Optimal Eigenvalue Estimate for the Dirac–Witten Operator on Bounded Domains with Smooth Boundary

Abstract

Eigenvalue estimate for the Dirac–Witten operator is given on bounded domains (with smooth boundary) of spacelike spin hypersurfaces satisfying the dominant energy condition, under four natural boundary conditions (MIT, APS, modified APS and chiral conditions). Roughly speaking, any eigenvalue of the Dirac–Witten operator satisfies $$\left|\lambda\right|^{2} \,\geq\, \frac{n}{n-1} \frak R_{0} ,$$ where \({\frak R_{0}}\) is the infimum of (the opposite of) the Lorentzian norm of the constraints vector. Equality cases are also investigated and lead to interesting geometric situations.