The displaced disks problem via symplectic topology

TL;DR

This paper proves that small area-preserving homeomorphisms on closed surfaces cannot displace large disks if the mass flow vanishes, solving a longstanding problem in symplectic topology.

Contribution

It provides a solution to the displaced disks problem using techniques from symplectic topology, a novel approach in this context.

Findings

Small area-preserving homeomorphisms cannot displace large disks with zero mass flow

Resolution of the displaced disks problem in symplectic topology

New insights into the behavior of surface homeomorphisms

Abstract

We prove that a $C^0$--small area preserving homeomorphism of a closed surface with vanishing mass flow can not displace a topological disk of large area. This resolves the displaced disks problem posed by F. B\'eguin, S. Crovisier, and F. Le Roux.