Isoparametric foliations and complex structures

TL;DR

This paper constructs explicit complex structures on certain isoparametric hypersurfaces and focal submanifolds in spheres, classifies invariant complex structures on homogeneous cases, and explores their geometric properties.

Contribution

It introduces new explicit complex structures on specific isoparametric hypersurfaces and focal submanifolds, and provides a complete classification of invariant complex structures on homogeneous cases.

Findings

Constructed complex structures on isoparametric hypersurfaces with g=4, m=1 in spheres.

Classified invariant complex structures on homogeneous isoparametric hypersurfaces with g=4.

Discussed geometric properties of the constructed complex structures.

Abstract

Explicit representations of complex structures on closed manifolds are valuable, but relatively rare in the literature. Using isoparametric theory, we construct complex structures on isoparametric hypersurfaces with $g=4, m=1$ in the unit sphere, as well as on focal submanifolds $M_+$ of OT-FKM type with $g=4$, $m=2$ and $m=4$ in the definite case. Furthermore, we investigate the existence and non-existence of invariant (almost) complex structures on homogeneous isoparametric hypersurfaces with $g=4$, providing a complete classification of those that admit such structures. Finally, we discuss the geometric properties of the complex structures arising from isoparametric theory.