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A spinorial characterization of hyperspheres

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Abstract

Let M be a compact orientable n-dimensional hypersurface, with nowhere vanishing mean curvature H, immersed in a Riemannian spin manifold \({\overline{M}}\) admitting a non trivial parallel spinor field. Then the first eigenvalue \({\lambda_1(D_{M}^{H})}\) (with the lowest absolute value) of the Dirac operator \({D_{M}^{H}}\) corresponding to the conformal metric \({\langle\;,\;\rangle^{H}=H^{2}\,\langle\;,\;\rangle}\), where \({\langle\;,\;\rangle}\) is the induced metric on M, satisfies \({\left|\lambda_1(D_{M}^{H})\right|\le \frac{n}{2}}\). By applying the Bourguignon-Gauduchon first variational formula, we obtain a necessary condition for \({\left|\lambda_1(D_{M}^{H})\right|=\frac{n}{2}}\). As a consequence, we prove that round hyperspheres are the only hypersurfaces of the Euclidean space satisfying the equality in the Bär inequality

$$\lambda_1(D_{M})^{2}\le \frac{n^{2}}{4{vol}(M)}\int_{M} H^{2}\, dV,$$

where D M stands now for the Dirac operator of the induced metric.

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References

  1. Agricola I., Friedrich T.: Upper bounds for the first eigenvalue of the Dirac operator on surfaces. J. Geom. Phys. 30(1), 1–22 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  2. Alexandrov A.D.: A characteristic property of spheres. Ann. Mat. Pura Appl. 58, 303–315 (1962)

    Article  MathSciNet  Google Scholar 

  3. Ammann B.: The Willmore conjecture for immersed tori with small curvature integral. Manuscr. Math. 101, 1–22 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  4. Ammann B.: Ambient Dirac eigenvalue estimates and the Willmore functional. In: Bourguignon, J.P., Branson, T., Chamseddine, A., Hijazi, O., Stanton, R.J. (eds.) Dirac Operators: Yesterday and Today, pp. 221–228. International Press, Boston (2005)

    Google Scholar 

  5. Bär C.: Real Killing spinors and holonomy. Commun. Math. Phys. 154, 509–521 (1993)

    Article  MATH  Google Scholar 

  6. Bär C.: Metrics with harmonic spinors. Geom. Funct. Anal. 6, 899–942 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bär C.: Extrinsic bounds of the Dirac operator. Ann. Glob. Anal. Geom. 16, 573–596 (1998)

    Article  MATH  Google Scholar 

  8. Baum H.: An upper bound for the first eigenvalue of the Dirac operator on compact spin manifolds. Math. Z. 206, 409–422 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  9. Baum, H., Friedrich, T., Grünewald, R., Kath, I.: Twistor and Killing Spinors on Riemannian Manifolds. Seminarbericht 108. Humboldt-Universität zu Berlin (1990)

  10. Bleecker D., Weiner J.: Extrinsic bound of λ1 of Δ on a compact manifold. Comment. Math. Helv. 51, 601–609 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  11. Bourguignon J.P., Gauduchon P.: Spineurs, opérateurs de Dirac et variations de métriques. Commun. Math. Phys. 144, 581–599 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  12. Bourguignon, J.P., Hijazi, O., Milhorat, J.L., Moroianu, A.: A Spinorial Approach to Riemannian and Conformal Geometry. Monograph (in preparation)

  13. Bureš J.: Dirac operators on hypersurfaces. Comment. Math. Univ. Carol. 34(2), 313–322 (1993)

    MATH  Google Scholar 

  14. Burstall, F., Ferus, D., Leschke, K., Pedit, F., Pinkall, U.: Conformal Geometry of Surfaces in S 4 and Quaternions. Lecture Notes in Mathematics, vol. 1772 (2002)

  15. Chen B.Y.: On the total curvature of immersed manifolds IV: spectrum and total mean curvature. Bull. Math. Acad. Sin. 7, 301–311 (1979)

    MATH  Google Scholar 

  16. Ferus D., Leschke K., Pedit F., Pinkall U.: Quaternionic holomorphic geometry: Plücker formula, Dirac eigenvalue estimates and energy estimates of harmonic 2-tori. Invent. Math. 146, 507–593 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  17. Friedrich, T.: Dirac Operators in Riemannian Geometry. A.M.S. Graduate Studies in Mathematics, vol. 25 (2000)

  18. Friedrich T., Kim E.C.: The Einstein-Dirac equation on Riemannian spin manifolds. J. Geom. Phys. 33, 128–172 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  19. Gallot S.: Équations différentielles caractéristiques de la sphère. Ann. Scient. Éc. Norm. Sup. 12, 235–267 (1979)

    MathSciNet  Google Scholar 

  20. Ginoux, N.: The Dirac Spectrum. Lecture Notes in Mathematics, vol. 1976 (2009)

  21. Gordon C., Webb D.L., Wolpert S.: One cannot hear the shape of a drum. Bull. Am. Math. Soc. 27, 134–138 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  22. Hijazi O.: A conformal lower bound for the smallest eigenvalue of the Dirac operator and Killing spinors. Commun. Math. Phys. 104, 151–162 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  23. Hijazi O.: Lower bounds for the eigenvalues of the Dirac operator. J. Geom. Phys. 16, 27–38 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  24. Hijazi O., Montiel S.: Extrinsic Killing spinors. Math. Z. 244, 337–347 (2003)

    MathSciNet  MATH  Google Scholar 

  25. Hijazi O., Montiel S., Zhang X.: Dirac operator on embedded hypersurfaces. Math. Res. Lett. 8, 195–208 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  26. Hijazi O., Montiel S., Zhang X.: Eigenvalues of the Dirac operator on manifolds with boundary. Commun. Math. Phys. 221, 255–265 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  27. Hijazi O., Montiel S., Roldán A.: Eigenvalue boundary problems for the Dirac operator. Commun. Math. Phys. 231, 375–390 (2002)

    Article  MATH  Google Scholar 

  28. Hijazi O., Montiel S., Zhang X.: Conformal lower bounds for the Dirac operator of embedded hypersurfaces. Asian J. Math. 6, 23–36 (2002)

    MathSciNet  MATH  Google Scholar 

  29. Hijazi O., Montiel S., Roldán A.: Dirac operators on hypersurfaces of manifolds with negative scalar curvature. Ann. Glob. Anal. Geom. 23, 247–264 (2003)

    Article  MATH  Google Scholar 

  30. Hitchin N.: Harmonic spinors. Adv. Math. 14, 1–55 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  31. Hopf, H.: Differential Geometry in the Large. Lecture Notes in Mathematics, vol. 1000 (1983)

  32. Hsiang W.Y., Teng Z.H., Yu W.C.: New examples of constant mean curvature immersions of (2k−1)-spheres into Euclidean 2k-space. Ann. Math. 117, 609–625 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  33. Kac M.: Can one hear the shape of a drum?. Am. Math. Monthly 73, 1–23 (1966)

    Article  MATH  Google Scholar 

  34. Kapouleas N.: Compact constant mean curvature surfaces in Euclidean three-space. J. Differ. Geom. 33, 683–715 (1991)

    MathSciNet  MATH  Google Scholar 

  35. Kapouleas N.: Constant mean curvature surfaces constructed by fusing Wente tori. Invent. Math. 119, 443–518 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  36. Lawson H.B., Michelsohn M.L.: Spin Geometry. Princeton Mathematical Series, vol. 38. Princeton University Press, Princeton (1989)

    Google Scholar 

  37. Montiel S.: Unicity of constant mean curvature hypersurfaces in some Riemannian manifolds. Indiana Univ. Math. J. 48, 711–748 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  38. Montiel S., Ros A.: Minimal immersions of surfaces by the first eigenfunctions and conformal area. Invent. Math. 83, 153–166 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  39. Ogiue, K.: Some open problems in differential geometry. In: Proceedings of Symposia in Pure Mathematics, vol. XXVII, pp. 407–411 (1973)

  40. O’Neill B.: Semi-Riemannian Geometry. Academic Press, New York (1983)

    MATH  Google Scholar 

  41. Reilly R.C.: On the first eigenvalue of the Laplacian for compact submanifolds of Euclidean space. Comment. Math. Helv. 52, 525–533 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  42. Taimanov I.A.: Dirac operator and conformal invariants of tori in 3-space. Tr. Mat. Inst. Steklova 244, 249–280 (2004)

    MathSciNet  Google Scholar 

  43. Trautman A.: The Dirac operator on hypersurfaces. Acta Phys. Plon. B 26, 1283–1310 (1995)

    MathSciNet  MATH  Google Scholar 

  44. Wang M.Y.: Parallel spinors and parallel forms. Ann. Glob. Anal. Geom. 7, 59–68 (1989)

    Article  MATH  Google Scholar 

  45. Wang M.Y.: On non-simply connected manifolds with non-trivial parallel spinors. Ann. Glob. Anal. Geom. 13, 31–42 (1995)

    Article  MATH  Google Scholar 

  46. Wente H.C.: Counterexample to a conjecture of H. Hopf. Pacific J. Math. 121, 193–243 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  47. Yau S.T.: Problem section. Ann. Math. Stud. 102, 669–706 (1982)

    Google Scholar 

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Authors and Affiliations

  1. Institut Élie Cartan, Université de Lorraine, Nancy, B.P. 239, 54506, Vandèuvre-Lès-Nancy Cedex, France

    Oussama Hijazi

  2. Departamento de Geometría y Topología, Universidad de Granada, 18071, Granada, Spain

    Sebastián Montiel

Authors
  1. Oussama Hijazi
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  2. Sebastián Montiel
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Correspondence to Sebastián Montiel.

Additional information

Communicated by A. Malchiodi.

Dédié à Jean pierre Bourguignon en témoignage de notre reconnaissance et amitié.

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Hijazi, O., Montiel, S. A spinorial characterization of hyperspheres. Calc. Var. 48, 527–544 (2013). https://doi.org/10.1007/s00526-012-0560-x

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