Zusammenfassung
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. For compact manifolds,
various techniques exist for understanding this. However, for non-compact
manifolds, no general techniques exist; controlling growth is a major
obstruction.
We introduce a new way of dealing with the gauge group of non-compact spaces
by solving a nonlinear system of PDEs and proving optimal bounds for solutions.
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. The construction relies on sharp
polynomial growth bounds for a linear system operator. The growth estimates
hold in remarkable generality without assumptions on asymptotic decay. In
particular, it holds for all singularities.
The techniques and ideas apply to many problems. We use them to solve a well
known open problem in Ricci flow. By dimension reduction, the most prevalent
singularities for a Ricci flow are expected to be $\SS^2 \RR^n-2 $;
followed by cylinders with an $\RR^n-3$ factor. We show that these, and more
general singularities, are isolated in a very strong sense; any singularity
sufficiently close to one of them on a large, but compact, set must itself be
one of these. This is key for applications since blowups only converge on
compact subsets. The proof relies on the above gauge fixing and several other
new ideas. One of these shows "propagation of almost splitting", which gives
significantly more than pseudo locality.
Nutzer