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Abstract

Let
$M\subset {\mathbb{\mathbb{S}}}^{n+1}\subset {\mathbb{R}}^{n+2}$
be a compact CMC rotational hypersurface of the
$\left(n+1\right)$dimensional Euclidean
unit sphere. Denote by
$A{}^{2}$
the square of the norm of the second fundamental form and
$J\left(f\right)=\Delta fnfA{}^{2}f$ 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
${Z}_{m}$ in its group of
isometries. If
$\mathit{\theta}$
denotes the change of the angle of the fundamental piece when written in polar coordinates,
then
$l=\frac{m\mathit{\theta}}{2\pi}$
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
$\left(2l1\right)n+\left(2m1\right)$.

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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

