%0 Generic %1 colding2021singularities %A Colding, Tobias Holck %A Minicozzi II, William P. %D 2021 %K Colding MInicozzi Ricci flow %T Singularities of Ricci flow and diffeomorphisms %U http://arxiv.org/abs/2109.06240 %X 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.

@misc{colding2021singularities, abstract = {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} \times \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.}, added-at = {2021-09-15T17:31:21.000+0200}, author = {Colding, Tobias Holck and Minicozzi II, William P.}, biburl = {https://www.bibsonomy.org/bibtex/27742249da870dd0f656b397b6ac427d6/gzhou}, description = {Singularities of Ricci flow and diffeomorphisms}, interhash = {7ccc9f5282ffa21426b10ba37cd6b1dc}, intrahash = {7742249da870dd0f656b397b6ac427d6}, keywords = {Colding MInicozzi Ricci flow}, note = {cite arxiv:2109.06240}, timestamp = {2021-09-15T17:31:21.000+0200}, title = {Singularities of Ricci flow and diffeomorphisms}, url = {http://arxiv.org/abs/2109.06240}, year = 2021 }