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?

possibilityis 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. $\endgroup$1more comment