Advertisement

Mathematische Zeitschrift

, Volume 290, Issue 3–4, pp 1115–1143 | Cite as

The spectral rigidity of complex projective spaces, revisited

Article

Abstract

A classical question in spectral geometry is, for each pair of nonnegative integers (pn) such that \(p\le 2n\), if the eigenvalues of the Laplacian on p-forms of a compact Kähler manifold are the same as those of \({\mathbb {C}}P^n\) equipped with the Fubini-Study metric, then whether or not this Kähler manifold is holomorphically isometric to \({\mathbb {C}}P^n\). For every positive even number p, we affirmatively solve this problem in all dimensions n with at most two possible exceptions. We also clarify in this paper some gaps in previous literature concerned with this question, among which one is related to the volume estimate of Fano Kähler–Einstein manifolds.

Keywords

Spectrum Rigidity Complex projective space Fano Kähler–Einstein manifold Volume 

Mathematics Subject Classification

58J50 58C40 53C55 

Notes

Acknowledgements

I would like to thank Yinhe Peng and Wei Xu for finding out the paper [15] from Canada and sending it to me.

References

  1. 1.
    Apte, M.: Sur certaines classes caract\(\acute{e}\)ristiques des vari\(\acute{e}\)t \(\acute{e}\)s Kähl\(\acute{e}\)riennes compactes. C. R. Acad. Sci. Paris 240, 149–151 (1955)MathSciNetGoogle Scholar
  2. 2.
    Berger, M.: Sur les spectgre d’une variété riemannienne. C. R. Acad. Sci. Paris 263, 13–16 (1963)Google Scholar
  3. 3.
    Berger, M.: A Panoramic View of Riemannian Geometry. Springer, Berlin (2003)CrossRefGoogle Scholar
  4. 4.
    Berman, R.J., Berndtsson, B.: The volume of Kähler–Einstein Fano varieties and convex bodies. J. Reine Angew. Math. 723, 127–152 (2017)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Besse, A.L.: Einstein Manifolds, Ergebnisse der Math. Springer, Berlin (1987)CrossRefGoogle Scholar
  6. 6.
    Bishop, R.L., Crittenden, R.J.: Geometry of Manifolds. Academic Press, New York (1964)zbMATHGoogle Scholar
  7. 7.
    Bishop, R.L., Goldberg, S.I.: On the topology of positively curved Kaehler manifolds II. Tohoku Math. J. 17, 310–318 (1965)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Bochner, S.: Curvature and Betti numbers, II. Ann. Math. 50, 77–93 (1949)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Chen, B.-Y., Vanhecke, L.: The spectrum of the Laplacian of Kähler manifolds. Proc. Am. Math. Soc. 79, 82–86 (1980)zbMATHGoogle Scholar
  10. 10.
    Debarre, O.: Higher-Dimensional Algebraic Geometry. Universitext, Springer, New York (2001)CrossRefGoogle Scholar
  11. 11.
    Dickson, L.E.: History of the Theory of Numbers. Vol II: Diophantine Analysis. Chelsea Publishing Co., New York (1966)zbMATHGoogle Scholar
  12. 12.
    Fujita, K.: Optimal bounds for the volumes of Kähler–Einstein Fano manifolds. arXiv:1508.04578 (to appear in Am. J. Math.)
  13. 13.
    Gauchman, H., Goldberg, S.I.: Spectral rigidity of compact Kaehler and contact manifolds. Tohoku Math. J. (2) 38, 563–573 (1986)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Gauntlett, J.P., Martelli, D., Sparks, J., Yau, S.-T.: Obstructions to the existence of Sasaki–Einstein metrics. Commun. Math. Phys. 273, 803–827 (2007)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Goldberg, S.I.: A characterization of complex projective spaces. C. R. Math. Rep. Acad. Sci. Can. 6, 193–198 (1984)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Griffiths, P., Harris, J.: Principles of Algebraic Geometry, Pure and Applied Mathematics. Wiley, New York (1978)zbMATHGoogle Scholar
  17. 17.
    Hirzebruch, F., Kodaira, K.: On the complex projective spaces. J. Math. Pures Appl. 36, 201–216 (1957)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Iskovskikh, V.A., Prokhorov, Y.G.: Fano varieties. In: Parshin, A.N., Shafarevich, I.R. (eds.) Algebraic Geometry, V, 1C247, Encyclopedia Math. Sci., vol. 47. Springer, Berlin (1999).Google Scholar
  19. 19.
    Kobayashi, S.: Topology of positively pinched Kaehler manifolds. Tohoku. Math. J. (2) 15, 121–139 (1963)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Kobayashi, S., Ochiai, T.: Characterizations of complex projective spaces and hyperquadrics. J. Math. Kyoto Univ. 13, 31–47 (1973)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Lascoux, A., Berger, M.: Variétés Kaehleriennes Compactes, Lecture Notes in Math., vol. 154. Springer, Berlin (1970)CrossRefGoogle Scholar
  22. 22.
    LeBrun, C., Salamon, S.: Strong rigidity of positive quaternion-Kähler manifolds. Invent. Math. 118, 109–132 (1994)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Li, P.: Some remarks on the uniqueness of the complex projective spaces. Bull. Lond. Math. Soc. 48, 379–385 (2016)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Li, P.: An integral inequality for constant scalar curvature metrics on Kähler manifolds. Adv. Math. 287, 774–787 (2016)MathSciNetCrossRefGoogle Scholar
  25. 25.
    McKean, H.F., Singer, I.M.: Curvature and the eigenvalues of the Laplacian. J. Differ. Geom. 1, 43–69 (1967)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Milnor, J.: Eigenvalues of the Laplace operator on certain manifolds. Proc. Natl. Sci. USA 51, 542 (1964)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Mori, S.: Projective manifolds with ample tangent bundles. Ann. Math. (2) 110, 593–606 (1979)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Patodi, V.K.: Curvature and the fundamental solution of the heat operator. J. Indian Math. Soc. 34, 269–285 (1970)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Patodi, V.K.: Curvature and the eigenforms of the Laplace operator. J. Differ. Geom. 5, 233–249 (1971)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Perrone, D.: On the spectral rigidity of \({\mathbb{C}}P^n\). Proc. Am. Math. Soc. 104, 871–875 (1988)zbMATHGoogle Scholar
  31. 31.
    Perrone, D.: Intrinsic characterizations of complex quadrics by the spectrum of the Laplacian on 2-forms. Simon Stevin. 63, 339–356 (1989)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Schoen, R., Yau, S.-T.: Lectures on Differential Geometry, Conference Proceedings and Lecture Notes in Geometry and Topology 1. International Press, Cambridge (1994)Google Scholar
  33. 33.
    Siu, Y.-T., Yau, S.-T.: Compact Kähler manifolds of positive bisectional curvature. Invent. Math. 59, 189–204 (1980)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Tanno, S.: Eigenvalues of the Laplacian of Riemannian manifolds. Tokoku Math. J. 25, 391–403 (1973)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Tanno, S.: The spectrum of the Laplacian for \(1\)-forms. Proc. Am. Math. Soc. 45, 125–129 (1974)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Tian, G.: Canonical Metrics in Kähler Geometry, Notes Taken by Meike Akveld, Lectures in Mathematics ETH Zürich. Birkhäuser, Basel (2000)CrossRefGoogle Scholar
  37. 37.
    Tosatti, V.: Uniqueness of \({\mathbb{C}}P^n\). Expo. Math. 35, 1–12 (2017)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Yau, S.-T.: Calabi’s conjecture and some new results in algebraic geometry. Proc. Natl. Acad. Sci. USA 74, 1798–1799 (1977)CrossRefGoogle Scholar
  39. 39.
    Yau, S.T.: Nonlinear analysis in geometry. Enseign. Math. (2) 33, 109–158 (1987)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Zheng, F.: Complex Differential Geometry, AMS/IP Studies in Advanced Mathematics 18. American Mathematical Society, Providence (2000)Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematical SciencesTongji UniversityShanghaiChina

Personalised recommendations