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

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.

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