Isoparametric foliations and critical sets of eigenfunctions

2012/03/09 by Tang, Zizhou, Yan, Wenjiao
#Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1203.2089

Abstract

Jakobson and Nadirashvili \citeJN constructed a sequence of eigenfunctions on T² with a bounded number of critical points, answering in the negative the question raised by Yau \citeYau1 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 Mⁿ in Sⁿ⁺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.

Related

• Isoparametric foliation and Yau conjecture on the first eigenvalue
• Isoparametric foliation and Yau conjecture on the first eigenvalue, II
• Eigenfunctions with infinitely many isolated critical points
• Eigenfunctions with prescribed nodal sets
• Some remarks on critical sets of Laplace eigenfunctions