Total mean curvature and first Dirac eigenvalue

TL;DR

This paper establishes an optimal upper bound for the first Dirac eigenvalue of hypersurfaces in Euclidean space, using positive mass theorems and quasi-spherical metrics, with applications to asymptotically flat and hyperbolic spaces.

Contribution

It introduces a novel method combining positive mass theorem and quasi-spherical metrics to bound the first Dirac eigenvalue in various geometric settings.

Findings

Derived an optimal upper bound for the first Dirac eigenvalue in Euclidean hypersurfaces.

Provided asymptotic expansion for eigenvalues on large spheres in asymptotically flat manifolds.

Extended the analysis to small geodesic spheres and hyperbolic spaces.

Abstract

In this note, we prove an optimal upper bound for the first Dirac eigenvalue of some hypersurfaces in 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.