Surfaces of small diameter with large width

TL;DR

This paper constructs specific 2D surfaces with fixed small diameter but arbitrarily large dividing cycle length, demonstrating the non-existence of a universal bound relating 1-width to diameter.

Contribution

It provides a method to construct surfaces with prescribed diameter and large dividing cycle length, answering a question posed by Sabourau.

Findings

Existence of surfaces with diameter 1 and arbitrarily large 1-cycle length

No universal inequality bounds 1-width by diameter for 2D surfaces

Constructive approach to control surface geometry and topology

Abstract

Given a 2-dimensional surface M and a constant C we construct a Riemannian metric g, so that diameter diam(M,g)=1 and every 1-cycle dividing M into two regions of equal area has length >C. It follows that there exists no universal inequality bounding 1-width of M in terms of its diameter. This answers a question of Stephane Sabourau.