  Based on the Atiyah-Patodi-Singer index formula, we construct an obstruction
to positive scalar curvature metrics with mean convex boundaries on spin
manifolds of infinite K-area. We also characterize the extremal case. Next we
show a general deformation principle for boundary conditions of metrics with
lower scalar curvature bounds. This implies that the relaxation of boundary
conditions often induces weak homotopy equivalences of spaces of such metrics.
This can be used to refine the smoothing of codimension-one singularites a la
Miao and the deformation of boundary conditions a la Brendle-Marques-Neves,
among others. Finally, we construct compact manifolds for which the spaces of
positive scalar curvature metrics with mean convex boundaries have nontrivial
higher homotopy groups.
