Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 19 (2023), 029, 14 pages      arXiv:2210.13037      https://doi.org/10.3842/SIGMA.2023.029
Contribution to the Special Issue on Differential Geometry Inspired by Mathematical Physics in honor of Jean-Pierre Bourguignon for his 75th birthday

Total Mean Curvature and First Dirac Eigenvalue

Simon Raulot
Laboratoire de Mathématiques R. Salem, UMR $6085$ CNRS-Université de Rouen, Avenue de l'Université, BP.12, Technopôle du Madrillet, 76801 Saint-Étienne-du-Rouvray, France

Received October 25, 2022, in final form May 09, 2023; Published online May 25, 2023

Abstract
In this note, we prove an optimal upper bound for the first Dirac eigenvalue of some hypersurfaces in the Euclidean space by combining a positive mass theorem and the construction of quasi-spherical metrics. As a direct consequence of this estimate, we obtain an asymptotic expansion for the first eigenvalue of the Dirac operator on large spheres in three-dimensional asymptotically flat manifolds. We also study this expansion for small geodesic spheres in a three-dimensional Riemannian manifold. We finally discuss how this method can be adapted to yield similar results in the hyperbolic space.

Key words: Dirac operator; total mean curvature; scalar curvature; mass.

pdf (397 kb)   tex (22 kb)  

References

  1. Andersson L., Dahl M., Scalar curvature rigidity for asymptotically locally hyperbolic manifolds, Ann. Global Anal. Geom. 16 (1998), 1-27, arXiv:dg-ga/9707017.
  2. Bär C., Lower eigenvalue estimates for Dirac operators, Math. Ann. 293 (1992), 39-46.
  3. Bär C., Extrinsic bounds for eigenvalues of the Dirac operator, Ann. Global Anal. Geom. 16 (1998), 573-596.
  4. Bartnik R., The mass of an asymptotically flat manifold, Comm. Pure Appl. Math. 39 (1986), 661-693.
  5. Bartnik R., Quasi-spherical metrics and prescribed scalar curvature, J. Differential Geom. 37 (1993), 31-71.
  6. Bartnik R.A., Chruściel P.T., Boundary value problems for Dirac-type equations, J. Reine Angew. Math. 579 (2005), 13-73, arXiv:math.DG/0307278.
  7. Bourguignon J.-P., Hijazi O., Milhorat J.L., Moroianu A., Moroianu S., A spinorial approach to Riemannian and conformal geometry, EMS Monogr. Math., Eur. Math. Soc. (EMS), Zürich, 2015.
  8. Bray H.L., Proof of the Riemannian Penrose inequality using the positive mass theorem, J. Differential Geom. 59 (2001), 177-267, arXiv:math.DG/9911173.
  9. Brendle S., Hung P.-K., Wang M.-T., A Minkowski inequality for hypersurfaces in the anti-de Sitter-Schwarzschild manifold, Comm. Pure Appl. Math. 69 (2016), 124-144, arXiv:1209.0669.
  10. Chruściel P., Boundary conditions at spatial infinity from a Hamiltonian point of view, in Topological Properties and Global Structure of Space-Time (Erice, 1985), NATO Adv. Sci. Inst. Ser. B Phys., Vol. 138, Plenum, New York, 1986, 49-59.
  11. Chruściel P.T., Herzlich M., The mass of asymptotically hyperbolic Riemannian manifolds, Pacific J. Math. 212 (2003), 231-264, arXiv:math.DG/0110035.
  12. Eichmair M., Miao P., Wang X., Extension of a theorem of Shi and Tam, Calc. Var. Partial Differential Equations 43 (2012), 45-56, arXiv:0911.0377.
  13. Fan X.-Q., Shi Y., Tam L.-F., Large-sphere and small-sphere limits of the Brown-York mass, Comm. Anal. Geom. 17 (2009), 37-72, arXiv:0711.2552.
  14. Friedrich Th., Der erste Eigenwert des Dirac-Operators einer kompakten, Riemannschen Mannigfaltigkeit nichtnegativer Skalarkrümmung, Math. Nachr. 97 (1980), 117-146.
  15. Ge Y., Wang G., Wu J., Hyperbolic Alexandrov-Fenchel quermassintegral inequalities II, J. Differential Geom. 98 (2014), 237-260, arXiv:1304.1417.
  16. Ginoux N., Une nouvelle estimation extrinsèque du spectre de l'opérateur de Dirac, C. R. Math. Acad. Sci. Paris 336 (2003), 829-832.
  17. Herzlich M., A Penrose-like inequality for the mass of Riemannian asymptotically flat manifolds, Comm. Math. Phys. 188 (1997), 121-133.
  18. Herzlich M., Minimal surfaces, the Dirac operator and the Penrose inequality, in Séminaire de Théorie Spectrale et Géométrie, Année 2001-2002, Sémin. Théor. Spectr. Géom., Vol. 20, Université de Grenoble I, Saint-Martin-d'Hères, 2002, 9-16.
  19. Hijazi O., A conformal lower bound for the smallest eigenvalue of the Dirac operator and Killing spinors, Comm. Math. Phys. 104 (1986), 151-162.
  20. Hijazi O., Première valeur propre de l'opérateur de Dirac et nombre de Yamabe, C. R. Acad. Sci. Paris Sér. I Math. 313 (1991), 865-868.
  21. Hijazi O., Montiel S., Raulot S., A holographic principle for the existence of imaginary Killing spinors, J. Geom. Phys. 91 (2015), 12-28, arXiv:1502.04091.
  22. Hijazi O., Montiel S., Raulot S., A positive mass theorem for asymptotically hyperbolic manifolds with inner boundary, Internat. J. Math. 26 (2015), 1550101, 17 pages.
  23. Hijazi O., Montiel S., Roldán A., Dirac operators on hypersurfaces of manifolds with negative scalar curvature, Ann. Global Anal. Geom. 23 (2003), 247-264.
  24. Hijazi O., Montiel S., Zhang X., Dirac operator on embedded hypersurfaces, Math. Res. Lett. 8 (2001), 195-208, arXiv:math.DG/0012262.
  25. Kwong K.-K., On the positivity of a quasi-local mass in general dimensions, Comm. Anal. Geom. 21 (2013), 847-871, arXiv:1207.7333.
  26. Nirenberg L., The Weyl and Minkowski problems in differential geometry in the large, Comm. Pure Appl. Math. 6 (1953), 337-394.
  27. Pogorelov A.V., Regularity of a convex surface with given Gaussian curvature, Mat. Sb. 31 (1952), 88-103.
  28. Schoen R., Yau S.-T., Proof of the positive mass theorem. II, Comm. Math. Phys. 79 (1981), 231-260.
  29. Schoen R., Yau S.-T., Positive scalar curvature and minimal hypersurface singularities, in Surveys in Differential Geometry 2019, Differential geometry, Calabi-Yau Theory, and General Relativity. Part 2, Surv. Differ. Geom., Vol. 24, Int. Press, Boston, MA, 2019, 441-480, arXiv:1704.05490.
  30. Shi Y., Tam L.-F., Positive mass theorem and the boundary behaviors of compact manifolds with nonnegative scalar curvature, J. Differential Geom. 62 (2002), 79-125, arXiv:math.DG/0301047.
  31. Shi Y., Tam L.-F., Rigidity of compact manifolds and positivity of quasi-local mass, Classical Quantum Gravity 24 (2007), 2357-2366.
  32. Wang M.T., Yau S.-T., A generalization of Liu-Yau's quasi-local mass, Comm. Anal. Geom. 15 (2007), 249-282, arXiv:math.DG/0602321.
  33. Witten E., A new proof of the positive energy theorem, Comm. Math. Phys. 80 (1981), 381-402.

Previous article  Next article  Contents of Volume 19 (2023)