Generic metrics and the mass endomorphism on spin 3-manifolds

Let (M, g, σ) be a closed Riemannian spin manifold of dimension n ≥ 2. We define the two quantities λ±min(M, [g], σ) := infg∈[g] |λ ± 1 (g)|Vol(M, g) , where λ+1 (g) is the smallest positive eigenvalue of the Dirac operator D and λ−1 (g) is its largest negative eigenvalue. If S is the sphere S with the standard metric, it is known that the two inequalities λ±min(M, [g], σ) ≤ λ + min(S ) hold. In the case n = 3 we show that for generic metrics one of these inequalities becomes a strict inequality.