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

Michael Bleher, Denis Brazke, Sebastian Nill

TL;DR

This work extends non-local isoperimetric energies to curved geometries by introducing a Riemannian autocorrelation function $c_\Omega$ and characterising finite perimeter via a BV framework expressed through geodesic difference quotients. The approach replaces Euclidean translations with geodesic flows, leveraging Liouville invariance to relate the non-local term to $c_\Omega$, and proving that $c_\Omega$ is Lipschitz if and only if $\Omega$ has finite perimeter, with $c'_\Omega(0)$ proportional to the perimeter. On the round sphere, the authors prove a Gamma-convergence result as $\varepsilon\to0$ for $0<\gamma<\gamma_{\mathrm{crit}}$, with $\gamma_{\mathrm{crit}}=1$, and show the energy localises to a multiple of the perimeter, precluding fine-scale pattern formation in the subcritical regime. Overall, the paper provides a geometry-aware framework for analyzing pattern formation on curved membranes by linking BV theory, geodesic dynamics, and non-local interactions via the Riemannian autocorrelation function.

Abstract

We study a family of non-local isoperimetric energies $E_{γ,\varepsilon}$ on the round sphere $M = S^n$, where the non-local interaction kernel $K_\varepsilon$ is the fundamental solution of the Helmholtz operator $1 - \varepsilon^2 Δ$. To analyse these energies, we introduce a Riemannian autocorrelation function $c_Ω$ associated to a measurable set $Ω\subset M$, defined on any compact, connected, oriented Riemannian manifold without boundary $(M^n,g)$ of dimension $n\ge2$. 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 $E_{γ,\varepsilon}$ admits a reformulation in terms of $c_Ω$, which allows us to compute the limit as $\varepsilon \to 0$ in a variational sense, that is, in the framework of $Γ$-convergence.

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

TL;DR

This work extends non-local isoperimetric energies to curved geometries by introducing a Riemannian autocorrelation function and characterising finite perimeter via a BV framework expressed through geodesic difference quotients. The approach replaces Euclidean translations with geodesic flows, leveraging Liouville invariance to relate the non-local term to , and proving that is Lipschitz if and only if has finite perimeter, with proportional to the perimeter. On the round sphere, the authors prove a Gamma-convergence result as for , with , and show the energy localises to a multiple of the perimeter, precluding fine-scale pattern formation in the subcritical regime. Overall, the paper provides a geometry-aware framework for analyzing pattern formation on curved membranes by linking BV theory, geodesic dynamics, and non-local interactions via the Riemannian autocorrelation function.

Abstract

We study a family of non-local isoperimetric energies on the round sphere , where the non-local interaction kernel is the fundamental solution of the Helmholtz operator . To analyse these energies, we introduce a Riemannian autocorrelation function associated to a measurable set , defined on any compact, connected, oriented Riemannian manifold without boundary of dimension . This function is intimately linked to Matheron's set covariogram from convex geometry. By establishing a characterisation of functions of bounded variation in terms of geodesic difference quotients, we show that has finite perimeter if and only if is Lipschitz, and we relate the Lipschitz constant to the perimeter of . We show that on the round sphere admits a reformulation in terms of , which allows us to compute the limit as in a variational sense, that is, in the framework of -convergence.
Paper Structure (16 sections, 32 theorems, 63 equations)

This paper contains 16 sections, 32 theorems, 63 equations.

Table of Contents

  1. Introduction
  2. Riemannian manifolds and the geodesic flow
  3. The unit tangent bundle and averaged integrals
  4. The geodesic flow
  5. The Jacobian of the exponential map
  6. Functions of Bounded Variation on Riemannian Manifolds
  7. Basic definitions and properties
  8. The mean geodesic variation
  9. The mean geodesic difference quotient
  10. Upper bound via smooth approximation
  11. Lower bound via radial mollifiers
  12. Total variation as limit of the geodesic difference quotient
  13. The Riemannian Autocorrelation Function
  14. Application to Pattern Formation
  15. Kernel on the Sphere
  16. ...and 1 more sections

Key Result

Theorem A

Let $(M^n,g)$ be a compact, connected, oriented Riemannian manifold without boundary of dimension $n\ge2$ and $\Omega \subset M$ measurable. Then the following are equivalent: In that case, the right-sided derivative at $r=0$ is and coincides with the Lipschitz constant, i.e. $\|c'_\Omega\|_{L^\infty(\mathbbm{R}_{\ge0})} = \frac{k_n}{2} \operatorname{Per}(\Omega)$.

Theorems & Definitions (36)

  • Theorem A
  • Theorem B
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • Corollary 1
  • Definition 3.1
  • Proposition 6
  • ...and 26 more