|
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
- Andersson L., Dahl M., Scalar curvature rigidity for asymptotically locally hyperbolic manifolds, Ann. Global Anal. Geom. 16 (1998), 1-27, arXiv:dg-ga/9707017.
- Bär C., Lower eigenvalue estimates for Dirac operators, Math. Ann. 293 (1992), 39-46.
- Bär C., Extrinsic bounds for eigenvalues of the Dirac operator, Ann. Global Anal. Geom. 16 (1998), 573-596.
- Bartnik R., The mass of an asymptotically flat manifold, Comm. Pure Appl. Math. 39 (1986), 661-693.
- Bartnik R., Quasi-spherical metrics and prescribed scalar curvature, J. Differential Geom. 37 (1993), 31-71.
- 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.
- 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.
- 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.
- 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.
- 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.
- Chruściel P.T., Herzlich M., The mass of asymptotically hyperbolic Riemannian manifolds, Pacific J. Math. 212 (2003), 231-264, arXiv:math.DG/0110035.
- 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.
- 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.
- Friedrich Th., Der erste Eigenwert des Dirac-Operators einer kompakten, Riemannschen Mannigfaltigkeit nichtnegativer Skalarkrümmung, Math. Nachr. 97 (1980), 117-146.
- Ge Y., Wang G., Wu J., Hyperbolic Alexandrov-Fenchel quermassintegral inequalities II, J. Differential Geom. 98 (2014), 237-260, arXiv:1304.1417.
- Ginoux N., Une nouvelle estimation extrinsèque du spectre de l'opérateur de Dirac, C. R. Math. Acad. Sci. Paris 336 (2003), 829-832.
- Herzlich M., A Penrose-like inequality for the mass of Riemannian asymptotically flat manifolds, Comm. Math. Phys. 188 (1997), 121-133.
- 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.
- Hijazi O., A conformal lower bound for the smallest eigenvalue of the Dirac operator and Killing spinors, Comm. Math. Phys. 104 (1986), 151-162.
- 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.
- 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.
- 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.
- Hijazi O., Montiel S., Roldán A., Dirac operators on hypersurfaces of manifolds with negative scalar curvature, Ann. Global Anal. Geom. 23 (2003), 247-264.
- Hijazi O., Montiel S., Zhang X., Dirac operator on embedded hypersurfaces, Math. Res. Lett. 8 (2001), 195-208, arXiv:math.DG/0012262.
- Kwong K.-K., On the positivity of a quasi-local mass in general dimensions, Comm. Anal. Geom. 21 (2013), 847-871, arXiv:1207.7333.
- Nirenberg L., The Weyl and Minkowski problems in differential geometry in the large, Comm. Pure Appl. Math. 6 (1953), 337-394.
- Pogorelov A.V., Regularity of a convex surface with given Gaussian curvature, Mat. Sb. 31 (1952), 88-103.
- Schoen R., Yau S.-T., Proof of the positive mass theorem. II, Comm. Math. Phys. 79 (1981), 231-260.
- 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.
- 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.
- Shi Y., Tam L.-F., Rigidity of compact manifolds and positivity of quasi-local mass, Classical Quantum Gravity 24 (2007), 2357-2366.
- 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.
- Witten E., A new proof of the positive energy theorem, Comm. Math. Phys. 80 (1981), 381-402.
|
|