Uryson width and volume

2019/09/09 by Papasoglu, Panos
#Differential Geometry (math.DG) #FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.1909.03738

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 vol(S³,g)=1 and for any map f:S³→ ℝ² there is some x∈ ℝ² for which diam(f^-1(x))>C-answering a question of Guth.

Related

• Volumes of balls in Riemannian manifolds and Uryson width
• Growth in the universal cover under large simplicial volume
• The Diffeomorphism Type of Manifolds with Almost Maximal Volume
• The volume and lengths on a three sphere
• A genus-zero surface with bounded curvature enclosing less volume than the unit sphere