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

Nifa, Anass

doi:10.5281/zenodo.22914243

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

Anass Nifa9 September 2026

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Abstract

For the Carlotto–Schulz minimal embeddings of $\mathbb{S}^{n-1}\times\mathbb{S}^{n-1}\times\mathbb{S}^1$ in $\mathbb{S}^{2n}$ , 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 $n\ge3$ ; 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 $n\ge3$ 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 $\mathbb{S}^4$ there are at least twenty-one such three-tori.

MSC 2020

53C42: Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
58J50: Spectral problems; spectral geometry; scattering theory on manifolds
34B24: Sturm–Liouville theory
65G20: Algorithms with automatic result verification
58E12: Variational problems concerning minimal surfaces (problems in two independent variables)

Mathematics Subject Classification 2020

Record

Version DOI: 10.5281/zenodo.22914243
All versions: 10.5281/zenodo.22914242
Subjects: Minimal hypersurface; Morse index; Jacobi operator; first eigenvalue; shooting method; interval arithmetic
Language: English

Cite this paper

Anass Nifa. Uniqueness, Morse index, and nullity of Carlotto–Schulz minimal hypersurfaces. Preprint, Zenodo, v1 (2026). doi:10.5281/zenodo.22914243

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References

As listed in the manuscript bibliography.

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DOI
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DOI · arXiv v4
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DOI · arXiv v3
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DOI
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arXiv v1
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arXiv v2
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arXiv v1

[reference-1] [1]
R. Bott, Nondegenerate critical manifolds, Ann. of Math. (2) 60 (1954), 248–261. doi:10.2307/1969631.
DOI

[reference-2] [2]
A. Carlotto and M. B. Schulz, Minimal hypertori in the four-dimensional sphere, Ars Inveniendi Analytica (2023), Paper No. 8, 33 pp. doi:10.15781/gtrt-v523. arXiv:2109.11768v3.
DOI · arXiv v3

[reference-3] [3]
A. Carlotto, M. B. Schulz and D. Wiygul, Spectral estimates for free boundary minimal surfaces via Montiel–Ros partitioning methods, Anal. PDE 18 (2025), 1715–1768. doi:10.2140/apde.2025.18.1715. arXiv:2301.03055v2.
DOI · arXiv v2

[reference-4] [4]
A. Carlotto, M. B. Schulz and D. Wiygul, Index growth not imputable to topology, Proc. Amer. Math. Soc. 153 (2025), no. 4, 1787–1801. doi:10.1090/proc/17151. arXiv:2407.11147v2.
DOI · arXiv v2

[reference-5] [5]
H. I. Choi and A. N. Wang, A first eigenvalue estimate for minimal hypersurfaces, J. Differential Geom. 18 (1983), no. 3, 559–562.

[reference-6] [6]
A. J. Duncan, Y. Sire and J. Spruck, An improved eigenvalue estimate for embedded minimal hypersurfaces in the sphere, Int. Math. Res. Not. IMRN (2024), no. 18, 12556–12567. arXiv:2308.12235.
arXiv

[reference-7] [7]
B. Firester and R. Tsiamis, New minimal surfaces in the sphere from capillary minimal cones, preprint (2026), arXiv:2602.20124v1.
arXiv v1

[reference-8] [8]
A. Hatcher, Algebraic Topology, Cambridge University Press, Cambridge, 2002. Author's electronic edition.
Source

[reference-9] [9]
A. Hatcher, Vector Bundles and K-Theory, version 2.2, November 2017, Proposition 1.17 and Theorem 3.9. Author's electronic text.
Source

[reference-10] [10]
W.-Y. Hsiang and H. B. Lawson, Jr., Minimal submanifolds of low cohomogeneity, J. Differential Geom. 5 (1971), 1–38. doi:10.4310/jdg/1214429775.
DOI

[reference-11] [11]
IEEE Computer Society, IEEE Standard for Floating-Point Arithmetic, IEEE Std 754-2019 (2019). doi:10.1109/IEEESTD.2019.8766229.
DOI

[reference-12] [12]
N. Kapouleas and D. Wiygul, \emph{The index and nullity of the Lawson surfaces $\xi_{g,1}$}, Cambridge J. Math. 8 (2020), no. 2, 363–405. doi:10.4310/CJM.2020.v8.n2.a3. arXiv:1904.05812v2.
DOI · arXiv v2

[reference-13] [13]
N. Kapouleas and J. Zou, Index and nullity of minimal surface doublings, I, preprint (2025), arXiv:2512.07734v1.
arXiv v1

[reference-14] [14]
M. Koiso, P. Piccione and T. Shoda, On bifurcation and local rigidity of triply periodic minimal surfaces in $ℝ^3$, Ann. Inst. Fourier (Grenoble) 68 (2018), no. 6, 2743–2778. doi:10.5802/aif.3222.
DOI

[reference-15] [15]
J. Lai and G. Wei, Embedded constant mean curvature hypertori in the $2n$-sphere, preprint (2025), arXiv:2503.19297v1.
arXiv v1

[reference-16] [16]
Korea Superintelligence Labs (Machina Mathematica), \emph{The stability index of the Carlotto–Schulz minimal hypertorus $X^n_{CS}$ in $S^{2n}$: what Perdomo's conjecture $(n^3+9n^2+11n+3)/3$, and $27$ at $n=2$, actually rests on—with an erratum for the printed shape operator}, Ideosphere Research, version 1, 17 pp., online research record (2026). Version-specific PDF; record and provenance. The displayed source snapshot is 7 September 2026; this is not used as an independently established first-publication date.
Source · Source

[reference-17] [17]
J. Milnor, Morse Theory, Annals of Mathematics Studies, vol. 51, Princeton University Press, Princeton, NJ, 1963.

[reference-18] [18]
S. Montiel and A. Ros, Schrödinger operators associated to a holomorphic map, in Global Differential Geometry and Global Analysis (Berlin, 1990), Lecture Notes in Math., vol. 1481, Springer, Berlin, 1991, pp. 147–174. doi:10.1007/BFb0083639.
DOI

[reference-19] [19]
N. S. Nedialkov, K. R. Jackson and G. F. Corliss, Validated solutions of initial value problems for ordinary differential equations, Appl. Math. Comput. 105 (1999), 21–68. doi:10.1016/S0096-3003(98)10083-8.
DOI

[reference-20] [20]
A. Nifa, Dirichlet deformations, Morse index, and nonlinear symmetry of Ricci-flat Böhm metrics, preprint, Zenodo (2026), doi:10.5281/zenodo.22810800.
DOI

[reference-21] [21]
P. Pavešić, Topological complexity of real Grassmannians, Proc. Roy. Soc. Edinburgh Sect. A 151 (2021), 2013–2029, doi:10.1017/prm.2020.92. arXiv:2011.13750v1.
DOI · arXiv v1

[reference-22] [22]
O. Perdomo, The stability index and Yau's conjecture for Carlotto–Schulz minimal hypertori, preprint (2025), arXiv:2508.09104v2.
arXiv v2

[reference-23] [23]
O. Perdomo, The stability index and Yau's conjecture for Carlotto–Schulz minimal hypertori, part II, preprint (2026), arXiv:2606.16980v1.
arXiv v1

[reference-24] [24]
O. Perdomo, New explicit eigenfunctions of the stability operator on some minimal hypersurfaces, preprint (2026). arXiv:2607.04917v1.
arXiv v1

[reference-25] [25]
T. Takahashi, Minimal immersions of Riemannian manifolds, J. Math. Soc. Japan 18 (1966), no. 4, 380–385. doi:10.2969/jmsj/01840380.
DOI

[reference-26] [26]
H. Tran, Index characterization for free boundary minimal surfaces, Commun. Anal. Geom. 28 (2020), no. 1, 189–222, doi:10.4310/CAG.2020.v28.n1.a6. arXiv:1609.01651v4.
DOI · arXiv v4

[reference-27] [27]
J. Weber, Classical Morse theory revisited I—Backward $\lambda$-lemma and homotopy type, Topol. Methods Nonlinear Anal. 47 (2016), no. 2, 641–646. doi:10.12775/TMNA.2016.020. arXiv:1410.0995v3, p. 3, restates Milnor's Theorem 3.2.
DOI · arXiv v3

[reference-28] [28]
B. White, The space of minimal submanifolds for varying Riemannian metrics, Indiana Univ. Math. J. 40 (1991), no. 1, 161–200, doi:10.1512/iumj.1991.40.40008.
DOI

[reference-29] [29]
B. White, On the bumpy metrics theorem for minimal submanifolds, Amer. J. Math. 139 (2017), no. 4, 1149–1155. arXiv:1503.01803v1.
arXiv v1

[reference-30] [30]
S.-T. Yau (ed.), Seminar on Differential Geometry, Annals of Mathematics Studies, vol. 102, Princeton University Press, Princeton, NJ, 1982.

[reference-31] [31]
J. Yu, On the first eigenvalue of embedded minimal hypersurfaces in the unit sphere, preprint (2026). arXiv:2606.10962v2.
arXiv v2

[reference-32] [32]
L. Zeng, The first eigenvalue of embedded minimal hypersurfaces in the unit sphere I: Yau's conjecture, preprint (2025), arXiv:2508.06123v1.
arXiv v1