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Mathematische Annalen

, Volume 355, Issue 4, pp 1255–1299 | Cite as

A perturbation method for spinorial Yamabe type equations on \(S^m\) and its application

  • Takeshi IsobeEmail author
Article

Abstract

For \(m\ge 2\), we prove the existence of non-trivial solutions for a certain kind of nonlinear Dirac equations with critical Sobolev nonlinearities on \(S^m\) via a perturbative variational method. For the special case \(m=2\), this establishes the existence of a conformal immersion \(S^2\rightarrow \mathbb R ^3\) with prescribed mean curvature \(H\) which is close to a positive constant under an index counting condition on \(H\).

Mathematics Subject Classification (2000)

35Q41 53C42 57R70 58E05 

Notes

Acknowledgments

I would like to express my gratitude to the anonymous referee for drawing my attention to the four vertex theorem and the recent result of M. Anderson [11]. He also gave me a lot of helpful suggestions that made this paper much easier to read.

References

  1. 1.
    Adams, R.: Sobolev Spaces. Academic Press, New York (1975)zbMATHGoogle Scholar
  2. 2.
    Ambrosetti, A., Badiale, M.: Homoclinics: Poincaré-Melnikov type results via a variational approach. Ann. Inst. H. Poincaré Anal. Nonlinéaire 15, 233–252 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Ambrosetti, A., Badiale, M.: Variational perturbative methods and bifurcation of bound states from the essential spectrum. Proc. R. Soc. Edinburgh Sect. A 128, 1131–1161 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Ambrosetti, A., Garcia, A.J.: Perturbation of \(\Delta u+u^{N+2/N-2}=0\), the scalar curvature problem in \(\mathbb{R}^N\), and related topics. J. Funct. Anal. 165, 117–149 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Ambrosetti, A., Malchiodi, A.: Perturbation methods and semilinear elliptic problems on \(\mathbb{R}^n\). In: Progress in Mathematics, vol. 240. Birkhäuser, Basel (2006)Google Scholar
  6. 6.
    Ammann, B.: A spin-conformal lower bound of the first positive Dirac eigenvalue. Diff. Geom. Appl. 18, 21–32 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Ammann, B.: A variational problem in conformal spin geometry. Universität Hamburg, Habilitationsschift (2003)Google Scholar
  8. 8.
    Ammann, B.: The smallest Dirac eigenvalue in a spin-conformal class and cmc-immersions. Commun. Anal. Geom. 17, 429–479 (2009)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Ammann, B., Humbert, E., Grosjean, J.-F.: Mass endomorphism and spinorial Yamabe type problem on conformally flat manifolds. Commun. Anal. Geom. 14, 163–182 (2006)zbMATHGoogle Scholar
  10. 10.
    Ammann, B., Humbert, E., Ahmedou, M.O.: An obstruction for the mean curvature of a conformal immersion \(S^n\rightarrow \mathbb{R}^{n+1}\). Proc. AMS. 135, 489–493 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Anderson, M.: Conformal immersions of prescribed mean curvature in \(\mathbb{R}^3\). arXiv:1204.5225Google Scholar
  12. 12.
    Atiyah, M.F., Bott, R.: The Yang-Mills equations over Riemann surfaces. Philos. Trans. R. Soc. Lond. Ser. A 308, 523–615 (1983)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Aubin, T.: Some nonlinear problems in Riemannian geometry. In: Springer Monographs in Mathematics. Springer, Berlin (1998)Google Scholar
  14. 14.
    Bahri, A., Coron, J.M.: The scalar curvature problem on the standard three-dimensional sphere. J. Funct. Anal. 95, 106–172 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Bott, R.: Nondegenerate critical manifolds. Ann. Math. 60(2), 248–261 (1954)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Bourguignon, J.-P., Ezin, J.P.: Scalar curvature functions in a conformal class of metrics and conformal transformations. Trans. Am. Math. Soc. 301, 723–736 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Banyaga, A., Hurtubise, D.: Lectures on Morse homology. In: Texts in the Mathematical Sciences. Kluwer, Dordrecht (2005)Google Scholar
  18. 18.
    Chang, K.L.: Infinite dimensional Morse theory and multiple solution problems. Prog. Nonlinear Diff. Equ. Appl. 6 (2006)Google Scholar
  19. 19.
    Chang, S.-Y.A.: Non-linear elliptic equations in conformal geometry. In: Zurich Lectures in Advanced Mathematics. EMS, San Francisco (2004)Google Scholar
  20. 20.
    Chavel, I.: Eigenvalues in Riemannian geometry. In: Pure and Applied Mathematics, vol. 115. Academic Press, London (1984)Google Scholar
  21. 21.
    Chen, Q., Jost, J., Wang, G.: Nonlinear Dirac equations on Riemann surfaces. Ann. Global Anal. Geom. 33, 253–270 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Chang, S.Y.A., Yang, P.: A perturbation result in prescribing scalar curvature on \(S^n\). Duke Math. J. 64, 27–69 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Chang, S.-Y.A., Gursky, M., Yang, P.: The scalar curvature equation on 2- and 3-spheres. Calc. Var. 1, 205–229 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Dahlberg, B.: The converse of the four vertex theorem. Proc. Am. Math. Soc. 133, 2131–2135 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    DeTurck, D., Gluck, H., Pomerleano, D., Shea, V.D.: The four vertex theorem and its converse. Not. AMS 54, 192–207 (2007)zbMATHGoogle Scholar
  26. 26.
    Friedrich, T.: On the spinor representation of surfaces in Euclidean 3-space. J. Geom. Phys. 28, 143–157 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Friedrich, T.: Dirac operators in Riemannian geometry. In: Graduate Studies in Mathematics, vol. 25. American Mathematical Society, New York (2000)Google Scholar
  28. 28.
    Ginoux, N.: The Dirac spectrum. In: Lecture Notes in Mathematics, vol. 1976. Springer, Berline (2009)Google Scholar
  29. 29.
    Hitchin, N.: Harmonic spinors. Adv. Math. 14, 1–55 (1974)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Isobe, T.: Existence results for solutions to nonlinear Dirac equations on compact spin manifolds. Manuscr. Math. 135, 329–360 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Isobe, T.: Nonlinear Dirac equations with critical nonlinearities on compact spin manifolds. J. Funct. Anal. 260, 253–307 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Isobe, T.: Spinorial Yamabe type equations on \(S^m\) via Conley index (2012) (preprint)Google Scholar
  33. 33.
    Kusner, R., Schmitt, N.: The spinor representation of surfaces in space. dg-ga/9610005.Google Scholar
  34. 34.
    Kazdan, J., Warner, F.: Existence and conformal deformation of metrics with prescribed Gaussian and scalar curvature. Ann. Math. 101, 317–331 (1975)MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Li, Y.Y.: Prescribing scalar curvature on \(S^3\), \(S^4\) and related problems. J. Funct. Anal. 118, 43–118 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Li, Y.Y.: Prescribing scalar curvature on \(S^n\) and related topics, Part I. J. Differ. Eq. 120, 319–410 (1995)zbMATHCrossRefGoogle Scholar
  37. 37.
    Li, Y.Y.: Prescribing scalar curvature on \(S^n\) and related topics, Part II. Commun. Pure Appl. Math. 49, 437–477 (1996)CrossRefGoogle Scholar
  38. 38.
    Lawson, H.B., Michelson, M.: Spin Geometry. Princeton University Press, Princeton (1989)zbMATHGoogle Scholar
  39. 39.
    Malchiodi, A.: The scalar curvature problem on \(S^n\): an approach via Morse theory. Calc. Var. 14, 429–445 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    Morse, M., Van Schaak, G.: The critical point theory under general boundary conditions. Ann. Math. 35, 545–571 (1934)CrossRefGoogle Scholar
  41. 41.
    Raulot, S.: A Sobolev-like inequality for the Dirac operator. J. Funct. Anal. 26, 1588–1617 (2009)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.Department of Mathematics, Graduate School of Science and EngineeringTokyo Institute of TechnologyTokyoJapan

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