The Riemannian Penrose Inequality in higher dimensions I am reading the proof of the Riemannian Penrose Inequality by Huisken and Ilmamen in The Inverse Mean Curvature Flow and the Riemannian Penrose Inequality and I was wondering why they restrict their proof to the dimension $n=3$.
I thought it might be because of the definition of the Geroch-Hawking mass, or the monotonicity of such a mass, and I was told that it works only in dimension $n=3$ because the Geroch-Hawking mass monotonicity formula relies on the Gauss-Bonnet Theorem. But the latter can be generalized to higher dimensions (for an even dimension), right (wikipedia: Generalized Gauss-Bonnet Theorem)?
Then which argument restricts their proof to $n=3$?
 A: This may be more of an extended comment rather than an answer, but I suspect the reason it works in $n=3$ is because, in their proof, Huisken and Ilmamen consider boundaries ($\partial M$) of the 3-space $M$, which is essentially the black hole exterior (the minimal surface in question). The 2-dimensional case $(\partial M)$ is very special because the Euler-Characteristic $\chi$ gives you $\textit{all}$ of the topological information about the surface.
Essentially, the theorem of Hawking and Ellis telling you that the black hole horizon has Euler characteristic $\chi=2$ gives you more information than you would have in higher dimensions. This 'free' information is essential in Huiseken and Ilmamen's proof.
An independent proof of the Riemmanian-Penrose inequality appeared due to Bray in 2001 (H.L. Bray, $\textit{Proof of the Riemannian Penrose inequality using the positive mass theorem}$) using conformal flows. These techniques can be extended to higher dimensions, and indeed it was shown by Bray and Lee that the Riemmanian-Penrose inequality holds for (at least) $n < 8$ using these techniques.
It may be possible to employ the methods of H&I by invoking the extended topology results due to Galloway and Schoen, regarding the Yamabe classification of black hole horizons in higher dimensions, and then the generalised Gauss-Bonnet theorem. I believe this has not been done before (?).
