  The central object of study of this thesis is inverse mean curvature vector
flow of two-dimensional surfaces in four-dimensional spacetimes. Being a system
of forward-backward parabolic PDEs, inverse mean curvature vector flow equation
lacks a general existence theory. Our main contribution is proving that there
exist infinitely many spacetimes, not necessarily spherically symmetric or
static, that admit smooth global solutions to inverse mean curvature vector
flow. Prior to our work, such solutions were only known in spherically
symmetric and static spacetimes. The technique used in this thesis might be
important to prove the Spacetime Penrose Conjecture, which remains open today.
  Given a spacetime $(N^{4}, \overline{g})$ and a spacelike hypersurface $M$.
For any closed surface $\Sigma$ embedded in $M$ satisfying some natural
conditions, one can "steer" the spacetime metric $\overline{g}$ such that the
mean curvature vector field of $\Sigma$ becomes tangential to $M$ while keeping
the induced metric on $M$. This can be used to construct more examples of
smooth solutions to inverse mean curvature vector flow from smooth solutions to
inverse mean curvature flow in a spacelike hypersurface.
