Abstract
In this paper, we investigate singular minimal hypersurfaces of the product of spheres \(\mathbb {S}^{n_1}(r_1) \times \mathbb {S}^{n_2}(r_2)\). 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.
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Acknowledgements
The authors would like to thank the referee for providing careful and thoughtful feedback which helped improve this work.
Funding
The first author was partially supported by the Brazilian National Council for Scientific and Technological Development, Brazil [Grant: 308440/2021-8 and 405468/2021-0], by Alagoas Research Foundation [Grant: E:60030.0000001758/2022], and both authors were partially supported by Coordination for the Improvement of Higher Education Personnel [Finance code - 001].
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Batista, M., Martins, M.B. A Gap in the Morse Index of Minimal Hypersurfaces in the Product of Spheres. J Geom Anal 34, 213 (2024). https://doi.org/10.1007/s12220-024-01648-z
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DOI: https://doi.org/10.1007/s12220-024-01648-z