Uryson width and volume

TL;DR

This paper provides a simplified proof of Guth's theorem linking volume and Uryson width, extends the approach to Hausdorff content, and constructs a 3-sphere metric with specific diameter properties answering Guth's question.

Contribution

It offers a concise proof of a key theorem, applies the method to Hausdorff content, and constructs a Riemannian metric on the 3-sphere with prescribed diameter properties.

Findings

Simplified proof of Guth's volume-Uryson width theorem

Extension of approach to Hausdorff content and recent results

Construction of a 3-sphere metric with unbounded inverse image diameter

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 $\text{vol}(S^3,g)=1$ and for any map $f:S^3\to \mathbb{R}^2$ there is some $x\in \mathbb{R}^2$ for which $\text{diam}(f^{-1}(x))>C$-answering a question of Guth.