Spectrum of the Laplacian and the Jacobi operator on Generalized rotational minimal hypersurfaces of spheres

Let M S^n+1 be the hypersurface generated by rotating a hypersurface M₀ contained in the interior of the unit ball of R^n-k+1. More precisely, M=\ (1-|m|²\, y\, , m): y Sᵏ, \, m M₀\. We deduce the equation for the mean curvature of M in terms of the principal curvatures of M₀ and in the particular case when M₀ is a surface of revolution in R³, we provide a way to find the eigenvalues of the Laplace and the Stability operators. Numerical examples of embedded minimal hypersurface in S^n+1 will be provided for several values of n. To illustrate the method for finding the eigenvalues, we will compute all the eigenvalues of the Laplace operator smaller than 12 and we compute all non positive eigenvalues of the Stability operators for a particular minimal embedded hypersurface in S⁶. We show that the stability index (the number of negative eigenvalues of the stability operator counted with multiplicity) for this example is 77 and the nullity (the multiplicity of the eigenvalue =0 of the Stability operator) is 14. Similar results are found in the case where M₀ is a hypersurface in R^l+2 of the form (f₂ (u) z, f₁ (u) ) with z in the l-dimensional unit sphere Sˡ