Cohomologie $L^{p}$ et formes harmoniques

TL;DR

This paper investigates the conditions under which the Hodge-de Rham decomposition for $L^r$-forms holds on Riemannian manifolds with certain spectral and geometric properties, focusing on the range of $p$ between 1 and 2.

Contribution

It establishes new criteria linking spectral properties of the Laplacian and geometric conditions to the validity of $L^p$ Hodge decompositions.

Findings

Existence of $p$ in (1,2) for which the Hodge decomposition holds.

Conditions on the spectrum of the Laplacian determine the decomposition's validity.

The geometric boundedness of the universal cover influences the $L^p$ cohomology.

Abstract

We show that a if a Riemannian manifold admits a universal cover with bounded geometry and if 0 does not belong to the spectrum or is an isolated point in the spectrum of the Laplacian on $\ell$-forms, then there exists $1<p<2$ such that for all $p<r<p^{\prime}$ the Hodge - de Rham decomposition for $L^{r}$-forms holds ($p^{\prime}$ denotes the conjugate of $p$).