The Hofer conjecture on embedding symplectic ellipsoids

TL;DR

This paper proves Hofer's conjecture that a four-dimensional ellipsoid symplectically embeds into another if and only if its ECH capacities are no larger, using a novel approach inspired by recent contact homology results.

Contribution

It establishes the equivalence between ellipsoidal and ball embedding problems and proves Hofer's conjecture without relying on ECH capacities directly.

Findings

Ellipsoid embedding characterized by ECH capacities

Proof of Hofer's conjecture in four dimensions

Connection between ellipsoid and ball embedding problems

Abstract

In this note we show that one open four dimensional ellipsoid embeds symplectically into another if and only the ECH capacities of the first are no larger than those of the second. This proves a conjecture due to Hofer. The argument uses the equivalence of the ellipsoidal embedding problem with a ball embedding problem that was recently established by McDuff. Its method is inspired by Hutchings' recent results on embedded contact homology (ECH) capacities but does not use them.