Diederich–Fornæss and Steinness Indices for Abstract CR Manifolds


We propose the concept of Diederich–Fornæss and Steinness indices on compact pseudoconvex CR manifolds of hypersurface type in terms of the D’Angelo 1-form. When the CR manifold bounds a domain in a complex manifold, under certain additional non-degeneracy conditions, those indices are shown to coincide with the original Diederich–Fornæss and Steinness indices of the domain, and CR invariance of the original indices follows.

This is a preview of subscription content, access via your institution.


  1. 1.

    Adachi, M.: A local expression of the Diederich-Fornaess exponent and the exponent of conformal harmonic measures. Bull. Braz. Math. Soc. 46(1), 65–79 (2015)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Adachi, M.: A CR proof for a global estimate of the Diederich-Fornaess index of Levi-flat real hypersurfaces. In: Complex Analysis and Geometry, vol. 144, pp. 41–48. Springer Proc. Math. Stat. Springer, Tokyo (2015)

  3. 3.

    Adachi, M., Brinkschulte, J.: Curvature restrictions for Levi-flat real hypersurfaces in complex projective planes. Ann. Inst. Fourier 65(6), 2547–2569 (2015)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Boas, H.P., Straube, E.J.: de Rham cohomology of manifolds containing the points of infinite type, and Sobolev estimates for the \(\overline{\partial }\)-Neumann problem. J. Geom. Anal. 3(3), 225–235 (1993)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Brinkschulte, J.: On the normal bundle of Levi-flat real hypersurfaces. Math. Ann. 375(1–2), 343–359 (2019)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Brunella, M.: Codimension one foliations on complex tori. Ann. Fac. Sci. Toulouse Math. 19(2), 405–418 (2010)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Chen, B.-Y.: Bergman kernel and hyperconvexity index. Anal. PDE 10(6), 1429–1454 (2017)

    MathSciNet  Article  Google Scholar 

  8. 8.

    D’Angelo, J.P.: Finite type conditions for real hypersurfaces. J. Differ. Geom. 14(1), 59–66 (1979)

    MathSciNet  Article  Google Scholar 

  9. 9.

    D’Angelo, J.P.: Iterated commutators and derivatives of the Levi form. In: Complex Analysis, University Park, PA., 1986. Lecture Notes in Math., vol. 1268, pp. 103–110. Springer, Berlin (1987)

  10. 10.

    Diederich, K., Fornaess, J.F.: Pseudoconvex domains: bounded strictly plurisubharmonic exhaustion functions. Invent. Math. 39(2), 129–141 (1977)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Diederich, K., Ohsawa, T.: On the displacement rigidity of Levi flat hypersurfaces—the case of boundaries of disc bundles over compact Riemann surfaces. Publ. Res. Inst. Math. Sci. 43(1), 171–180 (2007)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Fu, S., Shaw, M.-C.: Bounded plurisubharmonic exhaustion functions and Levi-flat hypersurfaces. Acta Math. Sin. 34(8), 1269–1277 (2018)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Grauert, H.: On Levi’s problem and the imbedding of real-analytic manifolds. Ann. Math. 68, 460–472 (1958)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Grauert, H.: Über Modifikationen und exzeptionelle analytische Mengen. Math. Ann. 146, 331–368 (1962)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Harrington, P.S.: On Competing Definitions for the Diederich-Fornæss Index, Preprint, eprint arXiv:1907.03689

  16. 16.

    Harrington, P.S., Shaw, M.C.: The strong Oka’s lemma, bounded plurisubharmonic functions and the \(\overline{\partial }\)-Neumann problem. Asian J. Math. 11(1), 127–139 (2007)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Liu, B.: The Diederich-Fornæss index I: for domains of non-trivial index. Adv. Math. 353, 776–801 (2019)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Oka, K.: Sur les fonctions analytiques de plusieurs variables. VI. Domaines pseudoconvexes. Tôhoku Math. J. 49, 15–52 (1942)

    MathSciNet  MATH  Google Scholar 

  19. 19.

    Ohsawa, T., Sibony, N.: Bounded P.S.H. functions and pseudoconvexity in Kähler manifold. Nagoya Math. J. 149, 1–8 (1998)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Straube, E.J.: Lectures on the \({\cal{L}}^2\)-Sobolev theory of the \({\overline{\partial }}\)-Neumann problem, ESI Lectures in Mathematics and Physics, European Mathematical Society (EMS), Zürich, pp. viii+206 (2010)

  21. 21.

    Yum, J.: On the Steinness index. J. Geom. Anal. 29(2), 1583–1607 (2019)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Yum, J.: CR-invariance of the Steinness index. Math. Z. (2020). https://doi.org/10.1007/s00209-020-02545-1

    Article  Google Scholar 

Download references


The authors would like to thank Kengo Hirachi for suggesting Remark 2.3 and the referees for their careful reading and suggestions to improve the presentation of this article.

Author information



Corresponding author

Correspondence to Jihun Yum.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Masanori Adachi is partially supported by a JSPS KAKENHI Grant Number JP18K13422.

Jihun Yum is supported by the Institute for Basic Science (IBS-R032-D1).

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Adachi, M., Yum, J. Diederich–Fornæss and Steinness Indices for Abstract CR Manifolds. J Geom Anal (2021). https://doi.org/10.1007/s12220-020-00598-6

Download citation


  • Pseudoconvexity
  • Plurisubharmonic function
  • Diederich–Fornaess index
  • Steinness index
  • D’Angelo 1-form

Mathematics Subject Classification

  • 32T27
  • 32U10
  • 32V15