$L^p$-cohomology, heat semigroup and stratified spaces

Let $(M,g)$ be an incomplete Riemannian manifold of finite volume and let $2\le p<\mathrm{\infty }$. In the first part of this paper we prove that under certain assumptions the inclusion of the space of $L^p$-differential forms into that of $L^2$-differential forms gives rise to an injective/surjective map between the corresponding $L^p$ and $L^2$ cohomology groups. Then in the second part we provide various applications of these results to the curvature and the intersection cohomology of compact Thom-Mather stratified pseudomanifolds and complex projective varieties with only isolated singularities.