The First Eigenvalue for the Bi-Beltrami-Laplacian on Minimal Isoparametric Hypersurfaces of $\mathbb{S}^{n+1}(1)$

TL;DR

This paper determines the first eigenvalues of the bi-Beltrami-Laplacian on minimal isoparametric hypersurfaces in spheres using a variational approach, overcoming previous difficulties and providing a simpler proof.

Contribution

The paper introduces a straightforward variational method to compute the first eigenvalues of the bi-Beltrami-Laplacian on minimal isoparametric hypersurfaces, filling a gap in existing results.

Findings

First eigenvalues explicitly determined for specific hypersurfaces

Overcomes previous technical difficulties in the limit theorem

Provides a simpler proof method for eigenvalue calculation

Abstract

In this paper, we investigate the first eigenvalues of two closed eigenvalue problems of the bi-Beltrami-Laplacian on minimal embedded isoparametric hypersurface in the unit sphere $\mathbb{S}^{n+1}(1)$. Although many mathematicians want to derive the corresponding results for the first eigenvalues of bi-Beltrami-Laplacian, they encountered great difficulties in proving the limit theorem of the version of bi-Beltrami-Laplacian along with the strategy due to I. Chavel and E. A. Feldman(Journal of Functional Analysis, 30 (1978), 198-222) and S. Ozawa (Duke Mathematics Journal, 48 (1981),767-778). Therefore, as the author knows, there are no any results of Tang-Yan type( Journal of Differential Geometry, 94 (2013) 521-540). However, by the variational argument, we overcome the difficulties and determine the first eigenvalues of the bi-Beltrami-Laplacian in the sense of isoparametric hypersurfaces. We note that our proof is quite simple.