Spectrum of the Laplacian and the Jacobi operator on rotational cmc hypersurfaces of spheres

Abstract

Let $M\subset \mathbb{S}^{n+1}\subset\mathbb{R}^{n+2}$ be a compact cmc rotational hypersurface of the $(n+1)$-dimensional Euclidean unit sphere. Denote by $|A|^2$ the square of the norm of the second fundamental form and $J(f)=-\Delta f-nf-|A|^2f$ the stability or Jacobi operator. In this paper we compute the spectra of their Laplace and Jacobi operators in terms of eigenvalues of second order Hill's equations. For the minimal rotational examples, we prove that the stability index --the numbers of negative eigenvalues of the Jacobi operator counted with multiplicity -- is greater than $3 n+4$ and we also prove that there are at least 2 positive eigenvalues of the Laplacian of $M$ smaller than $n$. When $H$ is not zero, we have that every non-flat CMC rotational immersion is generated by rotating a planar profile curve along a geodesic called the axis of rotation. Let $m$ be the number of points where the maximal distance from this profile curve to the origin is achieved (we assume that the coordinates of the plane containing the profile curve has been set up so that the axis of rotation goes through the origin). Let $l$ be be the wrapping number of the profile curve. We show that the number of negative eigenvalues of the operator $J$ counted with multiplicity is at least $(2l-1)n+(2m-1)$. This result was proven for the case $n=2$ by Rossman and Sultana. They called $m$ the number of bulges or the number of necks. We will slightly change the definition of $l$ to include immersed examples that contain the axis of rotation.