Diastolic and isoperimetric inequalities on surfaces

Balacheff, Florent; Sabourau, Stéphane

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 43 (2010) no. 4, pp. 579-605.

Nous démontrons une inégalité universelle entre la diastole, définie par un procédé de minimax sur l’espace des $1$-cycles, et l’aire d’une surface riemannienne fermée. De manière informelle, nous prouvons que toute surface riemannienne fermée peut être balayée par une famille de multi-lacets dont les longueurs sont contrôlées par l’aire de la surface. Cette inégalité diastolique, qui repose sur une majoration de la constante de Cheeger, fournit en particulier un procédé effectif pour trouver de courtes géodésiques fermées sur une $2$-sphère. Nous déduisons que toute surface riemannienne peut être décomposée en deux domaines de même aire dont la longueur du bord commun est majorée à l’aide de l’aire de la surface. Nous comparons également divers invariants riemanniens sur la $2$-sphère afin de souligner le rôle spécial joué par la diastole.

We prove a universal inequality between the diastole, defined using a minimax process on the one-cycle space, and the area of closed Riemannian surfaces. Roughly speaking, we show that any closed Riemannian surface can be swept out by a family of multi-loops whose lengths are bounded in terms of the area of the surface. This diastolic inequality, which relies on an upper bound on Cheeger's constant, yields an effective process to find short closed geodesics on the two-sphere, for instance. We deduce that every Riemannian surface can be decomposed into two domains with the same area such that the length of their boundary is bounded from above in terms of the area of the surface. We also compare various Riemannian invariants on the two-sphere to underline the special role played by the diastole.

Classification : 53C23, 53C20, 58E10
Keywords: Cheeger constant, closed geodesics, curvature-free inequalities, diastole, isoperimetric inequalities, one-cycles
Mot clés : constante de Cheeger, géodésiques fermées, inégalités sans courbure, diastole, inégalités isopérimétriques, $1$-cycles

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