  Motivated by the conjectured Penrose inequality and by the work of Hawking,
Geroch, Huisken and Ilmanen in the null and the Riemannian case, we examine
necessary conditions on flows of two-surfaces in spacetime under which the
Hawking quasilocal mass is monotone. We focus on a subclass of such flows which
we call uniformly expanding, which can be considered for null as well as for
spacelike directions. In the null case, local existence of the flow is
guaranteed. In the spacelike case, the uniformly expanding condition leaves a
1-parameter freedom, but for the whole family, the embedding functions satisfy
a forward-backward parabolic system for which local existence does not hold in
general. Nevertheless, we have obtained a generalization of the weak
(distributional) formulation of this class of flows, generalizing the
corresponding step of Huisken and Ilmanen's proof of the Riemannian Penrose
inequality.
