# Generalized Hawking Mass

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?

• Have you tried asking Carla? (If you are in Tuebingen her office should be somewhere near.) Oct 10 '14 at 15:47
• Good answer! I did, but she doesn't know, that's why I decided to ask math overflow. Oct 10 '14 at 16:23
• 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. Oct 13 '14 at 7:31
• 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 ... Oct 13 '14 at 8:12
• ... that term will probably need either a serious replacement or some physical justification why it is the genus that matters and not anything else. Oct 13 '14 at 8:12

As mentioned in the comments, you can replace the $16\pi$ term by applying the Gauss-Bonnet theorem which gives you a Einstein-Hilbert functional term. Why not just use that for higher dimensions? Then you can test its convergence to the ADM mass (up to a constant multiple, of course). This is actually done by Miao, Tam and Xie, Quasi-local mass integrals and the total mass (arxiv).