Advertisement

Paneitz operator for metrics near \(S^{3}\)

  • Fengbo HangEmail author
  • Paul C. Yang
Article

Abstract

We derive the first and second variation formula for the Green’s function pole’s value of Paneitz operator on the standard three sphere. In particular it is shown that the first variation vanishes and the second variation is nonpositively definite. Moreover, the second variation vanishes only at the direction of conformal deformation. We also introduce a new invariant of the Paneitz operator and illustrate its close relation with the second eigenvalue and Sobolev inequality of Paneitz operator.

Mathematics Subject Classification

Primary 53A30 Secondary 35J08 58J05 

Notes

Acknowledgements

We would like to thank the refree for his/her careful reading of the article and many comments that improved the presentation of the paper. The research of Yang is supported by NSF Grant DMS 1104536 and 1509505.

References

  1. 1.
    Besse, A.L.: Einstein Manifolds. Results in Mathematics and Related Areas, vol. 10. Springer-Verlag, Berlin (1987)CrossRefGoogle Scholar
  2. 2.
    Branson, T.: Differential operators canonically associated to a conformal structure. Math. Scand. 57(2), 293–345 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Chang, S.Y., Gursky, M.J., Yang, P.C.: An equation of Monge–Ampere type in conformal geometry, and four-manifolds of positive Ricci curvature. Ann. Math. (2) 155(3), 709–787 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Gursky, M.J., Hang, F.B., Lin, Y.J.: Riemannian manifolds with positive Yamabe invariant and Paneitz operator. Int. Math. Res. Not. IMRN 2016(5), 1348–1367 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Gursky, M.J., Malchiodi, A.: A strong maximum principle for the Paneitz operator and a nonlocal flow for the \(Q\) curvature. J. Eur. Math. Soc. 17, 2137–2173 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Hang, F.B.: On the higher order conformal covariant operators on the sphere. Commun. Contemp. Math. 9(3), 279–299 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Hang, F.B., Yang, P.C.: The Sobolev inequality for Paneitz operator on three manifolds. Calc. Var. PDE 21, 57–83 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Hang, F.B., Yang, P.C.: Sign of Green’s function of Paneitz operators and the \(Q\) curvature. Int. Math. Res. Not. IMRN 2015(19), 9775–9791 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Hang, F.B., Yang, P.C.: \(Q\) curvature on a class of \(3\)-manifolds. Commun. Pure Appl. Math. 69(4), 734–744 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Hang, F.B., Yang, P.C.: \(Q\)-curvature on a class of manifolds with dimension at least \(5\). Commun. Pure Appl. Math. 69(8), 1452–1491 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Hang, F.B., Yang, P.C.: Lectures on the fourth-order \(Q\) curvature equation. Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap. 31, 1–33 (2016)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Humbert, E., Raulot, S.: Positive mass theorem for the Paneitz–Branson operator. Cal. Var. PDE 36(4), 525–531 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Lee, J.M., Parker, T.H.: The yamabe problem. Bull. AMS 17(1), 37–91 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Paneitz, S.M.: A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds. SIGMA Symmetry Integrability Geom. Methods Appl. 4(036), 3 (2008)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Yang, P.C., Zhu, M.J.: On the Paneitz energy on standard three sphere. ESAIM Control Optim. Cal. Var. 10, 211–223 (2004)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Courant InstituteNew York UniversityNew YorkUSA
  2. 2.Department of MathematicsPrinceton UniversityPrincetonUSA

Personalised recommendations