Klaus Kröncke: Lp cohomology and Hodge decomposition for ALE manifolds

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 \(k\notin\left\{1,n-1\right\}\), 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). This is joint work with Baptiste Devyver.