Isoparametric functions on exotic spheres

This paper extends widely the work in [11]. 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 [26] (only exotic Kervaire spheres admit cohomogeneity one actions besides the standard spheres). As an application, we improve a beautiful result of Bérard-Bergery [2] (see also pp. 234-235 of [3]).