Skip to main content
SpringerLink
Log in

Optimal Eigenvalue Estimate for the Dirac–Witten Operator on Bounded Domains with Smooth Boundary

  • Published:
Letters in Mathematical Physics Aims and scope Submit manuscript

Abstract

Eigenvalue estimate for the Dirac–Witten operator is given on bounded domains (with smooth boundary) of spacelike spin hypersurfaces satisfying the dominant energy condition, under four natural boundary conditions (MIT, APS, modified APS and chiral conditions). Roughly speaking, any eigenvalue of the Dirac–Witten operator satisfies

$$\left|\lambda\right|^{2} \,\geq\, \frac{n}{n-1} \frak R_{0} ,$$

where \({\frak R_{0}}\) is the infimum of (the opposite of) the Lorentzian norm of the constraints vector. Equality cases are also investigated and lead to interesting geometric situations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bartnik R.: The mass of an asymptotically flat manifold. Commun. Pure Appl. Math. 39, 661–693 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  2. Chruściel P.T., Herzlich M.: The mass of asymptotically hyperbolic Riemannian manifolds. Pac. J. Math. 212(2), 231–264 (2003)

    Article  Google Scholar 

  3. Chruściel P.T., Maerten D., Tod P.: Rigid upper bounds for the angular momentum and centre of mass of non-singular asymptotically anti-de Sitter space-times. J. High Energy Phys 11(084), 42 (2006)

    Google Scholar 

  4. Friedrich T.: Der erste Eigenwert des Dirac–Operators einer kompakten Riemannschen Mannigfaltigkeit nichtnegativer Skalarkrümmung. Math. Nach. 97(2), 117–146 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  5. Gibbons G.W., Hawking S.W., Horrowitz G.T., Perry M.J.: Positive mass theorem for black holes. Commun. Math. Phys. 88, 295–308 (1983)

    Article  ADS  Google Scholar 

  6. Herzlich M.: The positive mass theorem for black holes revisited. J. Geom. Phys. 26, 97–111 (1998)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  7. Herzlich M.: A Penrose-like inequality for the mass of Riemannian asymptotically flat manifolds. Commun. Math. Phys. 188, 121–133 (1998)

    Article  ADS  MathSciNet  Google Scholar 

  8. Hijazi, O.: Opérateurs de Dirac sur les variétés riemanniennes. Minoration des valeurs propres. Thèse de 3ème cycle, Ecole Polytechnique (1984)

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

    Article  MATH  ADS  MathSciNet  Google Scholar 

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

    Article  MATH  ADS  Google Scholar 

  11. Hijazi O., Zhang X.: The Dirac–Witten operator on spacelike hypersurfaces. Comm. Anal. Geom. 11(4), 737–750 (2003)

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  ADS  MathSciNet  Google Scholar 

  13. Leitner F.: Imaginary Killing spinors in Lorentzian geometry. J. Math. Phys. 44(11), 4795–4806 (2003)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  14. Maerten D.: The positive energy-momentum theorem in AdS-asymptotically hyperbolic manifolds. Ann. Henri Poincaré7(5), 975–1011 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  15. Maerten D.: A Penrose-like inequality for maximal asymptotically flat spin initial data sets. Ann. Global Anal. Geom. 32(4), 391–414 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  16. Parker T., Taubes C.: On Witten’s proof of the positive energy theorem. Commun. Math. Phys. 84, 223–238 (1982)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  17. Raulot S.: Optimal eigenvalues estimate for the Dirac operator on domains with boundary. Lett. Math. Phys. 73(2), 135–145 (2005)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  18. Wald, R.: General Relativity. University Press of Chicago, Chicago

  19. Witten E.: A simple proof of the positive energy theorem. Commun. Math. Phys. 80, 381–402 (1981)

    Article  ADS  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. LMPT, Fédération Denis Poisson, Faculté des Sciences, Parc de Grandmont, 37200, Tours, France

    Daniel Maerten

Authors
  1. Daniel Maerten
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Daniel Maerten.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Maerten, D. Optimal Eigenvalue Estimate for the Dirac–Witten Operator on Bounded Domains with Smooth Boundary. Lett Math Phys 86, 1–18 (2008). https://doi.org/10.1007/s11005-008-0267-2

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11005-008-0267-2

Mathematics Subject Classification (2000)

Keywords

Search

Navigation