[Christian Remling](https://mathoverflow.net/questions/253129/what-is-the-newtonian-capacity-of-a-subset-of-sn#comment622887_253129) is right; in Chapter 6, section 2.4 of Schoen and Yau's [_Lectures of Differential Geometry_](http://www.ams.org/mathscinet-getitem?mr=1333601), 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 $\mathbb{R}^{n+1}$.