Foliated Hopf hypersurfaces in complex hyperbolic quadrics

TL;DR

This paper constructs the first known examples of homogeneous Hopf hypersurfaces with integrable maximal complex subbundle in a non-compact Hermitian symmetric space of rank two, advancing understanding in contact geometry and complex hyperbolic quadrics.

Contribution

It provides explicit constructions of a one-parameter family of such hypersurfaces in a non-compact Hermitian symmetric space, filling a gap in the existence problem.

Findings

First examples of such hypersurfaces in irreducible Kahler manifolds.

Explicit one-parameter family constructed.

Advances understanding of contact geometry in complex hyperbolic spaces.

Abstract

This paper deals with a limiting case motivated by contact geometry. The limiting case of a tensorial characterization of contact hypersurfaces in Kahler manifolds leads to Hopf hypersurfaces whose maximal complex subbundle of the tangent bundle is integrable. It is known that in non-flat complex space forms and in complex quadrics such real hypersurfaces do not exist, but the existence problem in other irreducible Kahler manifolds is open. In this paper we construct explicitly a one-parameter family of homogeneous Hopf hypersurfaces, whose maximal complex subbundle of the tangent bundle is integrable, in a Hermitian symmetric space of non-compact type and rank two. These are the first known examples of such real hypersurfaces in irreducible Kahler manifolds.