2019/02/19 by Óscar Perdomo, Perdomo, Oscar · 1 citation
Mathematics · #53C42-34L16 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry and complex manifolds
paper · pdf · doi:10.48550/arxiv.1902.07348
openalex publication_date 2019/02/19 · openalex created_date 2022/07/29 · openalex updated_date 2026/07/28
Let M\⊂ mathbbSn+1\⊂\ℝn+2 be a compact cmc\nrotational hypersurface of the (n+1)-dimensional Euclidean unit sphere.\nDenote by |A|2 the square of the norm of the second fundamental form and\nJ(f)=-\Δ f-nf-|A|2f the stability or Jacobi operator. In this paper we\ncompute the spectra of their Laplace and Jacobi operators in terms of\neigenvalues of second order Hill's equations.\n For the minimal rotational examples, we prove that the stability index --the\nnumbers of negative eigenvalues of the Jacobi operator counted with\nmultiplicity -- is greater than 3 n+4 and we also prove that there are at\nleast 2 positive eigenvalues of the Laplacian of M smaller than n. When H\nis not zero, we have that every non-flat CMC rotational immersion is generated\nby rotating a planar profile curve along a geodesic called the axis of\nrotation. Let m be the number of points where the maximal distance from this\nprofile curve to the origin is achieved (we assume that the coordinates of the\nplane containing the profile curve has been set up so that the axis of rotation\ngoes through the origin). Let l be be the wrapping number of the profile\ncurve. We show that the number of negative eigenvalues of the operator J\ncounted with multiplicity is at least (2l-1)n+(2m-1). This result was proven\nfor the case n=2 by Rossman and Sultana. They called m the number of bulges\nor the number of necks. We will slightly change the definition of l to\ninclude immersed examples that contain the axis of rotation.\n