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In this paper, we investigate the first eigenvalues of two closed eigenvalueproblems of the bi-Beltrami-Laplacian on minimal embedded isoparametrichypersurface in the unit sphere $\mathbb{S}^{n+1}(1)$. Although manymathematicians want to derive the corresponding results for the firsteigenvalues of bi-Beltrami-Laplacian, they encountered great difficulties inproving the limit theorem of the version of bi-Beltrami-Laplacian along withthe 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 thevariational argument, we overcome the difficulties and determine the firsteigenvalues of the bi-Beltrami-Laplacian in the sense of isoparametrichypersurfaces. We note that our proof is quite simple.

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