> The reason I suspect n=7 is as high as it goes is because the volume to surface area ratio of an n-dimensional sphere is r/n, which means that the higher n yo

> The reason I suspect n=7 is as high as it goes is because the volume to surface area ratio of an n-dimensional sphere is r/n, which means that the higher n you have, an n-sphere of radius r has a much larger boundary for the same volume. No, I haven't read the paper in detail either, but the reason is very likely the same for why Schoen & Yau (mentioned in the article) could initially only prove their Positive-Mass Theorem up to spatial dimension n=7: In n=8 spatial dimensions minimal hypersurfaces can suddenly develop singularities[0] and you can no longer treat them as manifolds but need tools from geometric measure theory (so-called minimal currents) to describe them. This makes proofs relying on minimal-surface theory in dimensions n > 7 much, much harder (though not necessarily impossible). [0]: The singularities, when zooming in, essentially look like the tip of the 7-dimensional Simons' cone, https://encyclopediaofmath.org/wiki/Simons_cone https://encyclopediaofmath.org/wiki/Simons_cone , making the latter the prototypical example. In fact, the existence of Simons' cone is the whole reason why (i.e. essential to the proof that) stable minimal hypersurfaces in n > 7 can have singularities but in lower dimensions they cannot and are instead smooth: There are no singular stable minimal hypercones in lower dimensions.