Singularities of Ricci flow and diffeomorphisms

TL;DR

This paper addresses the gauge problem in Ricci flow by developing a general PDE-based method for fixing diffeomorphisms, leading to a proof of strong rigidity of cylinders and advancing understanding of singularities.

Contribution

It introduces a novel PDE approach to solve the gauge problem in Ricci flow, enabling the proof of strong rigidity of cylinders and related singularity analysis.

Findings

Solved the gauge problem using nonlinear PDEs to fix diffeomorphisms.

Proved strong rigidity of cylinders in Ricci flow.

Developed new bounds and propagation techniques for singularity analysis.

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. 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.