Large-volume fractional isoperimetry on products of closed manifolds and Euclidean spaces

Abstract

Let $M$ be a closed connected Riemannian manifold and let $\mathcal P_s$ be the fractional perimeter on $M\times\mathbb R^k$ defined through its heat semigroup, where $k\ge1$ and $0<s<1$. We prove that the straight cylinder $M\times B_R$, with $B_R\subset\mathbb R^k$ a ball of radius $R$, uniquely minimizes $\mathcal P_s$ among all measurable sets of the same volume, up to translations in $\mathbb R^k$ and null sets, whenever $R\sqrt{λ_1(M)}$ 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.