Annals of Global Analysis and Geometry

, Volume 53, Issue 2, pp 233–249 | Cite as

Eigenvalues of the complex Laplacian on compact non-Kähler manifolds

  • Gabriel J. H. Khan


We consider \(\lambda \) is the principle eigenvalue of the complex Laplacian on a compact Hermitian manifold M. We prove that \(\lambda \ge C\) where C depends only on the dimension n, the diameter d, the Ricci curvature of the Levi-Civita connection on M, and a norm, expressed in curvature, that determines how much M fails to be Kähler. We first estimate the principal eigenvalue of a drift Laplacian and then study the structure of Hermitian manifolds using recent results due to Yang and Zheng (on curvature tensors of Hermitian manifolds, 2016. arXiv:1602.01189). We combine these results to obtain the main estimate. We also discuss several special cases in which one can obtain a lower bound solely in terms of the Riemannian geometry.


Hermitian manifold drift Laplacian Riemannian geometry Torsion estimates 



We owe many thanks to Bo Guan, Bo Yang, Adrian Lam, and Fangyang Zheng for their insights and help in deriving these results. Finally, thanks are due to Kori Brady and Fangyang Zheng for their edits and help in making the writing more clear.


  1. 1.
    Andrews, B., Ni, L.: Eigenvalue comparison on Bakry–Emery manifolds. Commun. Partial Differ. Equ. 37(11), 2081–2092 (2012)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Borisov, L., Salamon, S., Viaclovsky, J.: Twistor geometry and warped product orthogonal complex structures. Duke Math. J. 156(1), 125–166 (2011)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Chen, M.: General estimate of the first eigenvalue on manifolds. Front. Math. China 6(6), 1025–1043 (2011)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Chen, Q., Jost, J., Qiu, H.: Existence and Liouville theorems for V -harmonic maps from complete manifolds. Ann. Glob. Anal. Geom. 42(4), 565–584 (2012)Google Scholar
  5. 5.
    Donnelly, H.: A spectral condition determining the Kaehler property. In: Proceedings of the American Mathematical Society, 47 (1975)Google Scholar
  6. 6.
    Futaki, A., Li, H., Li, X.: On the first eigenvalue of the Witten–Laplacian and the diameter of compact shrinking solitons. Ann. Glob. Anal. Geom. 44(2), 105–114 (2012)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Gauduchon, P.: Structures complexes sur une varieté conforme de type negative. In: Complex Analysis and Geometry. Lecture Notes in Pure and Applied Mathematics, 173, Marcel Dekker (1995)Google Scholar
  8. 8.
    Gilkey, P.: Spectral geometry and the Kaehler condition for complex manifolds. Invent. Math. 26(3), 231–258 (1974)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Gonzalez, B., Negrin, E.: Gradient estimates for positive solutions of the Laplacian with drift. Proc. Am. Math. Soc. 127(2), 619–625 (1999)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Hamel, F., Nadirashvili, N., Russ, E.: An isoperimetric inequality for the principal eigenvalue of the Laplacian with drift. C. R. Math. 340(5), 347–352 (2005)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Khan, G., Yang, B., Zheng, F.: The set of all orthogonal complex structures on the flat 6-tori. Arxiv preprint. (2016)
  12. 12.
    Hernandez-lamoneda, L.: Curvature vs Almost Hermitian structures. Geom. Dedic. 79(2), 205–218 (2000)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Matsuo, K., Takahashi, T.: On compact astheno-Kähler manifolds. Colloq. Math. 89(2), 213–221 (2001)Google Scholar
  14. 14.
    Park, J.: Spectral geometry and the Kaehler condition for Hermitian manifolds with boundary. Recent Adv. Riemannian Lorentzian Geom. Contemp. Math. 337, 121–128 (2003)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Salamon, S.M.: Orthogonal complex structures. In: Proceedings of the 6th International Conference on Differential Geometry, Brno (1995)Google Scholar
  16. 16.
    Streets, J., Tian, G.: A parabolic flow of pluriclosed metrics. Int. Math. Res. Not. 2010, 3101–3133 (2010)MathSciNetMATHGoogle Scholar
  17. 17.
    Streets, J., Tian, G.: Regularity results for pluriclosed flow. Geom. Topol. 17(4), 23892429 (2013). MR 3110582, Zbl 1272.32022,MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Schoen, R., Yau, S.: Lectures on Differential Geometry. International Press, Cambridge, MA (1994)MATHGoogle Scholar
  19. 19.
    Yang, B., Zheng, F.: On curvature tensors of Hermitian manifolds. (2016) Arxiv preprint, arXiv:1602.01189

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© Springer Science+Business Media B.V. 2017

Authors and Affiliations

  1. 1.ColumbusUSA

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