On the L 2-Poincaré duality for incomplete Riemannian manifolds: A general construction with applications

Let (M,g) be an open, oriented and incomplete Riemannian manifold of dimension m. Under some general conditions we show the existence of a Hilbert complex (L2ωi(M,g),d,i) such that its cohomology groups, labeled with H2,i(M,g), satisfy the following properties: H2,i(M,g) =ker(dmax,i)/im(dmin,i) H2,i(M,g)=H 2,m-i(M,g) (Poincaré duality holds) There exists a well-defined and nondegenerate pairing: H2,i(M,g) × H 2,m-i(M,g), M If (L2ωi(M,g),d,i) is a Fredholm complex, then every closed extension of the de Rham complex (ωci(M),d i) is a Fredholm complex and, for each i = 0,..,m, the quotient (dmax,i)/(dmin,i) is a finite dimensional vector space.