Negative Lichnerowicz modes on Einstein warped products and spheres

Nifa, Anass

doi:10.5281/zenodo.22681024

Preprint · v1

Negative Lichnerowicz modes on Einstein warped products and spheres

Anass NifaInitial manuscript date: 31 August 2026Manuscript revision: 2 September 2026

Manuscript revision dated 2 September 2026, as confirmed by the author. Date metadata clarified on 21 September 2026. Mathematical content and references are unchanged. An obsolete publication/submission-status statement was removed on 21 September 2026.

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Abstract

Every nonhomogeneous compact Böhm metric on a sphere or a product of spheres in dimensions five through nine has a negative eigenvalue of the full Lichnerowicz Laplacian on transverse-traceless tensors. More generally, for a nonround doubly warped Einstein sphere with factor dimensions $p, q\ge2$ , the inequalities $(p-1)(q-4)\le4$ and $(q-1)(p-4)\le4$ suffice. They also apply to Wang’s $SO(3)\times SO(9)$ -invariant Einstein metric on $S^{11}$ . For a positive Einstein warped product of closed connected manifolds, with base dimension $k\ge3$ , fibre dimension $m\ge2$ , and positive nonconstant warping function, the condition $(k-2)(m-4)\le4$ gives the same conclusion. Fibre dimensions two through four are therefore covered without a bound on the total dimension. The variational tool is a sharp divergence penalty on smooth trace-free tensors: on a closed connected positive Einstein manifold of dimension $n\ge3$ , subtracting $n/(n-1)$ times the squared divergence from the full Lichnerowicz quadratic form preserves its negative transverse-traceless index. For the sphere problem, this penalty cancels the quadratic radial-derivative term in a smooth interface construction; a weighted-radius rigidity theorem supplies the sign of the remaining interface contribution. Applications include finite quotients and ancient Ricci flows obtained from Kröncke’s nonlinear instability theorem. An appendix treats negative spectrum on Ricci-flat asymptotically conical manifolds.

MSC 2020

53C25: Special Riemannian manifolds (Einstein, Sasakian, etc.)
58J50: Spectral problems; spectral geometry; scattering theory on manifolds

Mathematics Subject Classification 2020

Record

Version DOI: 10.5281/zenodo.22681024
All versions: 10.5281/zenodo.22681023
Subjects: Einstein metrics; Spectral geometry
Research index: OpenAlex record
Language: English

Cite this paper

Anass Nifa. Negative Lichnerowicz modes on Einstein warped products and spheres. Preprint, Zenodo, v1 (2026). doi:10.5281/zenodo.22681024

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References

As listed in the manuscript bibliography.

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DOI
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DOI · arXiv
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C. Böhm, Inhomogeneous Einstein metrics on low-dimensional spheres and other low-dimensional spaces, Invent. Math. 134 (1998), no. 1, 145–176. doi:10.1007/s002220050261.
DOI

[reference-2] [2]
G. W. Gibbons, S. A. Hartnoll, and C. N. Pope, Bohm and Einstein–Sasaki metrics, black holes, and cosmological event horizons, Phys. Rev. D 67 (2003), 084024. doi:10.1103/PhysRevD.67.084024. Preprint arXiv:hep-th/0208031 (2002).
DOI · arXiv

[reference-3] [3]
K. Kröncke and Á. Szabó, Optimal coordinates for Ricci-flat conifolds, Calc. Var. Partial Differential Equations 63 (2024), Paper No. 188, 41 pp. doi:10.1007/s00526-024-02780-y.
DOI

[reference-4] [4]
G. Verger, Instability of Böhm’s Einstein metrics, preprint, arXiv:2608.25865v1 (26 August 2026).
arXiv v1 · arXiv

[reference-5] [5]
J. Nienhaus and M. Wink, Einstein metrics on the ten-sphere, J. Eur. Math. Soc. (2025), published online first. doi:10.4171/JEMS/1668. Preprint arXiv:2303.04832v3.
DOI · arXiv v3

[reference-6] [6]
Q. S. Wang, Doubly warped product Einstein metrics on spheres, arXiv:2606.05519v2, 6 June 2026, Theorem 1.1.
arXiv v2

[reference-7] [7]
W. Batat, S. J. Hall and T. Murphy, Destabilising compact warped product Einstein manifolds, Comm. Anal. Geom. 29 (2021), no. 5, 1061–1094. Preprint arXiv:1607.05766v3 (8 January 2019); theorem and section locators refer to this version.
arXiv v3

[reference-8] [8]
K. Kröncke, Stability of Einstein metrics under Ricci flow, Comm. Anal. Geom. 28 (2020), no. 2, 351–394; arXiv:1312.2224, Proposition 8.2.
arXiv

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K. Kröncke, Stability and instability of Ricci solitons, Calc. Var. Partial Differential Equations 53 (2015), 265–287; arXiv:1403.3721v2, Definition 1.1, Theorem 1.3 and Section 5.
arXiv v2 · arXiv

[reference-10] [10]
K. Kröncke, Stable and unstable Einstein warped products, Trans. Amer. Math. Soc. 369 (2017), 6537–6563; arXiv:1507.01782.
arXiv

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X. Dai, X. Wang and G. Wei, On the stability of Riemannian manifold with parallel spinors, Invent. Math. 161 (2005), 151–176; arXiv:math/0311253, Lemmas 2.3–2.4.
arXiv