A Gap in the Morse Index of Minimal Hypersurfaces in the Product of Spheres

In this paper, we investigate singular minimal hypersurfaces of the product of spheres \(\mathbb {S}^{n_1}(r_1) \times \mathbb {S}^{n_2}(r_2)\) ${\mathbb{S}}^{n_1}(r_1)\times {\mathbb{S}}^{n_2}(r_2)$ . Using special cutoff functions and an ingenious trick, we prove a gap in the Morse index of singular minimal hypersurfaces under suitable assumptions. In particular, we characterize the singular minimal hypersurfaces with a low index for some values of the radii and dimensions of the spheres. Finally, we compute the value of the k-width of the product of spheres for some values of k.