Mathematics Colloquium, Fall 2007
SPECIAL LECTURE

Isoperimetric inequalities and the large scale geometry of metric spaces

Isoperimetric inequalities appear in many branches of mathematics and play an important role notably in geometry, analysis, geometric group theory and geometric measure theory. The purpose of this talk is to discuss two results which exhibit relationships between the growth of isoperimetric functions and the asymptotic geometry of Riemannian manifolds (and more generally of singular spaces).

In a first theorem, we show that a geodesic metric space cannot have a quadratic isoperimetric inequality for (sufficiently long) curves with isoperimetric constant strictly smaller than 1/(4pi), unless it is Gromov hyperbolic. Gromov hyperbolic spaces should be thought of as spaces of negative curvature on a large scale. Our result is optimal (with Euclidean space as a borderline case) and new even for Riemannian manifolds. It generalizes and strengthens well-known results of Gromov, Bowditch, Drutu and Papasoglu. The proof of our theorem combines ideas from metric geometry and geometric measure theory. In particular, it makes use of the powerful theory of currents in metric spaces.

Our second theorem concerns the relationship between the Euclidean rank r(X) of a proper and cocompact metric space X of non-positive curvature and higher-dimensional isoperimetric inequalities in X. The Euclidean rank is the maximal dimension of a Euclidean space isometrically contained in X. We prove that the isoperimetric filling functions I_k for k-dimensional cycles in X have Euclidean behavior when k is smaller than r(X) and sub-Euclidean behavior when k is bigger or equal to r(X). From this we can conclude that the I_k detect the Euclidean rank of X. Our result forms a first step towards a conjecture of Gromov about isoperimetric inequalities above the rank. It can moreover be used to establish higher rank analogs of results from hyperbolic geometry such as the stability of quasi-geodesics.