Instability of Einstein warped products and a one-sided Weyl curvature gap

Nifa, Anass

doi:10.5281/zenodo.22851382

Preprint · v1

Instability of Einstein warped products and a one-sided Weyl curvature gap

Anass Nifa1 September 2026

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Abstract

Let $\operatorname{Ric}_g=\Lambda g$ , where $\Lambda>0$ . For Einstein warped products of closed manifolds, we prove that the least transverse-traceless Einstein eigenvalue is strictly less than $-(n-2)\Lambda/(n-1)$ , without an upper bound on the dimension. The same estimate holds for the smooth spherical-fibre completions specified here, unless the completed metric is round. A trace-free Hessian correction expresses the shifted quadratic form as a negative sum of squared base-curvature tensors and admits an $H^1$ extension across the collapsing fibres. In dimension four, there is a universal positive gap for $\| (\lambda_{\max}(W^+)-2\Lambda/3)_+\|_{L^2}$ on closed simply connected oriented positive Einstein manifolds. Below this gap the excess vanishes, and the metric is anti-self-dual or Kähler–Einstein in the given orientation. The proof uses Einstein orbifold compactness, extension of parallel real line bundles through anti-self-dual ALE limits, and LeBrun–Ozuch’s Kähler–Einstein desingularization theorem. For circle-warped four-metrics, the corrected quadratic form equals a negative multiple of either chiral Weyl energy. The small-excess TT-semistable metrics are the round sphere and the Fubini–Study plane.

MSC 2020

53C25: Special Riemannian manifolds (Einstein, Sasakian, etc.)
58J50: Spectral problems; spectral geometry; scattering theory on manifolds
53C23: Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces

Mathematics Subject Classification 2020

Record

Version DOI: 10.5281/zenodo.22851382
All versions: 10.5281/zenodo.22851381
Subjects: Einstein metric; warped product; transverse-traceless tensor; Weyl curvature; integral pinching; orbifold degeneration
Language: English

Cite this paper

Anass Nifa. Instability of Einstein warped products and a one-sided Weyl curvature gap. Preprint, Zenodo, v1 (2026). doi:10.5281/zenodo.22851382

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References

As listed in the manuscript bibliography.

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[2]
M. T. Anderson and J. Cheeger, Diffeomorphism finiteness for manifolds with Ricci curvature and 𝐿𝑛/2 -norm of curvature bounded, Geom. Funct. Anal. 1 (1991), 231–252.
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arXiv v1
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DOI · arXiv v2
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C. LeBrun, Einstein manifolds, self-dual Weyl curvature, and conformally Kähler geometry, arXiv:1908.01881v2, 2019.
arXiv v2
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C. LeBrun and T. Ozuch, Desingularizations of conformally Kähler, Einstein orbifolds, arXiv:2601.19215v2, 5 February 2026, Theorem A.
arXiv v2
[24]
H. Nakajima, Hausdorff convergence of Einstein 4-manifolds, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 35 (1988), 411–424.
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A. Newlander and L. Nirenberg, Complex analytic coordinates in almost complex manifolds, Ann. of Math. (2) 65 (1957), 391–404.
[26]
A. Nifa, Negative Lichnerowicz modes on Einstein warped products and spheres, preprint, September 2026. DOI: 10.5281/zenodo.22681023.
DOI
[27]
T. Ozuch, Noncollapsed degeneration of Einstein 4-manifolds, I, Geom. Topol. 26 (2022), 1483–1528.
[28]
J. Petean, Ricci curvature and Yamabe constants, arXiv:math/0510308, Theorem A.
arXiv
[29]
P. Petersen, Riemannian Geometry, third edition, Graduate Texts in Mathematics, vol. 171, Springer, Cham, 2016.
[30]
M. E. Taylor, Partial Differential Equations II: Qualitative Studies of Linear Equations, second edition, Applied Mathematical Sciences, vol. 116, Springer, New York, 2011.
[31]
G. Verger, Instability of Böhm’s Einstein metrics, preprint, arXiv:2608.25865v1, 2026.
arXiv v1
[32]
P. Wu, Einstein four-manifolds with self-dual Weyl curvature of nonnegative determinant, Int. Math. Res. Not. IMRN (2021), 1043–1054; arXiv:1903.11818v2.
arXiv v2

[reference-1] [1]
M. T. Anderson, Ricci curvature bounds and Einstein metrics on compact manifolds, J. Amer. Math. Soc. 2 (1989), 455–490.

[reference-2] [2]
M. T. Anderson and J. Cheeger, Diffeomorphism finiteness for manifolds with Ricci curvature and 𝐿𝑛/2 -norm of curvature bounded, Geom. Funct. Anal. 1 (1991), 231–252.

[reference-3] [3]
R. H. Bamler and B. Kleiner, Ricci flow and contractibility of spaces of metrics, arXiv:1909.08710v1, 18 September 2019, Theorem 1.3.
arXiv v1

[reference-4] [4]
S. Bando, Bubbling out of Einstein manifolds, Tohoku Math. J. (2) 42 (1990), 205–216; correction and addition, 587–588.

[reference-5] [5]
S. Bando, A. Kasue and H. Nakajima, On a construction of coordinates at infinity on manifolds with fast curvature decay and maximal volume growth, Invent. Math. 97 (1989), 313–349.

[reference-6] [6]
W. Batat, S. J. Hall and T. Murphy, Destabilising compact warped product Einstein manifolds, Comm. Anal. Geom. 29 (2021), no. 5, 1061–1094, DOI: 10.4310/CAG.2021.v29.n5.a2; arXiv:1607.05766v3.
DOI · arXiv v3

[reference-7] [7]
A. L. Besse, Einstein Manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 10, Springer-Verlag, Berlin, 1987.

[reference-8] [8]
O. Biquard and T. Ozuch, Instability of conformally Kähler, Einstein metrics, arXiv:2310.10109v3, 9 February 2025, Theorem A.
arXiv v3

[reference-9] [9]
J. E. Borzellino and S.-H. Zhu, The splitting theorem for orbifolds, Illinois J. Math. 38 (1994), no. 4, 679–691, DOI: 10.1215/ijm/1256060999, Theorem 1.
DOI

[reference-10] [10]
S. Brendle, Einstein manifolds with nonnegative isotropic curvature are locally symmetric, Duke Math. J. 151 (2010), no. 1, 1–21, DOI: 10.1215/00127094-2009-061; arXiv:0812.0335v3.
DOI · arXiv v3

[reference-11] [11]
G. Catino and D. Dameno, Conformal Kähler rigidity of Einstein four-manifolds, preprint, arXiv:2607.21538v1, 2026.
arXiv v1

[reference-12] [12]
A. Derdziński, Self-dual Kähler manifolds and Einstein manifolds of dimension four, Compositio Math. 49 (1983), 405–433.

[reference-13] [13]
G. W. Gibbons, S. A. Hartnoll and C. N. Pope, Böhm and Einstein–Sasaki metrics, black holes and cosmological event horizons, Phys. Rev. D 67 (2003), 084024.

[reference-14] [14]
M. J. Gursky and C. LeBrun, On Einstein manifolds of positive sectional curvature, Ann. Global Anal. Geom. 17 (1999), 315–328.

[reference-15] [15]
C. He, P. Petersen and W. Wylie, On the classification of warped product Einstein metrics, Comm. Anal. Geom. 20 (2012), 271–311.

[reference-16] [16]
N. J. Hitchin, Kählerian twistor spaces, Proc. London Math. Soc. (3) 43 (1981), 133–150.

[reference-17] [17]
J. C. M. Hughes, J. F. Jamal Beek and F. V. Kusmartsev, Warped product Einstein manifolds in four dimensions, preprint, arXiv:2606.08047v1, 2026.
arXiv v1

[reference-18] [18]
S. Ilias, Constantes explicites pour les inégalités de Sobolev sur les variétés riemanniennes compactes, Ann. Inst. Fourier (Grenoble) 33 (1983), no. 2, 151–165.

[reference-19] [19]
B. Kleiner and J. Lott, Geometrization of three-dimensional orbifolds via Ricci flow, Astérisque 365 (2014), 101–177; arXiv:1101.3733v3, Section 2.2 and Remark 3.4.
arXiv v3

[reference-20] [20]
B. Kleiner and J. Lott, Corrections to “Geometrization of three-dimensional orbifolds via Ricci flow”, author-maintained correction notice, https://math.berkeley.edu/~lott/corrections2.pdf.

[reference-21] [21]
K. Kröncke, Stable and unstable Einstein warped products, Trans. Amer. Math. Soc. 369 (2017), no. 9, 6537–6563, DOI: 10.1090/tran/6959; arXiv:1507.01782v2.
DOI · arXiv v2

[reference-22] [22]
C. LeBrun, Einstein manifolds, self-dual Weyl curvature, and conformally Kähler geometry, arXiv:1908.01881v2, 2019.
arXiv v2

[reference-23] [23]
C. LeBrun and T. Ozuch, Desingularizations of conformally Kähler, Einstein orbifolds, arXiv:2601.19215v2, 5 February 2026, Theorem A.
arXiv v2

[reference-24] [24]
H. Nakajima, Hausdorff convergence of Einstein 4-manifolds, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 35 (1988), 411–424.

[reference-25] [25]
A. Newlander and L. Nirenberg, Complex analytic coordinates in almost complex manifolds, Ann. of Math. (2) 65 (1957), 391–404.

[reference-26] [26]
A. Nifa, Negative Lichnerowicz modes on Einstein warped products and spheres, preprint, September 2026. DOI: 10.5281/zenodo.22681023.
DOI

[reference-27] [27]
T. Ozuch, Noncollapsed degeneration of Einstein 4-manifolds, I, Geom. Topol. 26 (2022), 1483–1528.

[reference-28] [28]
J. Petean, Ricci curvature and Yamabe constants, arXiv:math/0510308, Theorem A.
arXiv

[reference-29] [29]
P. Petersen, Riemannian Geometry, third edition, Graduate Texts in Mathematics, vol. 171, Springer, Cham, 2016.

[reference-30] [30]
M. E. Taylor, Partial Differential Equations II: Qualitative Studies of Linear Equations, second edition, Applied Mathematical Sciences, vol. 116, Springer, New York, 2011.

[reference-31] [31]
G. Verger, Instability of Böhm’s Einstein metrics, preprint, arXiv:2608.25865v1, 2026.
arXiv v1

[reference-32] [32]
P. Wu, Einstein four-manifolds with self-dual Weyl curvature of nonnegative determinant, Int. Math. Res. Not. IMRN (2021), 1043–1054; arXiv:1903.11818v2.
arXiv v2