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

Oscar Perdomo

Published 2019-02-19Version 1

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 explain how to compute the spectra of their Laplace and Jacobi operators. To illustrate the method we pick a $3$-dimensional rotational minimal hypersurface in $\mathbb{S}^4$ and prove that the first three eigenvalues of the Laplace operators are: 0, a number near $0.4404$ with multiplicity 2, and 3 with multiplicity 5. We also show that the negative eigenvalues of the Jacobi operator are: a number near $-8.6534$ with multiplicity 1, a number near $-8.52078$ with multiplicity 2, $-3$ with multiplicity 5, a number near -2.5596 with multiplicity 6, and a number number near $-1.17496$ with multiplicity 1. The stability index of this hypersurface is thus 15.