Real hypersurfaces with two principal curvatures in complex projective and hyperbolic planes

TL;DR

This paper constructs and classifies the first known examples of real hypersurfaces with two nonconstant principal curvatures in complex projective and hyperbolic planes, revealing their foliation structure and symmetry properties.

Contribution

It introduces the first examples and classification of such hypersurfaces, linking their geometry to foliations by Lagrangian flat surfaces and cohomogeneity two polar actions.

Findings

Hypersurfaces are foliated by equidistant Lagrangian flat surfaces.

Each hypersurface corresponds to principal orbits of a cohomogeneity two polar action.

The classification provides new insights into the geometry of real hypersurfaces in complex space forms.

Abstract

We find the first examples of real hypersurfaces with two nonconstant principal curvatures in complex projective and hyperbolic planes, and we classify them. It turns out that each such hypersurface is foliated by equidistant Lagrangian flat surfaces with parallel mean curvature or, equivalently, by principal orbits of a cohomogeneity two polar action.