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Eigenvalue estimate of the Dirac operator and Rigidity of Poincare–Einstein metrics

  • Daguang Chen
  • Fang Wang
  • Xiao Zhang
Article
  • 24 Downloads

Abstract

We re-visit the eigenvalue estimate of the Dirac operator on spin manifolds with boundary in terms of the first eigenvalues of conformal Laplace operator as well as the conformal mean curvature operator. These problems were studied earlier by Hijazi–Montiel–Zhang and Raulot and we re-prove them under weaker assumption that a boundary chirality operator exists. Moreover, on these spin manifolds with boundary, we show that any \(C^{3,\alpha }\) conformal compactification of some Poincare–Einstein metric must be the standard hemisphere when the first nonzero eigenvalue of the Dirac operator achieves its lowest value, and any \(C^{3,\alpha }\) conformal compactification of some Poincare–Einstein metric must be the flat ball in Euclidean space when the first positive eigenvalue of the boundary Dirac operator achieves certain value relating to the second Yamabe invariant. In two cases the Poincare–Einstein metrics are standard hyperbolic metric.

Keywords

Eigenvalue Dirac operator Boundary condition Yamabe invariant Poincare–Einstein metric 

Notes

Acknowledgements

The work of D. Chen was supported by NSF of China grant 11471180 and 11831005. The work of F. Wang was supported by NSF of China grant 11571233 and 11871331. The work of X. Zhang was supported by NSF of China grants 11571345, 11731001 and HLM, NCMIS, CEMS, HCMS of Chinese Academy of Sciences.

References

  1. 1.
    Atiyah, M.F., Patodi, V.K., Singer, I.M.: Spectral asymmetry and Riemannian geometry. I. Math. Proc. Cambr. Phil. Soc. 77, 43–69 (1975)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Atiyah, M.F., Patodi, V.K., Singer, I.M.: Spectral asymmetry and Riemannian geometry. II. Math. Proc. Cambr. Phil. Soc. 78, 405–432 (1975)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Atiyah, M.F., Patodi, V.K., Singer, I.M.: Spectral asymmetry and Riemannian geometry. III. Math. Proc. Cambr. Phil. Soc. 79, 71–99 (1976)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bär, C., Ballmann, W.: Boundary value problems for elliptic differential operators of first order, In: Memory of C.C. Hsiung—lectures given at the JDG Symposium, Surveys in Differential Geometry XVII, 1–78 (2012)Google Scholar
  5. 5.
    Bartnik, R., Chruściel, P.: Boundary value problems for Dirac-type equations. J. Reine Angew. Math. 579, 13–73 (2005)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Booß-Bavnvek, B., Wojciechoski, K.P.: Elliptic boundatry problems for Dirac operators. Birkhäuser, Boston (1993)CrossRefGoogle Scholar
  7. 7.
    Chruściel, P., Delay, E., Lee, J., Skinner, D.: Boundary regularity of conformal compact einstein metrics. J. Differ. Geom. 69, 111–136 (2005)CrossRefGoogle Scholar
  8. 8.
    Chen, X., Lai, M., Wang, F.: Escobar-Yamabe compactifications for Poincare–Einstein manifolds and rigidity theorems, arXiv:1712.02540 (2017)
  9. 9.
    Escobar, J.: The Yamabe problem on manifolds with boundary. J. Differ. Geom. 35, 21–84 (1992)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Escobar, J.: Conformal deformation of a Riemannian metric to a scalar flat metric with constant mean curvature on the boundary. Ann. Math. 136, 1–50 (1992)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Escobar, J.: Conformal deformation of a Riemannian metric to a constant scalar curvature metric with constant mean curvature on the boundary. Indiana Univ. Math. J. 45(4), 917–943 (1996)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Farinelli, S., Schwarz, G.: On the spectrum of the Dirac operator under boundary conditions. J. Geom. Phys. 28, 67–84 (1998)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Friedrich, T.: Der erste eigenwert des Dirac-operators einer kompakten Riemannschen Mannigfaltigkeit nichtnegativer Skalarkrümmmung. Math. Nachr. 97, 117–146 (1980)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Gibbons, G., Hawking, S., Horowitz, G., Perry, M.: Positive mass theorems for black holes. Commun. Math. Phys. 88, 295–308 (1983)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Graham, C.R.: Volume renormalization for singular Yamabe metrics. Proc. Am. Math. Soc. 145, 1781–1792 (2017)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Hijazi, O.: A conformal lower bound for the smallest eigenvalue of the Dirac operator and Killing spinors. Commun. Math. Phys. 25, 151–162 (1986)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Hijazi, O., Montiel, S., Roldan, A.: Eigenvalue boundary problems for the Dirac operator. Commun. Math. Phys. 231, 375–390 (2002)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Hijazi, O., Montiel, S., Zhang, X.: Eigenvalues of the Dirac operator on manifolds with boundary. Commun. Math. Phys. 221, 255–265 (2001)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Hijazi, O., Montiel, S., Zhang, X.: Dirac operator on embedded hypersurfaces. Math. Res. Lett 8, 195–208 (2001)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Hijazi, O., Montiel, S., Zhang, X.: Conformal lower bounds for the Dirac operator of embedded hypersurfaces. Asian J. Math. 6, 23–36 (2002)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Hijazi, O., Zhang, X.: Lower bounds for the eigenvalues of the Dirac operator I: The hypersurface Dirac operator. Ann. Glob. Anal. Geom. 19, 355–376 (2001)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Hijazi, O., Zhang, X.: The Dirac-Witten operator on spacelike hypersurfaces. Commun. Anal. Geom. 11, 737–750 (2003)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Hitchin, N.: Harmonic spinors. Adv. Math. 14, 1–55 (1974)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Lawson, H., Michelsohn, M.: Spin geometry. Princeton Univ. Press, Princeton (1989)zbMATHGoogle Scholar
  25. 25.
    Lichnerowicz, A.: Spineurs harmoniques. C.R. Acad. Sci. Paris 257, Sèrie I, 7–9 (1963)Google Scholar
  26. 26.
    Li, G., Qing, J., Shi, Y.: Gap phenomena and curvature estimates for conformally compact Einstein manifolds. Trans. Am. Math. Soc. 369, 4385–4413 (2017)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Obata, M.: Certain conditions for a Riemannian manifold to be isometric with a sphere. J. Math. Soc. Jpn. 14, 333–340 (1962)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Qing, J.: On the rigidity for conformally compact Einstein manifolds. Int. Math. Res. Not. 21, 1141–1153 (2003)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Raulot, S.: The Hijazi inequality on manifolds with boundary. J. Geom. Phys. 56, 2189–2202 (2006)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Shi, Y., Tian, G.: Rigidity of asymptotically hyperbolic manifolds. Commun. Math. Phys. 259, 545–559 (2005)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Witten, E., Yau, S.T.: Connectedness of the boundary in the AdS/CFT correspondence. Adv. Theor. Math. Phys. 3, 1635–1655 (1999)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematical SciencesTsinghua UniversityBeijingPeople’s Republic of China
  2. 2.School of Mathematical SciencesShanghai Jiao Tong UniversityShanghaiPeople’s Republic of China
  3. 3.Institute of Mathematics, Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijingPeople’s Republic of China
  4. 4.School of Mathematical SciencesUniversity of Chinese Academy of SciencesBeijingPeople’s Republic of China

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