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On spectral characterizations of minimal hypersurfaces in a sphere. (English) Zbl 0826.58009
Let \(M\) be a closed minimal hypersurface in a Euclidean \((n + 1)\)-sphere. If \(n = 3\), and \(M\) is isoparametric, then it is completely determined by its spectrum \(\text{Spec}^p(M)\), \(p \in \{0,1,2,3\}\). In higher dimensions it is shown that \(M\) is a Clifford torus \(M_{m,n-m}\), if it has that spectrum for \(p = 0,1\). For \(2m = n\), the coincidence of a suitable single-dimensional spectrum \(\text{Spec}^p (M)\), \(p = p(n)\) is sufficient.
Reviewer: D.Ferus (Berlin)

58C40 Spectral theory; eigenvalue problems on manifolds
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53C40 Global submanifolds
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