Singularities of Ricci flow and diffeomorphisms

Abstract

We solve a well-known open problem in Ricci flow: Strong rigidity of cylinders. Strong rigidity is an illustration of a 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. At the heart of this problem in Ricci flow is comparing and recognizing metrics. This can be rather complicated 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. Often it can be avoided if one uses some additional structure of the particular situation. The gauge problem is subtle for non-compact spaces without additional structure. We solve this 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.