Sobolev inequalities for differential forms and $L_{q,p}$-cohomology

Marc Troyanov
2006
Article

Résumé

We study the relation between Sobolev inequalities for differential forms on a Riemannian manifold $(M,g)$ and the $L_{q,p}$ -cohomology of that manifold. The $L_{q,p}$ -cohomology of $(M,g)$ is defined to be the quotient of the space of closed differential forms in $L^p(M)$ modulo the exact forms which are exterior differentials of forms in $L^q(M)$ .

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https://infoscience.epfl.ch/record/114957?ln=fr

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Marc Troyanov
2006
Article

Résumé

Official source

https://infoscience.epfl.ch/record/114957?ln=fr

À propos de ce résultat

Concepts associés (9)

Concepts associés

Chargement

Publications associées (2)

Publications associées

Chargement

$L_{q,p}$-cohomology of Riemannian Manifolds with Negative Curvature

Marc Troyanov

We consider the L_{q,p}-cohomology of a complete simply connected Riemannian manifold (M,g) with pinched negative curvature. The connection between the L_{q,p}-cohomology of (M,g) and Sobolev inequalities for differential forms on (M,g) was established by the authors in a previous publications.

Springer2009

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Lq,p-cohomology of Riemannian manifolds and simplicial complexes of bounded geometry

Stephen Ducret

The Lq,p-cohomology of a Riemannian manifold (M, g) is defined to be the quotient of closed Lp-forms, modulo the exact forms which are derivatives of Lq-forms, where the measure considered comes from the Riemannian structure. The Lq,p-cohomology of a simplicial complex K is defined to be the quotient of p-summable cocycles of K, modulo the coboundaries of q-summable cocycles. We introduce those two notions together with a variant for coarse cohomology on graphs, and we establish their main properties. We define the categories we work on, i.e. manifolds and simplicial complexes of bounded geometry, and we show how cohomology classes can be represented by smooth forms. The first result of the thesis is a de Rham type theorem: we prove that for an orientable, complete and (non compact) Riemannian manifold with bounded geometry (M, g) together with a triangulation K with bounded geometry, the Lq,p-cohomology of the manifold coincides with the Lq,p-cohomology of the triangulation. This is a generalization of an earlier result from Gol'dshtein, Kuz'minov and Shvedov. The second result is a quasi-isometry invariance one: we prove how this de Rham type isomorphism together with a result in coarse cohomology induces the fact that the Lq,p-cohomology of a Riemannian manifold depends only on its quasi-invariance class. This result was proved in the q = p case by Elek. We establish some consequences, such as monocity results for Lq,p-cohomology, and the quasi-isometry invariance of the existence of Sobolev inequalities.

EPFL2009

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