{ "id": "1902.07348", "version": "v1", "published": "2019-02-19T23:47:22.000Z", "updated": "2019-02-19T23:47:22.000Z", "title": "Spectrum of the Laplacian and the Jacobi operator on rotational cmc hypersurfaces of spheres", "authors": [ "Oscar Perdomo" ], "comment": "9 pages, 8 figures", "categories": [ "math.DG" ], "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 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.", "revisions": [ { "version": "v1", "updated": "2019-02-19T23:47:22.000Z" } ], "analyses": { "keywords": [ "jacobi operator", "rotational cmc hypersurfaces", "multiplicity", "dimensional rotational minimal hypersurface", "dimensional euclidean unit sphere" ], "note": { "typesetting": "TeX", "pages": 9, "language": "en", "license": "arXiv", "status": "editable" } } }