Skip to main content
SpringerLink
Log in

Prescribing the \({\overline{Q}}^{\prime }\)-curvature on pseudo-Einstein CR 3-manifolds

  • Published:
Nonlinear Differential Equations and Applications NoDEA Aims and scope Submit manuscript

Abstract

In this paper we study the problem of prescribing the \({\overline{Q}}^{\prime }\)-curvature on embeddable pseudo-Einstein CR 3-manifolds. In the first stage we study the problem in the compact setting and we show that under natural assumptions, one can prescribe any positive (resp. negative) CR pluriharmonic function, if \(\int _{M}Q'\ dv_{\theta }>0\) (resp. \(\int _{M}Q'\ dv_{\theta }<0\)). In the second stage, we study the problem in the non-compact setting of the Heisenberg group. Under mild assumptions on the prescribed function, we prove existence of a one parameter family of solutions. In fact, we show that one can find two kinds of solutions: normal ones that satisfy an isoperimetric inequality and non-normal ones that have a biharmonic leading term.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Beckner, W.: Sharp Sobolev inequalities on the sphere and the Moser–Trudinger inequality. Ann. Math. 138, 213–242 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  2. Billingsley, P.: Convergence of Probability Measures. Wiley, New York (1968)

    MATH  Google Scholar 

  3. Branson, T.: The functional determinant. In: Global Analysis Research Center Lecture Notes Series, vol. 4. Seoul National University, Seoul (1993)

  4. Branson, T., Fontana, L., Morpurgo, C.: Moser–Trudinger and Beckner–Onofri’s inequalities on the CR sphere. Ann. Math. (2) 177(1), 1–52 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  5. Burns, D., Epstein, C.L.: A global invariant for three-dimensional CR-manifolds. Invent. Math. 92(2), 333–348 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  6. Case, J.S., Chanillo, S., Yang, P.: A remark on the kernel of the CR Paneitz operator. Nonlinear Anal. 126, 153–158 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  7. Case, J.S., Cheng, J.H., Yang, P.: An integral formula for the q-prime curvature in 3-dimensional cr geometry. Proc. Am. Math. Soc. 147(4), 1577–1586 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  8. Case, J.S., Hsiao, C.-Y., Yang, P.C.: Extremal metrics for the \(Q^{\prime }\)-curvature in three dimensions. J. Eur. Math. Soc. 21(2), 585–626 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  9. Case, J.S., Yang, P.: A Paneitz-type operator for CR pluriharmonic functions. Bull. Inst. Math. Acad. Sin. N.S. 8(3), 285–322 (2013)

    MathSciNet  MATH  Google Scholar 

  10. Chanillo, S., Kiessling, M.K.-H.: Surfaces with prescribed Gauss curvature. Duke Math. J. 105, 309–353 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  11. Chanillo, S., Chiu, H.-L., Yang, P.: Embeddability for \(3\)-dimensional Cauchy–Riemann manifolds and CR Yamabe invariants. Duke Math. J. 161(15), 2909–2921 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  12. Cheng, J.H., Lee, J.M.: The Burns–Epstein invariant and deformation of CR structures. Duke Math. J. 60(1), 221–254 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  13. Ellis, R.S.: Entropy, Large Deviations, and Statistical Mechanics. Springer, New York (1985)

    Book  MATH  Google Scholar 

  14. Fefferman, C., Hirachi, K.: Ambient metric construction of Q-curvatures in conformal and CR geometry. Math. Res. Lett. 10, 819–831 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  15. Glimm, J., Jaffe, A.: Quantum Physics, 2nd edn. Springer, New York (1987)

    Book  MATH  Google Scholar 

  16. Gursky, M.: The Weyl functional, de Rham cohomology, and Kähler–Einstein metrics. Ann. Math. (2) 148(1), 315–337 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  17. Heinonen, J., Koskela, P., Shanmugalingam, S., Tyson, J.: Sobolev Spaces on Metric Measure Spaces. An Approach Based on Upper Gradients. New Mathematical Monographs, 27. Cambridge University Press, Cambridge (2015)

    Book  MATH  Google Scholar 

  18. Hewitt, E., Savage, L.: Symmetric measures on Cartesian products. Trans. Am. Math. Soc. 80, 470 (1955)

    Article  MathSciNet  MATH  Google Scholar 

  19. Ho, P.T.: Prescribing the Q’-curvature in three dimension. Discrete Contin. Dyn. Syst. 39, 2285–2294 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  20. Hirachi, K.: Scalar pseudo-Hermitian invariants and the Szegő kernel on three-dimensional CR manifolds. In: Complex Geometry (Osaka, 1990), volume 143 of Lecture Notes in Pure and Applied Mathematics, pp. 67–76. Dekker, New York (1993)

  21. Hsiao, C.Y.: On CR Paneitz operators and CR pluriharmonic functions. Math. Ann. 362, 903–929 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  22. Kiessling, M.K.-H.: Statistical mechanics of classical particles with logarithmic interactions. Commun. Pure Appl. Math. 46, 27–56 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  23. Kiessling, M.K.-H.: Statistical mechanics approach to some problems in conformal geometry. Phys. A 279, 353–368 (2000)

    Article  MathSciNet  Google Scholar 

  24. Kiessling, M.K.-H.: Typicality analysis for the Newtonian N-body problem on \(S^2\) in the \(N\rightarrow \infty \) limit. J. Stat. Mech. Theory Exp. 1, P01028 (2011)

  25. Lee, J.M.: Pseudo-Einstein structures on CR manifolds. Am. J. Math. 110, 157–178 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  26. Maalaoui, A.: Prescribing the Q-curvature on the sphere with conical singularities. Discrete Contin. Dyn. Syst. 36, 6307–6330 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  27. Messer, J., Spohn, H.: Statistical mechanics of the isothermal Lane–Emden equation. J. Stat. Phys. 29, 561–578 (1982)

    Article  MathSciNet  Google Scholar 

  28. Takeuchi, Y.: Nonnegativity of the CR Paneitz operator for embeddable CR manifolds. Duke Math. J. 169(18), 3417–3438 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  29. Wang, Y., Yang, P.: Isoperimetric inequality on CR manifolds with nonnegative \(Q^{\prime }\)-curvature. Annali Della Scuola Normale di Pisa XVIII(1), 66 (2018)

Download references

Acknowledgements

The author wants to express his gratitude to Prof. Paul Yang for the fruitful conversations and insight that helped improve this paper. Also, the author wants to extend his thanks and gratitude to the referees for their comments and suggestions that led to this improved version of the paper.

Author information

Authors and Affiliations

  1. Department of Mathematics, Clark University, Worcester, MA, 01610, USA

    Ali Maalaoui

Authors
  1. Ali Maalaoui
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ali Maalaoui.

Additional information

Publisher's Note

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

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Maalaoui, A. Prescribing the \({\overline{Q}}^{\prime }\)-curvature on pseudo-Einstein CR 3-manifolds. Nonlinear Differ. Equ. Appl. 30, 30 (2023). https://doi.org/10.1007/s00030-023-00841-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00030-023-00841-3

Keywords

Mathematics Subject Classification

Search

Navigation