Square pegs between two graphs

Greene, Joshua Evan; Lobb, Andrew

Abstract

We show that there exists an inscribed square in a Jordan curve given as the union of two graphs of functions of Lipschitz constant less than 1+ √ 2. We are motivated by Tao's result that there exists such a square in the case of Lipschitz constant less than 1. In the case of Lipschitz constant 1, we show that the Jordan curve inscribes rectangles of every similarity class. Our approach involves analysing the change in the spectral invariants of the Jordan Floer homology under perturbations of the Jordan curve.

Citation

Greene, J. E., & Lobb, A. (in press). Square pegs between two graphs. Commentarii Mathematici Helvetici: A Journal of the Swiss Mathematical Society,

• Journal Article Type: Article
• Acceptance Date: Mar 16, 2025
• Deposit Date: Apr 2, 2025
• Journal: Commentarii Mathematici Helvetici: A Journal of the Swiss Mathematical Society
• Print ISSN: 0010-2571
• Electronic ISSN: 1420-8946
• Publisher: EMS Press
• Peer Reviewed: Peer Reviewed
• Public URL: https://durham-repository.worktribe.com/output/3773700
• Publisher URL: https://ems.press/journals/cmh
• Related Public URLs: https://arxiv.org/abs/2407.07798