Vol. 308, No. 2, 2020

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

Oscar M. Perdomo

Vol. 308 (2020), No. 2, 419–433
Abstract

Let M 𝕊n+1 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) = Δf 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 3n + 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 nonflat CMC rotational immersion is generated by rotating a planar profile curve along a geodesic called the axis of rotation. We assume that the coordinates of this plane has been set up so that the axis of rotation goes through the origin. The planar profile curve is made up of m copies, each one of them is a is rigid motion of a single curve that we will call the fundamental piece. For this reason every nonflat rotational CMC hypersurface has Zm in its group of isometries. If 𝜃 denotes the change of the angle of the fundamental piece when written in polar coordinates, then l = m𝜃 2π is a nonnegative integer. For unduloids (a subfamily of the rotational CMC hypersurfaces that include all the known embedded examples), we show that the number of negative eigenvalues of the operator J counted with multiplicity is at least (2l 1)n + (2m 1).

Keywords
stability operator, Jacobi operator, stability index, CMC, spheres, rotational hypersurfaces
Mathematical Subject Classification 2010
Primary: 53C42
Milestones
Received: 5 October 2019
Accepted: 10 July 2020
Published: 9 December 2020
Authors
Oscar M. Perdomo
Department of Mathematical Sciences
Central Connecticut State University
New Britain, CT
United States