Growth of a primitive of a differential form

Sikorav, 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 -