A Twistor Construction of Hopf Real Hypersurfaces in Complex hyperbolic Space

TL;DR

This paper introduces a twistor-based method to construct and classify Hopf real hypersurfaces in complex hyperbolic space, solving longstanding existence problems and providing new examples beyond classical ones.

Contribution

It presents a unified twistor construction for Hopf hypersurfaces in complex hyperbolic space and addresses the existence of hypersurfaces with principal curvature 2.

Findings

Constructed Hopf hypersurfaces from horizontal submanifolds in twistor spaces.

Identified twistor spaces with sets of circles in geodesic lines.

Provided new examples with principal curvature 2.

Abstract

It is very well known that Hopf real hypersurfaces in the complex projective space can be locally characterized as tubes over complex submanifolds. This also holds true for some, but not all, Hopf real hypersurfaces in the complex hyperbolic space. The main goal of this paper is to show, in a unified way, how to construct Hopf real hypersurfaces in the complex hyperbolic space from a horizontal submanifold in one of the three twistor spaces of the indefinite complex $2$-plane Grassmannian with respect to the natural para-quaternionic K\"ahler structure. We also identify these twistor spaces with the sets of circles in totally geodesic complex hyperbolic lines in the complex hyperbolic space. As an application, we describe all classical Hopf examples. We also solve the remarkable and long-standing problem of the existence of Hopf real hypersurfaces in the complex hyperbolic space, different from the horosphere, such that the associated principal curvature is $2$. We exhibit a method to obtain plenty of them.