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

Oscar M. Perdomo

Vol. 308 (2020), No. 2, 419–433

DOI: 10.2140/pjm.2020.308.419

Abstract

Let $M\subset {\mathbb{𝕊}}^{n+1}\subset {ℝ}^{n+2}$ 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 $J\left(f\right)=-\Delta f-nf-|A{|}^{2}f$ 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 $𝜃$ denotes the change of the angle of the fundamental piece when written in polar coordinates, then $l=\frac{m𝜃}{2\pi }$ 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 $\left(2l-1\right)n+\left(2m-1\right)$.

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