Optimal eigenvalue estimate for the Dirac-Witten operator on bounded domains with smooth boundary

TL;DR

This paper provides eigenvalue estimates for the Dirac-Witten operator on bounded domains with smooth boundaries, extending Friedrich's inequality and analyzing limiting cases under various boundary conditions.

Contribution

It generalizes Friedrich's inequality to the Dirac-Witten operator on bounded domains with different boundary conditions.

Findings

Eigenvalue bounds are established under four boundary conditions.

The results extend classical inequalities for the Dirac operator.

Limiting cases are thoroughly investigated.

Abstract

Eigenvalue estimate for the Dirac-Witten operator is given on bounded domains (with smooth boundary) of spacelike hypersurfaces satisfying the dominant energy condition, under four natural boundary conditions (MIT, APS, modified APS, and chiral conditions). This result is a generalisation of Friedrich's inequality for the usual Dirac operator. The limiting cases are also investigated.