Dirac and Plateau billiards in domains with corners

Groping our way toward a theory of singular spaces with positive scalar curvatures we look at the Dirac operator and a generalized Plateau problem in Riemannian manifolds with corners. Using these, we prove that the set of C ²-smooth Riemannian metrics g on a smooth manifold X, such that scal _g (x) ≥ κ(x), is closed under C ⁰-limits of Riemannian metrics for all continuous functions κ on X. Apart from that our progress is limited but we formulate many conjectures. All along, we emphasize geometry, rather than topology of manifolds with their scalar curvatures bounded from below.