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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 |
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Vol. 18 (2025), No. 1, 25–78
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Abstract |
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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. |
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Keywords
symplectic embeddings in four dimensions, ellipsoid
embedding capacity function, infinite staircases, continued
fractions, quantitative symplectic geometry, polydisks
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Mathematical Subject Classification
Primary: 11A55, 53-04, 53D05, 53D35, 53D42
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Supplementary materialDiscussion 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
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Authors |
| Caden Farley | |
| Department of Mathematics University of Tennessee Knoxville, TN United States |
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| Tara S. Holm | |
| Department of Mathematics Cornell University Ithaca, NY United States |
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| Nicki Magill | |
| Department of Mathematics University of California Berkeley, California United States |
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| Jemma Schroder | |
| Department of Mathematics The University of Texas at Austin Austin, TX United States |
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| Zichen Wang | |
| Electrical Engineering and Computer
Science Department University of Michigan Ann Arbor, MI United States |
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| Morgan Weiler | |
| Department of Mathematics University of California Riverside, CA United States |
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| Elizaveta Zabelina | |
| Department of Mathematics Cornell University Ithaca, NY United States |
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| © 2025 MSP (Mathematical Sciences Publishers). |
