Total Mean Curvature and First Dirac Eigenvalue

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.