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

Abstract: Let $M\subset S^{n+1}$ be the hypersurface generated by rotating a hypersurface $M_0$ contained in the interior of the unit ball of $\mathbb{R}^{n-k+1}$. More precisely, $M={(\sqrt{1-|m|^2}\, y\, , m):y\in S^k,\, m\in M_0}$. We deduce the equation for the mean curvature of $M$ in terms of the principal curvatures of $M_0$ and in the particular case when $M_0$ is a surface of revolution in $\mathbb{R}^3$, 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 $S^6$. 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 $\lambda=0$ of the Stability operator) is 14. Similar results are found in the case where $M_0$ is a hypersurface in $\mathbb{R}^{l+2}$ of the form $(f_2(u) z, f_1(u))$ with $z$ in the $l$-dimensional unit sphere $S^l$