We study the dynamics of corotational wave maps from
at
threshold energy. It is known that topologically trivial wave maps with energy
are global
and scatter to a constant map. We prove that a corotational wave map with energy
equal to
is globally defined and scatters in one time direction, and in the other time direction,
either the map is globally defined and scatters, or the map breaks down in finite time
and converges to a superposition of two harmonic maps. The latter behavior stands
in stark contrast to higher equivariant wave maps with threshold energy,
which have been proven to be globally defined for all time. Using techniques
developed in this paper, we also construct a corotational wave map with energy
which
blows up in finite time. The blow-up solution we construct provides the first example
of a minimal topologically trivial nondispersing solution to the full wave map
evolution.
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