A Riemannian Autocorrelation Function and its Application to Non-Local Isoperimetric Energies

2026/01/15 by Michael Bleher, Denis Brazke, Sebastian Nill · 1 voice
Mathematics · #math.AP #math.DG

paper · pdf

Abstract

We study a family of non-local isoperimetric energies E_γ,ε on the round sphere M = Sⁿ, where the non-local interaction kernel K_ε is the fundamental solution of the Helmholtz operator 1 - ε² Δ. To analyse these energies, we introduce a Riemannian autocorrelation function c_Ω associated to a measurable set Ω⊂ M, defined on any compact, connected, oriented Riemannian manifold without boundary (Mⁿ,g) of dimension n≥2. This function is intimately linked to Matheron's set covariogram from convex geometry. By establishing a characterisation of functions of bounded variation BV(M) in terms of geodesic difference quotients, we show that Ω has finite perimeter if and only if c_Ω is Lipschitz, and we relate the Lipschitz constant to the perimeter of Ω. We show that on the round sphere E_γ,ε admits a reformulation in terms of c_Ω, which allows us to compute the limit as ε → 0 in a variational sense, that is, in the framework of Γ-convergence.

Citations

• Euler's elastica functional as a large mass limit of a two-dimensional non-local isoperimetric problem
• Γ-limit for a sharp interface model related to pattern formation on biomembranes
• Large mass minimizers for isoperimetric problems with integrable nonlocal potentials
• Second-order asymptotics of the fractional perimeter as s→ 1
• A Nonlocal Isoperimetric Problem with Dipolar Repulsion
• Fractional Sobolev norms and BV functions on manifolds
• Fundamental Solutions and Gegenbauer Expansions of Helmholtz Operators in Riemannian Spaces of Constant Curvature
• Domain formation in membranes near the onset of instability
• Axisymmetric critical points of a nonlocal isoperimetric problem on the two-sphere
• Heat kernel and analysis on manifolds
• The \mathbf Γ-limit of the two-dimensional Ohta-Kawasaki energy. I. Droplet density
• Small Volume Fraction Limit of the Diblock Copolymer Problem: I. Sharp Interface Functional
• Two-Dimensional Matter: Order, Curvature and Defects

Discussions

• New preprint! "A Riemannian Autocorrelation Function and its Application to Non-Local Isoperimetric Energies" w/ Denis Brazke and Sebastian Nill. arxiv.org/abs/2601.10481 Thread on what we did (and wh [bsky, 0 points, 1 comments]

Related

• Uniqueness results for critical points of a non-local isoperimetric problem via curve shortening
• Geodesic distance Riesz energy on the sphere
• Perimeter on a manifold, with applications to partial differential equations
• On the shape factor of interaction laws for a non-local approximation of the Sobolev norm and the total variation
• On the High Energy Behavior of Nonlinear Functionals of Random Eigenfunctions on \mathbb S^d