# Growth of a primitive of a differential form

Bulletin de la Société Mathématique de France (2001)

- Volume: 129, Issue: 2, page 159-168
- ISSN: 0037-9484

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topSikorav, Jean-Claude. "Growth of a primitive of a differential form." Bulletin de la Société Mathématique de France 129.2 (2001): 159-168. <http://eudml.org/doc/272316>.

@article{Sikorav2001,

abstract = {For an exact differential form on a Riemannian manifold to have a primitive bounded by a given function $f$, by Stokes it has to satisfy some weighted isoperimetric inequality. We show the converse up to some constants if $M$ has bounded geometry. For a volume form, it suffices to have the inequality ($|\Omega |\le \int _\{\partial \Omega \}\,f\{\rm d\}\sigma $ for every compact domain $\Omega \subset M$). This implies in particular the “well-known” result that if $M$ is the universal covering of a compact Riemannian manifold with non-amenable fundamental group, then the volume form has a bounded primitive. Thanks to a recent theorem of A.$\dot\{\rm Z\}$uk, we also obtain that if the fundamental group is infinite, the volume form always has a primitive with linear growth.},

author = {Sikorav, Jean-Claude},

journal = {Bulletin de la Société Mathématique de France},

keywords = {exact differential form; isoperimetric inequalities; bounded geometry},

language = {eng},

number = {2},

pages = {159-168},

publisher = {Société mathématique de France},

title = {Growth of a primitive of a differential form},

url = {http://eudml.org/doc/272316},

volume = {129},

year = {2001},

}

TY - JOUR

AU - Sikorav, Jean-Claude

TI - Growth of a primitive of a differential form

JO - Bulletin de la Société Mathématique de France

PY - 2001

PB - Société mathématique de France

VL - 129

IS - 2

SP - 159

EP - 168

AB - For an exact differential form on a Riemannian manifold to have a primitive bounded by a given function $f$, by Stokes it has to satisfy some weighted isoperimetric inequality. We show the converse up to some constants if $M$ has bounded geometry. For a volume form, it suffices to have the inequality ($|\Omega |\le \int _{\partial \Omega }\,f{\rm d}\sigma $ for every compact domain $\Omega \subset M$). This implies in particular the “well-known” result that if $M$ is the universal covering of a compact Riemannian manifold with non-amenable fundamental group, then the volume form has a bounded primitive. Thanks to a recent theorem of A.$\dot{\rm Z}$uk, we also obtain that if the fundamental group is infinite, the volume form always has a primitive with linear growth.

LA - eng

KW - exact differential form; isoperimetric inequalities; bounded geometry

UR - http://eudml.org/doc/272316

ER -

## References

top- [1] M. Gromov – Structures métriques pour les variétés riemanniennes, Textes Mathématiques [Mathematical Texts], vol. 1, CEDIC, Paris, 1981, Edited by J. Lafontaine and P. Pansu. Zbl0509.53034MR682063

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