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Equivariant CR Yamabe problem

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Abstract

As a generalization of the Yamabe problem, Hebey and Vaugon considered the equivariant Yamabe problem: for a subgroup G of the isometry group, find a G-invariant metric whose scalar curvature is constant in a given conformal class. In this paper, we introduce the equivariant CR Yamabe problem and prove some related results.

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References

  1. Aubin, T.: Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire. J. Math. Pures Appl. (9) 55, 269–296 (1976)

    MathSciNet  Google Scholar 

  2. Baues, O., Kamishima, Y.: A note on vanishing of equivariant differentiable cohomology of proper actions and application to CR-automorphism and conformal groups (2021). ar**V:2101.03831

  3. Cheng, J.H., Chiu, H.L., Yang, P.: Uniformization of spherical CR manifolds. Adv. Math. 255, 182–216 (2014)

    Article  MathSciNet  Google Scholar 

  4. Cheng, J.H., Malchiodi, A., Yang, P.: A positive mass theorem in three dimensional Cauchy–Riemann geometry. Adv. Math. 308, 276–347 (2017)

    Article  MathSciNet  Google Scholar 

  5. Dragomir, S., Tomassini, G.: Differential Geometry and Analysis on CR Manifolds, Progr. Math., vol. 246. Birkhäuser, Boston (2006)

  6. Folland, G., Stein, E.: Estimates for the \(\bar{\partial }_b\) complex and analysis on the Heisenberg group. Commun. Pure Appl. Math. 27, 429–522 (1974)

    Article  Google Scholar 

  7. Gamara, N.: The CR Yamabe conjecture–the case \(n=1\). J. Eur. Math. Soc. 3, 105–137 (2001)

    Article  MathSciNet  Google Scholar 

  8. Gamara, N., Yacoub, R.: CR Yamabe conjecture—the conformally flat case. Pac. J. Math. 201, 121–175 (2001)

    Article  MathSciNet  Google Scholar 

  9. Henry, G., Madani, F.: The equivariant second Yamabe constant. J. Geom. Anal. 28(4), 3747–3774 (2018)

    Article  MathSciNet  Google Scholar 

  10. Hebey, E., Vaugon, M.: Le problème de Yamabe équivariant. Bull. Sci. Math. 117, 241–286 (1993)

    MathSciNet  Google Scholar 

  11. Ho, P.T.: The Webster scalar curvature flow on CR sphere. Part I. Adv. Math. 268, 758–835 (2015)

    Article  MathSciNet  Google Scholar 

  12. Ho, P.T., Shin, J.: Equivariant Yamabe problem with boundary. Calc. Var. Partial Differ. Equ. 61(1), 37 (2022)

    Article  MathSciNet  Google Scholar 

  13. Jerison, D., Lee, J.M.: Extremals for the Sobolev inequality on the Heisenberg group and the CR Yamabe problem. J. Am. Math. Soc. 1, 1–13 (1988)

    Article  MathSciNet  Google Scholar 

  14. Jerison, D., Lee, J.M.: Intrinsic CR normal coordinates and the CR Yamabe problem. J. Differ. Geom. 29, 303–343 (1989)

    Article  MathSciNet  Google Scholar 

  15. Jerison, D., Lee, J.M.: The Yamabe problem on CR manifolds. J. Differ. Geom. 25, 167–197 (1987)

    Article  MathSciNet  Google Scholar 

  16. Lee, J.M.: CR manifolds with noncompact connected automorphism groups. J. Geom. Anal. 6(1), 79–90 (1996)

    Article  MathSciNet  Google Scholar 

  17. Lee, J.M.: The Fefferman metric and pseudo-Hermitian invariants. Trans. Am. Math. Soc. 296(1), 411–429 (1986)

    MathSciNet  Google Scholar 

  18. Lee, J.M., Parker, T.: The Yamabe problem. Bull. Am. Math. Soc. (N.S.) 17(1), 37–91 (1987)

    Article  MathSciNet  Google Scholar 

  19. Lelong-Ferrand, J.: Transformations conformes et quasi-conformes des variétés riemanniennes compactes. Mem. Acad. Royale de Belgique 39(5), 1–44 (1971)

    Google Scholar 

  20. Lichnerowicz, A.: Sur les transformations conformes d’une variété riemanniemme compacte. C. R. Acad. Sci. Paris 259, 697–700 (1964)

    MathSciNet  Google Scholar 

  21. Madani, F.: Equivariant Yamabe problem and Hebey–Vaugon conjecture. J. Funct. Anal. 258, 241–254 (2010)

    Article  MathSciNet  Google Scholar 

  22. Madani, F.: Hebey–Vaugon conjecture II. C. R. Math. Acad. Sci. Paris 350, 849–852 (2012)

    Article  MathSciNet  Google Scholar 

  23. Obata, M.: The conjectures on conformal transformations of Riemannian manifolds. J. Differ. Geom. 6(2), 247–258 (1971)

    Article  MathSciNet  Google Scholar 

  24. Schoen, R.: Conformal deformation of a Riemannian metric to constant scalar curvature. J. Differ. Geom. 20, 479–495 (1984)

    Article  MathSciNet  Google Scholar 

  25. Schoen, R.: On the conformal and CR automorphism groups. Geom. Funct. Anal. 5(2), 464–481 (1995)

    Article  MathSciNet  Google Scholar 

  26. Sung, C.: Surgery and equivariant Yamabe invariant. Differ. Geom. Appl. 24(3), 271–287 (2006)

    Article  MathSciNet  Google Scholar 

  27. Tanaka, N.: A differential geometric study on strongly pseudo-convex manifolds, Kinokuniya, Tokyo (1975)

  28. Trudinger, N.S.: Remarks concerning the conformal deformation of Riemannian structures on compact manifolds. Ann. Scuola Norm. Sup. Pisa (3) 22, 265–274 (1968)

    MathSciNet  Google Scholar 

  29. Webster, S.M.: On the transformation group of a real hypersurface. Trans. Am. Math. Soc. 231(1), 179–190 (1977)

    Article  MathSciNet  Google Scholar 

  30. Webster, S.M.: Pseudohermitian structures on a real hypersurface. J. Differ. Geom. 13, 25–41 (1978)

    Article  Google Scholar 

  31. Yamabe, H.: On a deformation of Riemannian structures on compact manifolds. Osaka Math. J. 12, 21–37 (1960)

    MathSciNet  Google Scholar 

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Acknowledgements

The author would like to thank Prof. Kamishima for sending him the paper [2]. This work is supported by the National Science and Technology Council (NSTC), Taiwan, with grant Number: 112-2115-M-032 -006 -MY2.

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  1. Department of Mathematics, Tamkang University, Tamsui, New Taipei City, 251301, Taiwan

    Pak Tung Ho

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  1. Pak Tung Ho
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Ho, P.T. Equivariant CR Yamabe problem. Annali di Matematica (2024). https://doi.org/10.1007/s10231-024-01484-6

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