Scalar and tensor eigenvalue bounds for conformally Kähler Einstein four-manifolds

Nifa, Anass

doi:10.5281/zenodo.22851442

Preprint · v1

Scalar and tensor eigenvalue bounds for conformally Kähler Einstein four-manifolds

Anass Nifa1 September 2026

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Abstract

We prove scalar and tensor eigenvalue estimates for conformally Kähler Einstein four-manifolds. For the Chen–LeBrun–Weber metric, $\operatorname{Ric}_g=\Lambda g$ , the first positive scalar eigenvalue satisfies $4\Lambda/3<\lambda_1^{\mathrm{sc}}(g)<1951\Lambda/1000$ . Thus this metric admits a destabilizing conformal variation, as asserted in Hall–Murphy’s Conjecture 5.3. The upper bound follows from boundary moments of an affine quotient on the moment polygon and a positive-coefficient polynomial identity. For any closed Einstein four-manifold conformal to a positive-scalar-curvature Kähler metric, put $m=b_2^->0$ and $\rho=\max s_k/\min s_k$ , using the complex orientation. We prove $\lambda_m^{\mathrm{TT}}\le(4\Lambda/3)(1-\rho^{-3})$ for the full Lichnerowicz operator, with strict inequality for a nonconstant conformal factor. For the Chen–LeBrun–Weber metric, this gives two TT eigenvalues below $7\Lambda/6$ , proves the instability of its normalized Ricci-flat cone, and, together with the scalar bound, gives at least three positive directions for the entropy Hessian modulo diffeomorphisms and scaling. Finally, under $\det W^+\ge0$ , the stable cones over closed oriented positive Einstein four-manifolds are precisely those over the round sphere and the Fubini–Study projective plane.

MSC 2020

53C25: Special Riemannian manifolds (Einstein, Sasakian, etc.)
58J50: Spectral problems; spectral geometry; scattering theory on manifolds
53C55: Global differential geometry of Hermitian and Kählerian manifolds
53E20: Ricci flows

The manuscript prints the legacy code 53C44 for geometric evolution equations. This page uses the current MSC 2020 code 53E20 (Ricci flows).

Mathematics Subject Classification 2020

Record

Version DOI: 10.5281/zenodo.22851442
All versions: 10.5281/zenodo.22851441
Subjects: Einstein metric; scalar Laplacian; Lichnerowicz Laplacian; conformal instability; toric Kähler metric; Ricci-flat cone
Language: English

Cite this paper

Anass Nifa. Scalar and tensor eigenvalue bounds for conformally Kähler Einstein four-manifolds. Preprint, Zenodo, v1 (2026). doi:10.5281/zenodo.22851442

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References

As listed in the manuscript bibliography.

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DOI · arXiv v1
[23]
C. LeBrun, On Einstein, Hermitian 4-manifolds, J. Differential Geom. 90 (2012), no. 2, 277–302. arXiv:1010.0238.
DOI · arXiv
[24]
C. LeBrun, Einstein metrics, conformal curvature, and anti-holomorphic involutions, Ann. Math. Québec 45 (2021), 391–405. arXiv:2007.01180.
DOI · arXiv
[25]
A. Nifa, Negative Lichnerowicz modes on Einstein warped products and spheres, preprint, 2026, 21 pp. DOI: 10.5281/zenodo.22681023.
DOI
[26]
M. Obata, Certain conditions for a Riemannian manifold to be isometric with a sphere, J. Math. Soc. Japan 14 (1962), 333–340.
DOI
[27]
D. N. Page, A compact rotating gravitational instanton, Phys. Lett. B 79 (1978), no. 3, 235–238.
[28]
P. Schwahn and U. Semmelmann, Einstein metrics, their moduli spaces and stability, arXiv:2507.18463v3, 2026.
arXiv v3
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P. Wu, Einstein four-manifolds with self-dual Weyl curvature of nonnegative determinant, Int. Math. Res. Not. IMRN 2021 (2021), no. 2, 1043–1054. arXiv:1903.11818v2.
DOI · arXiv v2

[reference-1] [1]
V. Apostolov and G. Maschler, Conformally Kähler, Einstein–Maxwell geometry, J. Eur. Math. Soc. 21 (2019), no. 5, 1319–1360. arXiv:1512.06391v1. Equation numbers cited here refer to the preprint.
DOI · arXiv v1

[reference-2] [2]
A. L. Besse, Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 10, Springer-Verlag, Berlin, 1987.

[reference-3] [3]
O. Biquard and T. Ozuch, Instability of conformally Kähler, Einstein metrics, J. Differential Geom. 133 (2026), no. 3, 335–346; arXiv:2310.10109v3. Numbered citations refer to version 3.
DOI · arXiv v3

[reference-4] [4]
O. Biquard and Y. Rollin, Wormholes in ACH Einstein manifolds, Trans. Amer. Math. Soc. 361 (2009), no. 4, 2021–2046; arXiv:math/0609558v4.
DOI · arXiv v4

[reference-5] [5]
H.-D. Cao, R. S. Hamilton and T. Ilmanen, Gaussian densities and stability for some Ricci solitons, arXiv:math/0404165, 2004.
arXiv

[reference-6] [6]
H.-D. Cao and M. Zhu, On second variation of Perelman’s Ricci shrinker entropy, Math. Ann. 353 (2012), no. 3, 747–763; arXiv:1008.0842v5.
DOI · arXiv v5

[reference-7] [7]
G. Catino and D. Dameno, Conformal Kähler rigidity of Einstein four-manifolds, arXiv:2607.21538v1, 23 July 2026.
arXiv v1

[reference-8] [8]
X. Chen, C. LeBrun and B. Weber, On conformally Kähler, Einstein manifolds, J. Amer. Math. Soc. 21 (2008), no. 4, 1137–1168; arXiv:0705.0710v2. Numbered citations refer to version 2.
DOI · arXiv v2

[reference-9] [9]
E. Delay, Conformal Laplacian on symmetric 2-tensors, Manuscripta Math. 174 (2024), no. 1–2, 73–86.
DOI

[reference-10] [10]
E. Delay, Two remarkable differential operators acting on symmetric 2-tensors, lecture slides, Lichnerowicz conference, Institut Henri Poincaré, 13 June 2025, 69 pp.; author’s conference slides.
Source

[reference-11] [11]
A. Derdziński, Self-dual Kähler manifolds and Einstein manifolds of dimension four, Compositio Math. 49 (1983), no. 3, 405–433; NUMDAM.
Source

[reference-12] [12]
S. K. Donaldson, Interior estimates for solutions of Abreu’s equation, Collect. Math. 56 (2005), no. 2, 103–142; arXiv:math/0407486.
arXiv

[reference-13] [13]
P. B. Gilkey, Invariance theory, the heat equation, and the Atiyah–Singer index theorem, Mathematics Lecture Series, vol. 11, Publish or Perish, Wilmington, DE, 1984; Sections 1.3 and 1.6. Electronic edition.
Source

[reference-14] [14]
S. Hall, R. Haslhofer and M. Siepmann, The stability inequality for Ricci-flat cones, J. Geom. Anal. 24 (2014), no. 1, 472–494. arXiv:1111.4981v1, 21 November 2011.
DOI · arXiv v1

[reference-15] [15]
S. J. Hall and T. Murphy, On the spectrum of the Page and the Chen–LeBrun–Weber metrics, Ann. Global Anal. Geom. 46 (2014), 87–101.
DOI

[reference-16] [16]
S. J. Hall and T. Murphy, Erratum to: On the spectrum of the Page and the Chen–LeBrun–Weber metrics, Ann. Global Anal. Geom. 47 (2015), 223.
DOI

[reference-17] [17]
S. J. Hall and T. Murphy, Numerical approximations to extremal toric Kähler metrics with arbitrary Kähler class, Proc. Edinb. Math. Soc. (2) 60 (2017), no. 4, 893–910; arXiv:1407.1272v2. Numbered citations, including Conjecture 5.3, refer to version 2.
DOI · arXiv v2

[reference-18] [18]
S. J. Hall and T. Murphy, Bounding the first invariant eigenvalue of toric Kähler manifolds, Math. Res. Lett. 24 (2017), no. 1, 67–81; arXiv:1505.01075.
arXiv

[reference-19] [19]
R. A. Hennigar, H. K. Kunduri, K. T. B. Sievers and Y. Wang, Spectrum of the Laplacian on the Page metric, J. Phys. A: Math. Theor. 58 (2025), no. 40, 405204; arXiv:2412.19879v1.
DOI · arXiv v1

[reference-20] [20]
N. Koiso, A decomposition of the space  of Riemannian metrics on a manifold, Osaka J. Math. 16 (1979), no. 2, 423–429.
DOI

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

[reference-22] [22]
K. Kröncke and Á. Szabó, Optimal coordinates for Ricci-flat conifolds, Calc. Var. Partial Differential Equations 63 (2024), no. 7, Paper No. 188, 41 pp.; arXiv:2203.01711v1.
DOI · arXiv v1

[reference-23] [23]
C. LeBrun, On Einstein, Hermitian 4-manifolds, J. Differential Geom. 90 (2012), no. 2, 277–302. arXiv:1010.0238.
DOI · arXiv

[reference-24] [24]
C. LeBrun, Einstein metrics, conformal curvature, and anti-holomorphic involutions, Ann. Math. Québec 45 (2021), 391–405. arXiv:2007.01180.
DOI · arXiv

[reference-25] [25]
A. Nifa, Negative Lichnerowicz modes on Einstein warped products and spheres, preprint, 2026, 21 pp. DOI: 10.5281/zenodo.22681023.
DOI

[reference-26] [26]
M. Obata, Certain conditions for a Riemannian manifold to be isometric with a sphere, J. Math. Soc. Japan 14 (1962), 333–340.
DOI

[reference-27] [27]
D. N. Page, A compact rotating gravitational instanton, Phys. Lett. B 79 (1978), no. 3, 235–238.

[reference-28] [28]
P. Schwahn and U. Semmelmann, Einstein metrics, their moduli spaces and stability, arXiv:2507.18463v3, 2026.
arXiv v3

[reference-29] [29]
P. Wu, Einstein four-manifolds with self-dual Weyl curvature of nonnegative determinant, Int. Math. Res. Not. IMRN 2021 (2021), no. 2, 1043–1054. arXiv:1903.11818v2.
DOI · arXiv v2