Sobolev inequalities for differential forms andL q,p -cohomology

Gol’dshtein, Vladimir; Troyanov, Marc

doi:10.1007/bf02922133

Cited by 55 publications

(76 citation statements)

References 13 publications

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“…The paper [11] contains some foundational material and shows how Sobolev inequalities for differential forms can be interpreted in the framework of L q,p -cohomology , the paper [12] gives some applications to quasi-conformal geometry and [13] relates the L q,p -cohomology to more general classes of mappings. The paper [13] contains some computations for negatively curved Riemannian manifolds and the papers [22,24] study the relation between the L q,p -cohomology of a manifold and the L pHodge decomposition on that manifold.…”

Section: Introduction and Statement Of The Resultsmentioning

confidence: 99%

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The Hölder-Poincaré duality for L q,p -cohomology

Gol’dshtein

Troyanov

2011

Ann Glob Anal Geom

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Section: Introduction and Statement Of The Resultsmentioning

confidence: 99%

“…This result is Proposition 8.4 from [11], an example of its usefulness can be found in that paper where it is used to prove the non vanishing of the L q,p -cohomology of the hyperbolic plane. We give a short proof.…”

Section: The Conformal Cohomologymentioning

confidence: 91%

The Hölder-Poincaré duality for L q,p -cohomology

Gol’dshtein

Troyanov

2011

Ann Glob Anal Geom

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“…It follows from the results of [8] that, under suitable assumptions on p, q, the L q,p -cohomology of a Riemannian manifold can be expressed in terms of smooth forms.…”

Section: Qp -Cohomology and Smooth Formsmentioning

confidence: 99%

Reduced L_{q,p}-Cohomology of Some Twisted Products

Gol’dshtein¹,

Kopylov²

2016

Ann. math. Blaise Pascal

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Abstract. Vanishing results for reduced Lp,q-cohomology are established in the case of twisted products, which are a generalization of warped products. Only the case q ≤ p is considered. This is an extension of some results by Gol ′ dshtein, Kuz ′ minov and Shvedov about the Lp-cohomology of warped cylinders. One of the main observations is the vanishing of the "middledimensional" cohomology for a large class of manifolds.Mathematics Subject Classification. 58A10, 58A12.

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“…Ces inégalités sont apparues sous la plume de V. Gol'dshtein et M. Troyanov [GT06], et certaines des constantes qu'elles font intervenir sont des invariants conformes. Nous allons ici en donner une version « à poids » qui permettra l'application au laplacien de Witten.…”

Section: Minoration Du Spectre Dans Une Classe Conforme Pondérée Et Iunclassified

Sur la multiplicité des valeurs propres du Laplacien de Witten

Jammes¹

2012

Trans. Amer. Math. Soc.

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Résumé.-Sur toute variété compacte de dimension supérieure ou égale à 4, on prescrit le volume et le début du spectre du laplacien de Witten agissant sur les p-formes différentielles pour 0 < p < n. En particulier, on prescrit la multiplicité des premières valeurs propres. Sur les variétés de dimension 3, on construit des exemples de première valeur propre multiple pour les 1-formes, dont la multiplicité dépend du genre maximal des surfaces immergées dont toute la 1-cohomologie est induite par la cohomologie de la variété. En particulier, cette multiplicité est au moins égale à 3.Mots-clefs : laplacien de Witten, formes différentielles, multiplicité de valeurs propres.Abstract.-On any compact manifold of dimension greater than 4, we prescribe the volume and any finite part of the spectrum of the Witten Laplacian acting on p-form for 0 < p < n. In particular, we prescribe the multiplicity of the first eigenvalues. On 3-dimensional manifolds, we give examples of multiple first eigenvalue for 1-forms, whose multiplicity depends on the maximal genus of embedded surfaces all of whose 1-cohomology is induced by the cohomology of the manifold. In particular, this multiplicity is at least 3.

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Sobolev inequalities for differential forms andL q,p -cohomology

Cited by 55 publications

References 13 publications

The Hölder-Poincaré duality for L q,p -cohomology

The Hölder-Poincaré duality for L q,p -cohomology

Reduced L_{q,p}-Cohomology of Some Twisted Products

Sur la multiplicité des valeurs propres du Laplacien de Witten

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