Large-volume fractional isoperimetry on products of closed manifolds and Euclidean spaces
Abstract
Let M be a closed connected Riemannian manifold and let 𝒫s be the fractional perimeter on M×ℝk defined through its heat semigroup, where k≥1 and 0<s<1. We prove that the straight cylinder M× BR, with BR⊂ℝk a ball of radius R, uniquely minimizes 𝒫s among all measurable sets of the same volume, up to translations in ℝk and null sets, whenever R√(λ1(M)) is at least an explicit threshold depending only on k and s. Here λ1(M) is the first positive eigenvalue of the nonnegative Laplace--Beltrami operator −ΔM on M. Under a larger threshold, the perimeter deficit controls both the deviation of the slice volumes from their mean and the square of the asymmetry relative to straight cylinders. For flat tori this gives large-volume uniqueness of straight cylinders for the periodic fractional perimeter, including singly periodic sets in every ambient dimension. The proof combines spectral reduction on M with rearrangement of the Euclidean slices and a one-variable comparison between an interaction gain and a concavity loss.
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BibTeX
@misc{seo2026largevolume,
title = {{Large-volume fractional isoperimetry on products of closed manifolds and Euclidean spaces}},
author = {Juneyoung Seo},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.08556},
eprint = {2610.08556},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.08556},
}
RIS
TY - UNPB
TI - Large-volume fractional isoperimetry on products of closed manifolds and Euclidean spaces
AU - Juneyoung Seo
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.08556
UR - https://arxiv.org/abs/2610.08556
AB - Let M be a closed connected Riemannian manifold and let 𝒫_s be the fractional perimeter on M×ℝ^k defined through its heat semigroup, where k≥1 and 0<s<1. We prove that the straight cylinder M× B_R, with B_R⊂ℝ^k a ball of radius R, uniquely minimizes 𝒫_s among all measurable sets of the same volume, up to translations in ℝ^k and null sets, whenever R√(λ_1(M)) is at least an explicit threshold depending only on k and s. Here λ_1(M) is the first positive eigenvalue of the nonnegative Laplace--Beltrami operator −Δ_M on M. Under a larger threshold, the perimeter deficit controls both the deviation of the slice volumes from their mean and the square of the asymmetry relative to straight cylinders. For flat tori this gives large-volume uniqueness of straight cylinders for the periodic fractional perimeter, including singly periodic sets in every ambient dimension. The proof combines spectral reduction on M with rearrangement of the Euclidean slices and a one-variable comparison between an interaction gain and a concavity loss.
ER -