Total mean curvature and first Dirac eigenvalue. (English) Zbl 1530.58016
Author’s 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.”
These results are interesting and useful to study the properties of Dirac operators.
These results are interesting and useful to study the properties of Dirac operators.
Reviewer: Chuan-Fu Yang (Nanjing)
MSC:
| 58J50 | Spectral problems; spectral geometry; scattering theory on manifolds |
| 58C40 | Spectral theory; eigenvalue problems on manifolds |
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