# Riemannian Penrose Inequality for higher dimensions

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1. Dec 27, 2015

### Sasha_Tw

I am reading the proof of the Riemannian Penrose Inequality (http://en.wikipedia.org/wiki/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 $n=3$?

2. Dec 27, 2015

### bcrowell

Staff Emeritus
It seems unlikely to make sense for n=2, since the motivation had to do with black holes, which don't exist in 2+1 dimensions.

It may be that it holds for n>3, but with a trivial change in the geometrical factor of $16\pi$. Have you tried working out the case of the 4+1-dimensional Schwarzschild spacetime?

3. Dec 28, 2015

### Sasha_Tw

Thank you for your answer! The proof was generalized to higher dimensions, up to $n=8$ by Bray. But my question is about the Huisken and Ilmanen proof. I know there proof was restricted to dimension $n=3$ due to an argument linked to the Geroch monotonicity. I think it is linked to the fact that the Euler characteristic has to be less or equal than 2. Is that something that is valid only in dimension 3 ? Perhaps coming from the Hawking topology Theorem ? I am still looking into this !

4. Dec 28, 2015

### jkl71

There's a black hole solution in 3 dimensions (it does require a negative cosmological constant) http://arxiv.org/abs/hep-th/9204099

5. Dec 28, 2015

### bcrowell

Staff Emeritus
But that wouldn't be asymptotically flat, would it?

6. Dec 28, 2015

### jkl71

No, but I was only addressing the existence of 3-D black holes, not the inequality