Pólya's conjecture on \mathbbS¹ × \R

2025/06/04 by Pedro Freitas, Rui Wang, Freitas, Pedro +1
Mathematics · #Analytic and geometric function theory #FOS: Mathematics #Geometric Analysis and Curvature Flows #Holomorphic and Operator Theory #Spectral Theory (math.SP)

paper · pdf · doi:10.48550/arxiv.2506.04341

openalex publication_date 2025/06/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the area ranges where the two possible isoperimetric domains on the infinite cylinder \mathbbS¹× \R, namely, geodesic disks and cylindrical strips of the form \mathbbS¹× [0,h], satisfy Pólya'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 \mathbbS¹× \R corresponding to a given area to satisfy Pólya's conjecture. In the case of the cylindrical strip, we also provide a necessary and sufficient condition for the Li-Yau inequalities to hold.

Citations

• Pólya's conjecture for thin products
• Semiclassical estimates for eigenvalue means of Laplacians on spheres
• Families of non-tiling domains satisfying Pólya's conjecture
• Optimal unions of scaled copies of domains and Pólya's conjecture

Related

• Families of non-tiling domains satisfying Pólya's conjecture
• Pólya-type inequalities on spheres and hemispheres
• Improvement of Pólya's conjecture for balls and cylinders
• Finite variations on the isoperimetric problem
• A note on the convex body isoperimetric conjecture in the plane