Hodge theory and cohomology with compact supports
Edward L. Bueler and Igor Prokhorenkov
This paper constructs a Hodge theory of noncompact topologically tame manifolds $M$. The main result is an isomorphism between the de~Rham cohomology with compact supports of $M$ and the kernel of the Hodge–Witten–Bismut Laplacian $\triangle_\mu$ associated to a measure $d\mu$ which has sufficiently rapid growth at infinity on $M$. This follows from the construction of a space of forms associated to $\triangle_\mu$ which satisfy an "extension by zero" property. The "extension by zero" property is proved for manifolds with cylindrical ends possessing Gaussian growth measures.