Uryson width and volume

Abstract: The Uryson width of an $n$-manifold gives a way to describe how close is the manifold to an $n-1$ dimensional complex. It turns out that this is a useful tool to approach several geometric problems.

In this talk we will give a brief survey of some questions in `curvature free' geometry and sketch a novel approach to the classical systolic inequality of Gromov. Our approach follows up recent work of Guth relating Uryson width and local volume growth. For example we deduce also the following result of Guth: there is an $\epsilon _n>0$ such that for any $R>0$ and any compact aspherical $n$-manifold $M$ there is a ball $B(R)$ of radius $R$ in the universal cover of $M$ such that $vol(B(R))\geq \epsilon _n R^n$.

differential geometrygeometric topologymetric geometry

Audience: researchers in the topic

Topology and geometry: extremal and typical

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