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
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
].
The Geroch energy has interesting positivity and monotonicity properties along a special flow in
[150
, 219
]. This flow is the so-called inverse mean curvature flow defined as follows. Let
be
a smooth function such that
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 [207
, 208
], 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.
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