  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, especially 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. 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.
