Contracting thin disks

TL;DR

This paper proves that a Riemannian 2-disc with small area relative to its diameter can be contracted through a homotopy with controlled intermediate curve lengths, extending understanding of geometric filling properties.

Contribution

It provides a new bound on the homotopy filling of Riemannian 2-discs with small area, answering a question posed by Liokumovich-Nabutovsky-Rotman.

Findings

Homotopy length bounds depend on boundary length, diameter, and area.

Small-area discs can be contracted with controlled intermediate curve lengths.

The result extends previous geometric filling bounds.

Abstract

We answer a question of Liokumovich-Nabutovsky-Rotman showing that if D is a Riemannian 2-disc with boundary length L, diameter d and area A << d then D can be filled by a homotopy where the lengths of the intermediate curves are bounded by $L+2d+O(\sqrt A)$.