HODGE THEORY AND COHOMOLOGY WITH COMPACT SUPPORTS

EDWARD L. BUELER IGOR PROKHORENKOV

Abstract. 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 △_{μ} associated to a measure dμ which has sufficiently rapid growth at infinity on M. This follows from the construction of a space of forms associated to △_{μ} which satisfy an “extension by zero” property. The “extension by zero” property is proved for manifolds with cylindrical ends possessing gaussian growth measures.