In a recent preprint ( arXiv:1811.08519 ) E. Barbosa and F. Conrado derive for manifolds with boundary topological obstructions to the existence of non-negative scalar curvature metrics with mean convex boundaries.

The boundary of a Riemannian manifold is said to be mean convex, if the mean curvature of it with respect to the outward unit normal vector field is non-negative.

Instead of writing down here the results in their full glory, let us mention only the following example: the manifold \((S^1 \times T^\circ)\# N\), where \(T^\circ\) is the 2-dimensional torus with an open disc removed and \(N\) is a closed, connected and orientable 3-dimensional manifold, do not admit Riemannian metrics with non-negative scalar curvature and mean convex boundary.