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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)

##### MSC:
 58C40 Spectral theory; eigenvalue problems on manifolds 53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) 53C40 Global submanifolds
##### Keywords:
closed minimal hypersurface; spectrum; Clifford torus
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##### References:
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