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The first eigenvalue of homogeneous minimal hypersurfaces in a unit sphere \(S^{n+1}(1)\). (English) Zbl 0578.53043
The Laplacian on a Riemannian \(n\)-manifold \(M\) minimally immersed into a unit sphere \(S^{n+1}(1)\) has, due to a theorem of T. Takahashi [J. Math. Soc. Japan 18, 380–385 (1966; Zbl 0145.18601)], the first eigenvalue not greater than \(n\). With this fact in mind the present author computes the first eigenvalues for some homogeneous minimal hypersurfaces in a unit sphere \(S^{n+1}(1)\). It is to be noted that a homogeneous hypersurface of \(S^{n+1}(1)\) has constant principal curvatures so that it is isoparametric. For some of such hypersurfaces these eigenvalues were already computed by H. Muto, Y. Ohnita and H. Urakawa [Tôhoku Math. J. (2) 36, 253–267 (1984; Zbl 0539.53044)], but in the present paper the following theorem is proved:
If \(M\) is an \(n\)-dimensional compact homogeneous minimal hypersurface in a unit sphere with \(r\) distinct principal curvatures, then the first eigenvalue of the Laplacian on \(M\) is \(n\) unless \(r=4\).
The idea developed in [W. Y. Hsiang and H. B. Lawson jun., J. Differ. Geom. 5, 1–38 (1971; Zbl 0219.53045)] and [R. Takagi and T. Takahashi, Differential Geometry, in Honor of Kentaro Yano, 469–481 (1972; Zbl 0244.53042)] and the method used in the above-mentioned paper by H. Muto and others are applied in the author’s proof.
Reviewer: Y.Muto

MSC:
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
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