Generic doubling of rectangular pegs

TL;DR

This paper proves that for a generic class of smooth Jordan curves, each admits two distinct inscribed rectangles with a specific aspect ratio, using Floer theory and intersection theory, extending previous existence results.

Contribution

It provides a new Floer theoretical proof for the multiplicity of inscribed rectangles in Jordan curves, and extends the result to cyclic quadrilaterals, with detailed intersection computations.

Findings

Every smooth Jordan curve admits two geometrically distinct inscribed rectangles with a fixed aspect angle.

The proof translates the problem into intersection points of Lagrangian tori using Floer homology.

The multiplicity result is extended to cyclic quadrilaterals.

Abstract

We prove a multiplicity result for rectangular pegs that there is a generic class of smooth Jordan curves in which every curve admits two geometrically distinct similar inscribed rectangles with aspect angle in $(0,\frac{\pi}{2})$, based on the existence of rectangular pegs in any smooth Jordan curve, which is first proved by Greene and Lobb [GL21] and we give an alternative Floer theoretical proof in this paper. The key insight is that the rectangular peg problem is translated into finding intersection points of two Lagrangian tori. We present two distinct proofs for the multiplicity result: one involves Lagrangian Floer homology, and the other is differential topological in nature which employs a novel computation formula for the algebraic intersection number. Both rely crucially on certain generic geometric transversality of the two tori, and the correspondence between the intersection points and the inscribed rectangles. Moreover, such a generic doubling result can be further extended to cyclic quadrilateral pegs based on the existence result [GL23]. In Appendix B, written jointly with Urs Frauenfelder, we provide the detailed derivation of the computation formula for the algebraic intersection number.