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Four-periodic infinite staircases for four-dimensional polydisks

Caden Farley, Tara S. Holm, Nicki Magill, Jemma Schroder, Zichen Wang, Morgan Weiler and Elizaveta Zabelina

Vol. 18 (2025), No. 1, 25–78
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 (2019). Our work computes the function on infinitely many intervals and thereby indicates a method of proof for a conjecture of Usher.

Keywords
symplectic embeddings in four dimensions, ellipsoid embedding capacity function, infinite staircases, continued fractions, quantitative symplectic geometry, polydisks
Mathematical Subject Classification
Primary: 11A55, 53-04, 53D05, 53D35, 53D42
Supplementary material

Discussion and partial listing of the Python code for exploring ATFs

Milestones
Received: 5 December 2022
Revised: 8 August 2023
Accepted: 12 August 2023
Published: 24 January 2025

Communicated by Frank Morgan
Authors
Caden Farley
Department of Mathematics
University of Tennessee
Knoxville, TN
United States
Tara S. Holm
Department of Mathematics
Cornell University
Ithaca, NY
United States
Nicki Magill
Department of Mathematics
University of California
Berkeley, California
United States
Jemma Schroder
Department of Mathematics
The University of Texas at Austin
Austin, TX
United States
Zichen Wang
Electrical Engineering and Computer Science Department
University of Michigan
Ann Arbor, MI
United States
Morgan Weiler
Department of Mathematics
University of California
Riverside, CA
United States
Elizaveta Zabelina
Department of Mathematics
Cornell University
Ithaca, NY
United States