Remark on the limit case of positive mass theorem for manifolds with inner boundary

In (Comm. Math. Phys. 188 (1997) 121–133) 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)(greater than)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 Diracharmonic spinor.