Uryson Width and Volume

Panos Papasoglu

doi:10.1007/s00039-020-00533-5

Abstract

We give a short proof of a theorem of Guth relating volume of balls and Uryson width. The same approach applies to Hausdorff content implying a recent result of Liokumovich–Lishak–Nabutovsky–Rotman. We show also that for any C>0 there is a Riemannian metric g on a 3-sphere such that {hbox {vol}}(S^3,g)=1 and for any map f:S^3rightarrow {mathbb {R}}^2 there is some xin {mathbb {R}}^2 for which text {diam}(f^{-1}(x))>C, answering a question of Guth.

Highlights

The Uryson width is a notion of topological dimension theory that was brought to the realm of Riemannian Geometry by Gromov [Gro[83], Gro[88], Gro86]
We recall the precise definition: if X is a metric space we say that X has q-Uryson width ≤ W if there exists a q-dimensional simplicial complex Y and a continuous map π : X → Y such that every fiber π−1(y) has diameter ≤ W
Guth in [Gut17] discusses the relationship between classical topological dimension theory and the quantitative version of this theory for manifolds. In this spirit we show the following quantitative characterization of small Uryson width that resembles the inductive definition of classical topological dimension

Summary

Introduction

The Uryson width is a notion of topological dimension theory that was brought to the realm of Riemannian Geometry by Gromov [Gro[83], Gro[88], Gro86]. We recall the precise definition: if X is a metric space we say that X has q-Uryson width ≤ W if there exists a q-dimensional simplicial complex Y and a continuous map π : X → Y such that every fiber π−1(y) has diameter ≤ W. Guth conjectured something stronger that applies to general metric spaces and uses Hausdorff content instead of volume. Guth in [Gut17] discusses the relationship between classical topological dimension theory and the quantitative version of this theory for manifolds In this spirit we show the following quantitative characterization of small Uryson width that resembles the inductive definition of classical topological dimension. Using the co-area inequality (see Lemma 2.5) we show that there is a thickened grid that locally has volume growth much smaller than r2 so by induction it admits a map f to a 1-dimensional complex Σ with small fibers. I thank the referees for their constructive comments that made this paper more readable

Uryson Width of Compact Metric Spaces

The General Case and a Refinement

An Example