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The isospectrum problem of compact minimal hypersurfaces in \(S^{n+1}(1)\). (English) Zbl 1103.53017

In the framework of the problem of characterizing a Riemannian manifold \(M\) by the spectrum of the Laplace-Beltrami operator \(\Delta\) acting on \(p\)-forms of \(M\), the authors consider compact minimal hypersurfaces of the standard unit sphere \(S^{n+1}(1)\). More precisely, among all such hypersurfaces they characterize the Clifford tori \(M_{n_1,n_2}=S^{n_1}(\sqrt{\frac{n_1}{n}}) \times S^{n_2}(\sqrt{ \frac{n_2}{n}})\) (where \(n_1+n_2=n)\), by the spectra of \(\Delta\) acting on \(p\) and \(q\)-forms, for almost all \(0\leq p<q\leq n\), that is, for all of these integers satisfying two inequalities. The paper is written in a clear and readable way.

MSC:

53C20 Global Riemannian geometry, including pinching
53C40 Global submanifolds
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
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