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In their paper "Conformally flat Manifolds, Kleinian Groups and Scalar Curvature", Schoen and Yau repeatedly use the term "Newtonian capacity" for a subset of $S^n$.

I know the following definition: If $\Omega$ is a Riemannian manifold with boundary, then the capacity of a subset $K$ of the interior of $\Omega$ is the infimum over $$E(u) = \int_\Omega |d u|^2,$$ where $u$ varies over all $C^1$ functions with $u \equiv 1$ on $K$ and $u\equiv 0$ on $\partial M$.

However, setting $\Omega = S^n$ does not make sense since $\Omega$ does not have a boundary. So does anybody have an idea how this is meant?

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  • $\begingroup$ In wikipedia (scaled up one dimension) $K$ must be $n+1$-dimensional, namely the interior of $\Sigma$ (which is supposed to be a hypersurface of $\mathbb{R}^{n+1}$. So I cannot see how $K$ could be a subset of $S^n$... $\endgroup$
    – Matthias Ludewig
    Commented Oct 26, 2016 at 20:06

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Christian Remling is right; in Chapter 6, section 2.4 of Schoen and Yau's Lectures of Differential Geometry, which covers the material presented in your linked paper, we see that the authors define Hausdorff measure and Newton capacity relative to the the ambient embedding into $\mathbb{R}^{n+1}$.

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