Riemannian manifolds not quasi-isometric to leaves in codimension one foliations

Schweitzer, Paul A.

Annales de l'Institut Fourier, Tome 61 (2011) no. 4, pp. 1599-1631.

Chaque variété ouverte ${\displaystyle L}$ de dimension plus grande que 1 possède des métriques Riemanniennes complètes g avec géométrie bornée telles que ${\displaystyle (L,g)}$ n’est pas quasi-isométrique à une feuille d’un feuilletage de codimension un d’une variété fermée. Donc il n’y a pas de conditions sur la géométrie locale de ${\displaystyle (L,g)}$ qui suffisent pour qu’elle soit quasi-isométrique à une feuille de tel feuilletage. Nous introduisons la «  propriété d’homologie bornée  », une propriété semi-locale de ${\displaystyle (L,g)}$ qui est nécessaire pour qu’elle puisse être feuille d’un feuilletage de codimension 1 d’une variété compacte, à une quasi-isométrie près. Une étape essentielle de la démonstration utilise une généralisation partielle du théorème de la feuille fermée de Novikov aux dimensions plus grandes.

Every open manifold $L$ of dimension greater than one has complete Riemannian metrics $g$ with bounded geometry such that $(L,g)$ is not quasi-isometric to a leaf of a codimension one foliation of a closed manifold. Hence no conditions on the local geometry of $(L,g)$ suffice to make it quasi-isometric to a leaf of such a foliation. We introduce the ‘bounded homology property’, a semi-local property of $(L,g)$ that is necessary for it to be a leaf in a compact manifold in codimension one, up to quasi-isometry. An essential step involves a partial generalization of the Novikov closed leaf theorem to higher dimensions.

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