  We establish new mean curvature rigidity theorems of spin fill-ins with
non-negative scalar curvature using two different spinorial techniques. Our
results address two questions by Miao and Gromov, respectively. The first
technique is based on extending boundary spinors satisfying a generalized
eigenvalue equation via the Fredholm alternative for an APS boundary value
problem, while the second is a comparison result in the spirit of Llarull and
Lott using index theory. We also show that the latter implies a new Witten-type
integral inequality for the mass of an asymptotically Schwarzschild manifold
which holds even when the scalar curvature is not assumed to be non-negative.
