Remark on the Limit Case of Positive Mass Theorem for Manifolds with Inner Boundary

In [5] Herzlich proved a new positive mass theorem for Riemannian 3-manifolds $(N,g)$ whose mean curvature of the boundary allows some positivity. In this paper we study what happens to the limit case of the theorem when, at a point of the boundary, the smallest positive eigenvalue of the Dirac operator of the boundary is strictly larger than one-half of the mean curvature (in this case the mass $m(g)$ must be strictly positive). We prove that the mass is bounded from below by a positive constant $c(g),m(g)\ge c(g)$, and the equality $m(g)=c(g)$ holds only if, outside a compact set, $(N,g)$ is conformally flat and the scalar curvature vanishes. The constant $c(g)$ is uniquely determined by the metric $g$ via a Dirac-harmonic spinor.