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