Square pegs between two graphs

TL;DR

This paper proves the existence of inscribed squares in certain Jordan curves formed by two Lipschitz graphs with specific bounds, extending previous results and analyzing spectral invariants of Jordan Floer homology.

Contribution

It extends Tao's result by showing inscribed squares for curves with Lipschitz constant less than 1 + √2 and explores spectral invariants in Jordan Floer homology.

Findings

Existence of inscribed squares in Jordan curves formed by two Lipschitz graphs with constant less than 1 + √2

Jordan curves with Lipschitz constant 1 inscribe rectangles of all similarity classes

Spectral invariants of Jordan Floer homology change under curve perturbations

Abstract

We show that there always exists an inscribed square in a Jordan curve given as the union of two graphs of functions of Lipschitz constant less than $1 + \sqrt{2}$. 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.