$L^p$ cohomology and Hodge decomposition for ALE manifolds

Abstract

We relate the dimensions of $L^p$ reduced cohomology spaces in degree k of an ALE manifold to the dimension of some spaces of decaying harmonic forms, depending both on p and on k. In this class of manifolds, this provides an extension to $p\neq 2$ of the well-known result of Hodge. In particular, we prove that for fixed $k\notin\left\{1,n-1\right\}$, the dimension of the $L^p$ reduced cohomology spaces in degree k is independent of $p\in (1,\infty)$, while for $k\in\{1,n-1\}$, the dimension jumps exactly once by a factor N-1 (N being the number of ends) when $p$ varies in $(1,\infty)$. We also prove $L^p$ Hodge decompositions for k-forms on such manifolds, for the optimal values of k and p. When these are not available, we provide a substitute (a modified Hodge decomposition).