Skip to main content
Log in

On the prescribed Q-curvature problem in Riemannian manifolds

  • Published:
manuscripta mathematica Aims and scope Submit manuscript

Abstract

We prove the existence of metrics with prescribed Q-curvature under natural assumptions on the sign of the prescribing function and the background metric. In the dimension four case, we also obtain existence results for curvature forms requiring only restrictions on the Euler characteristic. Moreover, we derive a prescription result for open submanifolds which allow us to conclude that any smooth function on \({\mathbb {R}}^n\) can be realized as the Q-curvature of some Riemannian metric.

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

Access this article

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. Baird, P., Fardoun, A., Regbaoui, R.: Prescribed \(Q\)-curvature on manifolds of even dimension. J. Geom. Phys. 59(2), 221–233 (2009)

    Article  MathSciNet  Google Scholar 

  2. Berger, M., Ebin, D.: Some decompositions of the space of symmetric tensors on a Riemannian manifold. J. Differ. Geom. 3, 379–392 (1969)

    Article  MathSciNet  Google Scholar 

  3. Branson, T.P.: Differential operator scanonically associated to a conformal structure. Math. Scand. 57, 293–345 (1985)

    Article  MathSciNet  Google Scholar 

  4. Branson, T., Gilkey, P., Pohjanpelto, J.: Invariants of locally conformally flat manifolds. Trans. Am. Math. Soc. 347, 939–953 (1995)

    Article  MathSciNet  Google Scholar 

  5. Brendle, S.: Global existence and convergence for a higher order flow in conformal geometry. Ann. Math. (2) 158, 323–343 (2003)

    Article  MathSciNet  Google Scholar 

  6. Brendle, S.: Convergence of the \(Q\)-curvature flow on \({\mathbb{S}}^4\). Adv. Math. 205, 1–32 (2006)

    Article  MathSciNet  Google Scholar 

  7. Canzani, Y., Gover, R., Jakobson, D., Ponge, R.: Conformal invariants from nodal sets. I. Negative eigen values and curvature prescription. Int. Math. Res. Notice IMRN 9, 2356–2400 (2014)

    MATH  Google Scholar 

  8. Case, J., Lin, Y., Yuan, W.: Conformally variational riemannian invariants. Trans. Am. Math. Soc. 371(11), 8217–8254 (2019)

    Article  MathSciNet  Google Scholar 

  9. Chang, S.Y.A., Eastwood, M., Ørsted, B., Yang, P.: What is \(Q\)-curvature? Acta Appl. Math. 102(2–3), 119–125 (2008)

    Article  MathSciNet  Google Scholar 

  10. Chang, S.Y.A., Yang, P.C.: Extremal metrics of zeta function determinants on 4-manifolds. Ann. Math. 142, 171–212 (1995)

    Article  MathSciNet  Google Scholar 

  11. Chang, S.-Y.A., Gursky, M., Yang, P.: Remarks on a fourth order invariant in conformal geometry. Aspects Math. HKU. 353–372 (2019)

  12. Chtioui, H., Rigane, A.: On the prescribed \(Q\)-curvature problem on \({\mathbb{S}}^n\). J. Funct. Anal. 261, 2999–3043 (2011)

    Article  MathSciNet  Google Scholar 

  13. Djadli, Z., Malchiodi, A.: Existence of conformal metrics with constant \(Q\)-curvature. Ann. Math. 168(3), 813–858 (2008)

    Article  MathSciNet  Google Scholar 

  14. Delanoë, P., Robert, F.: On the local Nirenberg problem for the \(Q\)-curvatures. Pacific J. Math. 231, 293–304 (2007)

    Article  MathSciNet  Google Scholar 

  15. Fefferman, C., Graham, C.R.: \(Q\)-curvature and Poincaré metrics. Math. Res. Lett. 9, 139–151 (2002)

    Article  MathSciNet  Google Scholar 

  16. Fischer, A., Marsden, J.: Linearization stability of nonlinear partial differential equations. In: Proceedings of a Symposium in Pure Mathematics, vol. 27. American Mathematical Society, Providence, pp. 219–263 (1975)

  17. Fisher, A.: Mardsen, Linearization stability of nonlinear partial differential equations. In: Proceedings of a Symposium in Pure Mathematics, vol. 27, Part 2. pp. 219–263 (1975)

  18. Graham, C.R., Zworski, M.: Scattering matrix in conformal geometry. Invent. Math. 152, 89–118 (2003)

    Article  MathSciNet  Google Scholar 

  19. Graham, C.R., Jenne, R., Mason, L.J., Sparling, G.A.J.: Conformally invariant powers of the Laplacian I Existence. J. Lond. Math. Soc. (2) 46(3), 557–565 (1992)

    Article  MathSciNet  Google Scholar 

  20. Gursky, M.: The principal eigenvalue of a conormally invariant differential operator, with an application to semilinear elliptic PDE. Commun. Math. Phys. 207, 131–147 (1999)

    Article  Google Scholar 

  21. Kazdan, J.: Prescribing the Curvature of a Riemannian Manifold. American Mathematics Society, New York (1984) (CBMS Regional Conference Series 57)

  22. Kazdan, J., Warner, F.: Curvature functions for compact 2-manifold. Ann. Math. 99, 14–47 (1974)

    Article  MathSciNet  Google Scholar 

  23. Kazdan, J., Warner, F.: Curvature functions for open 2-manifold. Ann. Math. 99, 203–219 (1974)

    Article  MathSciNet  Google Scholar 

  24. Kazdan, J., Warner, F.: Existence and conformal deformation of metrics with prescribed Gaussian and scalar curvature. Ann. Math. (2) 101, 317–331 (1975)

    Article  MathSciNet  Google Scholar 

  25. Kazdan, J., Warner, F.: Scalar Curvature and conformal deformation of Riemannian structure. J. Differ. Geom. 10, 113–134 (1975)

    Article  MathSciNet  Google Scholar 

  26. Kazdan, J., Warner, F.: A direct approach to the determination of Gaussian and scalar curvature functions. Invent. Math. 28, 227–230 (1975)

    Article  MathSciNet  Google Scholar 

  27. Lin, J., Yuan, W.: A symmetric 2-tensor canonically associated to Q-curvature and its applications. Pacific J. Math. 291, 425–438 (2017)

    Article  MathSciNet  Google Scholar 

  28. Levy, T., Oz, Y.: Liouville conformal field theories in higher dimensions (2018). arXiv:1804.02283. [hep-th]

  29. Lin, Y.-J., Yuan, W.: Deformations of \(Q\)-curvature I. Calc. Var. Partial Differ. Equ. 55(4):Paper No. 101, 29 (2016)

  30. Malchiodi, A., Struwe, M.: \(Q\)-curvature flow on \({\mathbb{S}}^4\). J. Differ. Geom. 73, 1–44 (2006)

    Article  Google Scholar 

  31. Mazýa, V.G., Shaposhnikova, T.O.: Theory of Multipliers in Spaces of Differentiable Functions. Monographs and Studies in Mathematics, vol. 23. Pitman, Boston (1985)

    Google Scholar 

  32. Nakayama, Y.: Canceling the Weyl anomaly from a position-dependent coupling. Phys. Rev. D 97(4), 045008 (2018). https://doi.org/10.1103/PhysRevD.97.045008. arXiv:1711.06413

    Article  MathSciNet  Google Scholar 

  33. Ndiaye, C.B.: Constant \(Q\)-curvature metrics in arbitrary dimension. J. Funct. Anal. 251(1), 1–58 (2007)

    Article  MathSciNet  Google Scholar 

  34. Robert, F.: Admissible \(Q\)-Curvatures Under Isometries for the Conformal GJMS Operators. Contemporary Mathematics, vol. 540, pp. 241–259. American Mathematics Society, Providence (2011)

    MATH  Google Scholar 

  35. Wallach, N., Warner, F.: Curvature forms for 2-manifolds. Proc. Am. Math. Soc. 25, 712–713 (1970)

    MathSciNet  MATH  Google Scholar 

  36. Wei, J., Xu, X.: On conformal deformations of metrics on \({\mathbb{S}}^n\). J. Funct. Anal. 157, 292–325 (1998)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

T.C would like to thank C. Arezzo for his kind interest in this work. We thank the referee for valuable comments and suggestions.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Tiarlos Cruz.

Additional information

Publisher's Note

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

T. Cruz has been partially suported by CNPq/Brazil Grant 311803/2019-9.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Cruz, F.F., Cruz, T. On the prescribed Q-curvature problem in Riemannian manifolds. manuscripta math. 165, 121–133 (2021). https://doi.org/10.1007/s00229-020-01198-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00229-020-01198-y

Mathematics Subject Classification

Navigation