Four-periodic infinite staircases for four-dimensional polydisks

TL;DR

This paper identifies a new infinite staircase in the ellipsoid embedding function for four-dimensional polydisks, extending previous work and providing a method to analyze these functions on multiple intervals.

Contribution

It extends the understanding of infinite staircases in symplectic embedding functions to four-dimensional polydisks, building on Usher's earlier results.

Findings

Identified a new infinite staircase for four-dimensional polydisks.

Computed the embedding function on infinitely many intervals.

Suggested a proof method for Usher's conjecture.

Abstract

The ellipsoid embedding function of a symplectic four-manifold measures the amount by which its symplectic form must be scaled in order for it to admit an embedding of an ellipsoid of varying eccentricity. This function generalizes the Gromov width and ball packing numbers. In the one continuous family of symplectic four-manifolds that has been analyzed, one-point blowups of the complex projective plane, there is an open dense set of symplectic forms whose ellipsoid embedding functions are completely described by finitely many obstructions, while there is simultaneously a Cantor set of symplectic forms for which an infinite number of obstructions are needed. In the latter case, we say that the embedding function has an infinite staircase. In this paper we identify a new infinite staircase when the target is a four-dimensional polydisk, extending a countable family identified by Usher in 2019. Our work computes the function on infinitely many intervals and thereby indicates a method of proof for a conjecture of Usher.