### 4.4 Criticality in singularity formation without gravitational collapse

It is well known that the Yang–Mills field does not form singularities from smooth initial conditions in
3+1 dimensions [70], but Bizoń and Tabor [26] have shown singularity formation in 4+1 (the critical
dimension for this system from the point of view of energy scaling arguments) and 5+1 (the first
supercritical dimension). In 5+1 there is the countable family W _{n} of CSS solutions with n unstable modes,
such that W _{1} acts as a critical solution separating singularity (W _{0}) formation from dispersal to infinity. In
4+1 there are no self-similar solutions and the formation of singularities seems to proceed through adiabatic
shrinking of a static solution.
Completely parallel results can be found for wave maps, for which the critical dimension is 2+1. For the
wave map from 3+1 Minkowski to the 3-sphere, Bizoń [16] has shown that there is a countable family of
regular (before the CH) CSS solutions labeled by a nodal number , such that each solution has n
unstable modes. Simulations of collapse in spherical symmetry [23, 151] and in 3 dimensions [149] show
that n = 0 is a global attractor and the n = 1 solution is the critical solution (see also [67, 68] for
computations of the largest perturbation-eigenvalues of W _{0} and W _{1}). Again, for the wave map from 2+1
Minkowski to the 2-sphere generic singularity formation proceeds through adiabatic shrinking of a static
solution [24].

These results have led to the suggestion in [26] that criticality (in the sense of the existence of a
codimension-1 solution separating evolution towards qualitatively different end states) could be a generic
and robust feature of evolutionary PDE systems in supercritical dimensions, and not an effect particular of
gravity.

Garfinkle and Isenberg [88] examine the threshold between the round end state and pinching off in Ricci
flow for a familiy of spherically symmetric geometries on S^{3}. They have found intermediate approach to a
special “javelin” geometry, but have not investigated whether this is universal. See also [89] and [135].
Update

A scaling of the shape of the event horizon at the moment of merger in binary black hole mergers is
noted in [44], but this is really a kinematic effect.