Generalized Hawking Mass

Question

This is a fairly general question. Let $(M^3,g)$ be a Riemannian 3-manifold. Let $\Sigma^2$ be a dimension-2 submanifold of $M$. The Hawking mass of $\Sigma^2$ is defined as

$m(\Sigma^2) := \frac{|\Sigma^2|}{64\pi^{3/2}}(16\pi - \int_{\Sigma^2} H^2)$.

A lot is known about the Hawking mass. My question is, has there been any work done to generalize the Hawking mass to higher dimensions? Is there anything known about a higher-dimensional Hawking mass?

This post imported from StackExchange MathOverflow at 2015-03-27 18:53 (UTC), posted by SE-user Michael Pinkard

Comments

Have you tried asking Carla? (If you are in Tuebingen her office should be somewhere near.)

This post imported from StackExchange MathOverflow at 2015-03-27 18:53 (UTC), posted by SE-user Willie Wong

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Good answer! I did, but she doesn't know, that's why I decided to ask math overflow.

This post imported from StackExchange MathOverflow at 2015-03-27 18:53 (UTC), posted by SE-user Michael Pinkard

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One possibility is that you can start with the characterisation of the Hawking mass in spherical symmetry as the "flux relative to the Kodama vector field" and see if it leads you to anything. For the standard 3+1 case you can see the computations on my blog (scroll down a little to the section titled "Kodama vector field"). But whatever it is it should probably agree with the mass of higher dimensional Schwarzschild.

This post imported from StackExchange MathOverflow at 2015-03-27 18:53 (UTC), posted by SE-user Willie Wong

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For the usual formula, one thing you need to contend with is the $16\pi$ term inside the parentheses: more generally that term is/should be proportional to the Euler characteristic of your two surface $\Sigma$, and arises actually from Gauss-Bonnet and integrating scalar curvature (so the formula you gave is arguably not the correct definition for higher genus surfaces). The higher dimensional Gauss-Bonnet is more complicated, so ...

This post imported from StackExchange MathOverflow at 2015-03-27 18:53 (UTC), posted by SE-user Willie Wong

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... that term will probably need either a serious replacement or some physical justification why it is the genus that matters and not anything else.

This post imported from StackExchange MathOverflow at 2015-03-27 18:53 (UTC), posted by SE-user Willie Wong

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Sorry I didn't respond earlier. Thanks for the link to the blog post, it's interesting and I'll definitely dig into it further!

This post imported from StackExchange MathOverflow at 2015-03-27 18:53 (UTC), posted by SE-user Michael Pinkard

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What's H and what's \(|\Sigma^2|(area?)\)

What do you mean mean curvature, It a local quantity its being integrated over. I was referring to the \(|\Sigma^2|\) in the formula, which is some number probably area.

@dimension10 I am not creating a barrage of comments, Ill just update this comment and i hope you get a ping each time I do so.

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@Prathyush \(\Sigma^2\) is the closed two-dimensional surface you're calculating the Hawking mass of. H is the mean curvature of \(\Sigma^2\).

@Prathyush Yes, that's right, \(|\Sigma^2|\) is the area of \(\Sigma^2\) the surface. The mean curvature \(H\) is just another measure of curvature that is the mean of the principal curvatures, see the wikipedia page. To ping me with each edit, just place one more ping with each edit, like I did here in this comment.