometric Reeb flow

We classify real hypersurfaces with isometric Reeb flow in the complex hyperbolic quadrics [Formula: see text], [Formula: see text]. We show that [Formula: see text] is even, say [Formula: see text], and any such hypersurface becomes an open part of a tube around a [Formula: see text]-dimensional complex hyperbolic space [Formula: see text] which is embedded canonically in [Formula: see text] as a totally geodesic complex submanifold or a horosphere whose center at infinity is [Formula: see text]-isotropic singular. As a consequence of the result, we get the nonexistence of real hypersurfaces with isometric Reeb flow in odd-dimensional complex quadrics [Formula: see text], [Formula: see text].