### 6.2 The Geroch energy

#### 6.2.1 The definition

Suppose that the 2-surface for which is defined is embedded in the spacelike hypersurface . Let be the extrinsic curvature of in and the extrinsic curvature of in . (In Section 4.1.2 we denoted the latter by .) Then , by means of which

In the last step we used the Gauss-Bonnet theorem for . is known as the Geroch energy [150]. Thus it is not greater than the Hawking energy, and, in contrast to , it depends not only on the 2-surface , but the hypersurface as well.

The calculation of the small sphere limit of the Geroch energy was saved by observing [204] that, by Equation (41), the difference of the Hawking and the Geroch energies is proportional to . Since, however, - for the family of small spheres - does not tend to zero in the limit, in general this difference is . It is zero if is spanned by spacelike geodesics orthogonal to at . Thus, for general , the Geroch energy does not give the expected result. Similarly, in vacuum the Geroch energy deviates from the Bel-Robinson energy in order even if is geodesic at .

Since and since the Hawking energy tends to the ADM energy, the large sphere limit of in an asymptotically flat cannot be greater than the ADM energy. In fact, it is also precisely the ADM energy [150].

#### 6.2.2 Monotonicity properties

The Geroch energy has interesting positivity and monotonicity properties along a special flow in  [150219]. This flow is the so-called inverse mean curvature flow defined as follows. Let be a smooth function such that

1. its level surfaces, , are homeomorphic to ,
2. there is a point such that the surfaces are shrinking to in the limit , and
3. they form a foliation of .

Let be the lapse function of this foliation, i.e. if is the outward directed unit normal to in , then . Denoting the integral on the right hand side in Equation (41) by , we can calculate its derivative with respect to . In general this derivative does not seem to have any remarkable property. If, however, the foliation is chosen in a special way, namely if the lapse is just the inverse mean curvature of the foliation, where , furthermore is maximal (i.e. ) and the energy density of the matter is non-negative, then, as shown by Geroch [150], holds. Jang and Wald [219] modified the foliation slightly such that , and the surface was assumed to be future marginally trapped (i.e.  and ). Then they showed that, under the conditions above, . Since tends to the ADM energy as , these considerations were intended to argue that the ADM energy should be non-negative (at least for maximal ) and not less than (at least for time-symmetric ), respectively. Later Jang [217] showed that if a certain quasi-linear elliptic differential equation for a function on a hypersurface admits a solution (with given asymptotic behaviour), then defines a mapping between the data set on and a maximal data set (i.e. for which ) such that the corresponding ADM energies coincide. Then Jang shows that a slightly modified version of the Geroch energy is monotonic (and tends to the ADM energy) with respect to a new, modified version of the inverse mean curvature foliation of .

The existence and the properties of the original inverse mean curvature foliation of above were proven and clarified by Huisken and Ilmanen [207208], giving the first complete proof of the Riemannian Penrose inequality, and, as proved by Schoen and Yau [328], Jang’s quasi-linear elliptic equation admits a global solution.