Recent progress in isoparametric functions and isoparametric hypersurfaces

TL;DR

This survey reviews recent advances in the theory of isoparametric functions and hypersurfaces, focusing on their existence, non-existence, and the Yau conjecture related to minimal hypersurfaces in spheres.

Contribution

It summarizes recent progress in understanding isoparametric functions on Riemannian manifolds and updates on the Yau conjecture concerning eigenvalues of minimal hypersurfaces.

Findings

Existence and non-existence results for isoparametric functions on exotic spheres

Progress on the Yau conjecture for minimal isoparametric hypersurfaces

Historical overview of eigenvalue estimates in the context of isoparametric hypersurfaces

Abstract

This paper gives a survey of recent progress in isoparametric functions and isoparametric hypersurfaces, mainly in two directions. (1) Isoparametric functions on Riemannian manifolds, including exotic spheres. The existences and non-existences will be considered. (2) The Yau conjecture on the first eigenvalues of the embedded minimal hypersurfaces in the unit spheres. The history and progress of the Yau conjecture on minimal isoparametric hypersurfaces will be stated.