Spectrum of the Laplacian and the Jacobi operator on Generalized rotational minimal hypersurfaces of spheres

The paper states and (by analogy to the l=1 case) justifies that for M parametrized by φ(y,u,z)=(√(1−f1(u)^2−f2(u)^2) y, f2(u) z, f1(u)), the Laplacian decomposes as in Lemma IV.1 and the spectra of ∆ and J=∆+n+|A|^2 are the unions of the periodic Sturm–Liouville spectra obtained after separating variables with spherical harmonics on Sk and Sl (Theorem IV.2). The candidate solution reproduces this separation-of-variables argument and the block-diagonalization via the O(k+1)×O(l+1) symmetry, matching the paper’s formulae for ∆ and the principal curvatures (hence |A|^2 depends only on u) to obtain the 1D operators L_{ij} and S_{ij}. Minor technical slips in the candidate’s divergence-form/weight identification do not affect the main conclusion. See Lemma IV.1 and Theorem IV.2 for the explicit ∆ and spectral decomposition, and (IV.3) for principal curvatures, which depend only on u, ensuring |A|^2=|A|^2(u) and thus separation for J as well.