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-smooth Riemannian metrics g on a smooth manifold X, such that scalg(x) ≥ κ(x), is closed under C-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. 1 Setting the Stage. A closed subset P = P in a smooth n-manifold X is called a cornered or curvefaced polyhedral n-domain of depth d = 0,1, ..., n if the boundary of ∂P of P is decomposed into the union of a countable locally finite (e.g. finite) family of (possibly disconnected) (n − 1)-faces Qi = Qn−1 i with a distinguished set of adjacent pairs of faces (Qi,Qj), such that ● every face Qi is contained in a smooth hypersurface Yi = Y n−1 i ⊂X, where Yi is transversal to Yi for all adjacent pairs of (n − 1)-faces; ●● the boundary ∂Qi of each Qi ⊂ Yi equals the union of the intersections Qi∩Qj for all faces Qj that are adjacent to a given Qi, where the corresponding decompositions ∂Qi = ⋃j Qi ∩Qj give polyhedral (n − 1)-domain structures of depth d − 1 to all Qi. This defines the notion of a polyhedral domain structure by induction on d, where polyhedral domains of depth zero are non-empty closed subsets P ⊂ X with empty boundaries, i.e. just smooth manifolds with no extra structures and domains of depth one are those bounded by smooth hypersurfaces. A polyhedral domain P is called cosimplicial if the intersections of all ktuples of mutually non-equal (n − 1)-faces satisfy dim(Qi1 ∩Qi2 ∩ ... ∩Qik) ≤ n − k + 1. Polyhedra, Edges, Corners. Our attention will be focused on the boundaries of cornered domains that are unions of faces, ∪iQi. We call these boundaries curve-faced polyhedra or polyhedral hypersurfaces and may occasionally denote them P rather than ∂P . The codimension two faces of P that are (n-2)-dimensional intersections of faces Qi are called edges of P and the higher codimension faces are called corners. Thus, corners appear starting from d = 3, such as the ordinary corners of a cube in the 3-space.