Isoparametric functions on exotic spheres

TL;DR

This paper investigates the existence of isoparametric functions on exotic spheres, showing that all homotopy spheres of dimension greater than four admit such functions with two focal points, contrasting with known symmetry classifications.

Contribution

It extends previous work by establishing new existence and non-existence results of isoparametric functions on exotic spheres and related manifolds, revealing novel geometric properties.

Findings

Every homotopy n-sphere (n>4) admits an isoparametric function with two focal points.

Exotic Kervaire spheres are unique among exotic spheres for admitting cohomogeneity one actions.

Improves a classical result of Bérard-Bergery regarding geometric structures on exotic spheres.

Abstract

This paper extends widely the work in \cite{GT13}. Existence and non-existence results of isoparametric functions on exotic spheres and Eells-Kuiper projective planes are established. In particular, every homotopy $n$-sphere ($n>4$) carries an isoparametric function (with certain metric) with 2 points as the focal set, in strong contrast to the classification of cohomogeneity one actions on homotopy spheres \cite{St96} ( only exotic Kervaire spheres admit cohomogeneity one actions besides the standard spheres ). As an application, we improve a beautiful result of B\'{e}rard-Bergery \cite{BB77} ( see also pp.234-235 of \cite{Be78} ).