Isoparametric foliations and critical sets of eigenfunctions

TL;DR

This paper investigates the critical sets of specific eigenfunctions on isoparametric hypersurfaces in spheres, revealing unique configurations of critical points and submanifolds that challenge previous assumptions about eigenfunction behavior.

Contribution

It identifies explicit eigenfunctions on minimal isoparametric hypersurfaces with novel critical set structures, including isolated points and submanifolds, expanding understanding of eigenfunction geometry.

Findings

Eigenfunctions with critical sets of 8 points and submanifolds on isoparametric hypersurfaces.

Eigenvalues associated with these eigenfunctions are n, 2n, and 3n.

Similar critical set phenomena occur on focal submanifolds.

Abstract

Jakobson and Nadirashvili \cite{JN} constructed a sequence of eigenfunctions on $T^2$ with a bounded number of critical points, answering in the negative the question raised by Yau \cite{Yau1} which asks that whether the number of the critical points of eigenfunctions for the Laplacian increases with the corresponding eigenvalues. The present paper finds three interesting eigenfunctions on the minimal isoparametric hypersurface $M^n$ in $S^{n+1}(1)$. The corresponding eigenvalues are $n$, $2n$ and $3n$, while their critical sets consist of $8$ points, a submanifold(infinite many points) and $8$ points, respectively. On one of its focal submanifolds, a similar phenomenon occurs.