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 derive the equation for the mean curvature of $M$ in terms of the principal curvatures of $M_0$. For the particular case when $M_0$ is a surface of revolution in $\mathbb{R}^3$, we provide a method for finding the eigenvalues of the Laplace and stability operators. To illustrate this method, we consider an example of a minimal embedded hypersurface in $S^6$ and numerically compute all the eigenvalues of the Laplace operator less than 12, as well as all non-positive eigenvalues of the stability operators. For this example, we show that the stability index (the number of negative eigenvalues of the stability operator, counted with multiplicity) 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 $S^l$. Carlotto and Schulz have found examples of embedded minimal hypersurfaces in the case where $M_0=S^k\times S^1$.