P\'{o}lya's conjecture on $\mathbb{S}^1 \times \R$

TL;DR

This paper investigates the validity of Pólya's conjecture on the infinite cylinder 1, characterizing when isoperimetric domains like geodesic disks and cylindrical strips satisfy the conjecture and related inequalities.

Contribution

It provides a complete characterization of the conditions under which Pólya's conjecture holds for isoperimetric domains on 1, including bounds and criteria for strips and disks.

Findings

Upper bound on radius for Pólya's conjecture on geodesic disks.

Complete characterization of strip heights where the conjecture holds.

Necessary and sufficient conditions for Li-Yau inequalities on strips.

Abstract

We study the area ranges where the two possible isoperimetric domains on the infinite cylinder $\mathbb{S}^{1}\times \R$, namely, geodesic disks and cylindrical strips of the form $\mathbb{S}^1\times [0,h]$, satisfy P\'{o}lya's conjecture. In the former case, we provide an upper bound on the maximum value of the radius for which the conjecture may hold, while in the latter we fully characterise the values of $h$ for which it does hold for these strips. As a consequence, we determine a necessary and sufficient condition for the isoperimetric domain on $\mathbb{S}^{1}\times \R$ corresponding to a given area to satisfy P\'{o}lya's conjecture. In the case of the cylindrical strip, we also provide a necessary and sufficient condition for the Li-Yau inequalities to hold.