Notices of the International Consortium of Chinese Mathematicians
Volume 10 (2022)
Singularities and diffeomorphisms
Pages: 112 – 116
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.
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.
Dedicated to Blaine Lawson with admiration
The authors were partially supported by NSF DMS Grants 2104349 and 2005345.
Published 16 August 2022