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Riemannian Penrose Inequality for higher dimensions

  1. Dec 27, 2015 #1
    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. jcsd
  3. Dec 27, 2015 #2


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    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?
  4. Dec 28, 2015 #3
    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 !
  5. Dec 28, 2015 #4
    There's a black hole solution in 3 dimensions (it does require a negative cosmological constant) http://arxiv.org/abs/hep-th/9204099
  6. Dec 28, 2015 #5


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    But that wouldn't be asymptotically flat, would it?
  7. Dec 28, 2015 #6
    No, but I was only addressing the existence of 3-D black holes, not the inequality
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