Periodic Inscription of Isosceles Trapezoids

TL;DR

The work addresses a periodic peg-type problem by inscribing isosceles trapezoids into a pair of disjoint periodic plane curves. It reformulates balanced inscriptions as a Lagrangian intersection problem in a Weinstein-type symplectic manifold and proves nontrivial Lagrangian Floer homology for the associated noncompact Lagrangian cylinders, guaranteeing intersections and hence balanced inscriptions. The continuous case follows from a smooth approximation via a convergence argument, yielding at least two distinct balanced inscriptions for generic smooth pairs; a rectangle corollary arises from specializing the trapezoid parameters. The results generalize square and rectangle inscriptions for periodic curves and connect discrete geometric configurations with symplectic topology methods, highlighting a novel variant of the Toeplitz square peg problem.

Abstract

We prove that a pair of continuous disjoint periodic curves in $\mathbb{C}$ inscribes an isosceles trapezoid with any similarity type. The case of smooth curves can be identified with a Lagrangian intersection problem for a pair of Lagrangian cylinders in $\mathbb{R}\times S^1\times\mathbb{C}$, and the continuous case follows from the smooth one by a standard convergence argument.

Figures (1)

• Figure 1: An annotated isosceles trapezoid $Q$.

Theorems & Definitions (19)

• Definition 1.1
• Theorem 1.2
• Corollary 1.3
• Conjecture 1.4
• Remark
• Lemma 2.1
• Corollary 2.2
• proof
• Lemma 2.3
• proof