Isoparametric foliations and complex structures

TL;DR

The paper investigates explicit complex and almost complex structures on isoparametric hypersurfaces in unit spheres, focusing on the g=4 case and OT-FKM type focal submanifolds. It develops integrable complex structures on isoparametric hypersurfaces with $(g,m_1,m_2)=(4,1,l-2)$, and constructs several complex structures on focal submanifolds M_+ for OT-FKM type with $m=2$ and $m=4$, including an alternative Calabi–Eckmann embedding for $m=2$. It further analyzes homogeneous examples, establishing invariant complex-structure existence criteria via restricted root decompositions, and shows nonexistence of invariant structures in key cases (e.g., $M_+$ with $m=8$ under Spin(9)). The geometric properties are explored, revealing that Kähler structures occur only in the $(2,2)$ case, and that many constructed complex structures are not balanced, highlighting both obstructions and possibilities for explicit complex-geometric structures in isoparametric geometry.

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.

Theorems & Definitions (31)

• Proposition 1
• Theorem 1
• Theorem 2
• Proposition 2
• Remark 1
• Theorem 3
• Remark 2
• proof
• Proposition 3
• Lemma 2.1