  Comparing and recognizing metrics can be extraordinarily difficult because of
the group of diffeomorphisms. Two metrics, that could even be the same, could
look completely different in different coordinates. This is the gauge problem.
The general gauge problem is extremely subtle for non-compact spaces. Often it
can be avoided if one uses some additional structure of the particular
situation. However, in many problems there is no additional structure. Instead
we solve the gauge problem directly in great generality.
  The techniques and ideas apply to many problems. We use them to solve a
well-known open problem in Ricci flow: Strong rigidity of cylinders. Strong
rigidity is an illustration of a {\it shrinker principle} that uniqueness
radiates out from a compact set. It implies that if one tangent flow at a
future singular point is a cylinder, then all tangent flows are.
  We solve the gauge problem by solving a nonlinear system of PDEs. The PDE
produces a diffeomorphism that fixes an appropriate gauge in the spirit of the
slice theorem for group actions. We then show optimal bounds for the
displacement function of the diffeomorphism. Strong rigidity relies on gauge
fixing and several other new ideas. One of these is "propagation of almost
splitting", another is quadratic rigidity in the right gauge, and a third is an
optimal polynomial growth bound for PDEs that holds in great generality.
