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Uniqueness, Morse index, and nullity of Carlotto–Schulz minimal hypersurfaces

  • Anass Nifa
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Abstract

For the Carlotto–Schulz minimal embeddings of 𝕊n−1×𝕊n−1×𝕊1 in 𝕊2n, we prove uniqueness and nondegeneracy of the normalized monotone closing profile. The first positive Laplace eigenvalue is 2n−1, and its eigenspace consists of ambient coordinate functions. The full Morse index is $27$ for n=2 and (n3+9n2+11n+3)/3 for n≥3; the nullity is n2+2n, and every Jacobi field is induced by a rotation. The scalar argument also applies to graphical doubles with unequal spherical factors. Uniqueness for n≥3 follows from a monotone Jacobi-residual ratio and compact continuation. The kernel theorem gives local rigidity and, for nearby bumpy ambient metrics, minimal hypersurfaces of every index from the base index through that index plus n2+2n. In 𝕊4 there are at least twenty-one such three-tori.

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Cite

BibTeXRIS
BibTeX
@misc{nifa2026uniqueness,
  title = {{Uniqueness, Morse index, and nullity of Carlotto–Schulz minimal hypersurfaces}},
  author = {Anass Nifa},
  howpublished = {HAL},
  year = {2026},
  month = sep,
  doi = {10.5281/zenodo.22914243},
  url = {https://hal.science/hal-05769981v1},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Uniqueness, Morse index, and nullity of Carlotto–Schulz minimal hypersurfaces
AU  - Anass Nifa
PY  - 2026
DA  - 2026/09/09
PB  - HAL
DO  - 10.5281/zenodo.22914243
UR  - https://hal.science/hal-05769981v1
AB  - For the Carlotto–Schulz minimal embeddings of 𝕊^n−1×𝕊^n−1×𝕊^1 in 𝕊^2n, we prove uniqueness and nondegeneracy of the normalized monotone closing profile. The first positive Laplace eigenvalue is 2n−1, and its eigenspace consists of ambient coordinate functions. The full Morse index is $27$ for n=2 and (n^3+9n^2+11n+3)/3 for n≥3; the nullity is n^2+2n, and every Jacobi field is induced by a rotation. The scalar argument also applies to graphical doubles with unequal spherical factors. Uniqueness for n≥3 follows from a monotone Jacobi-residual ratio and compact continuation. The kernel theorem gives local rigidity and, for nearby bumpy ambient metrics, minimal hypersurfaces of every index from the base index through that index plus n^2+2n. In 𝕊^4 there are at least twenty-one such three-tori.
ER  -