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

SIGMA 19 (2023), 029, 14 pages      arXiv:2210.13037
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

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.

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