The vanishing mean curvature flow in Minkowski space is the
natural evolutionary generalisation of the minimal surface equation,
and has applications in cosmology as a model equation for cosmic
strings and membranes. The equation clearly admits initial data which
leads to singularity formation in finite time; Nguyen and Tian have
even shown stability of the singularity formation in low dimension. On
the other hand, Brendle and Lindblad separately have shown that all
"nearly flat" initial data leads to global existence of solutions. In
this talk, I describe an intermediate regime where global existence
of solutions can be proven on a codimension 1 set of initial data; and
where the codimension 1 condition is optimal --- The
catenoid, being a minimal surface in R^3, is a static solution to the
vanishing mean curvature flow. Its variational instability as a
minimal surface leads to a linear instability under the flow. By
appropriately "modding out" this unstable mode we can show the
existence of a stable manifold of initial data that gives rise to
solutions which scatters toward to the
catenoid. This is joint work with Roland Donninger, Joachim Krieger,
and Jeremy Szeftel. The preprint is available at http://arxiv.org/abs/1310.5606v1