Tobias Holck Colding

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.

There is a long history of parabolic monotonicity formulas that developed independently from several different fields and a much more recent elliptic theory. The elliptic theory can be localized and there are additional monotone quantities. There is also a surprising link: Taking a high-dimensional limit of the right elliptic monotonicity can give a parabolic one as a limit. Poincaré was the first to observe such a connection. We introduce two deficit functions, one elliptic and one parabolic, then show that the parabolic deficit is pointwise the limit of the elliptic and, that the elliptic satisfies an equation that converges to the equation for the parabolic. These pointwise quantities and their equations recover the monotonicities and leads to an elliptic proof of the log Sobolev inequality as well as new concentration of measure phenomena.

We prove a strong Frankel theorem for mean curvature flow shrinkers in all dimensions: Any two shrinkers in a sufficiently large ball must intersect. In particular, the shrinker itself must be connected in all large balls. The key to the proof is a strong Bernstein theorem for incomplete stable Gaussian surfaces.

We show how to use the arguments of [CM2] to get a stronger effective version of uniqueness of blowups that has a number of consequences.

We prove sharp lower bounds for eigenvalues of the drift Laplacian for a modified Ricci flow. The modified Ricci flow is a system of coupled equations for a metric and weighted volume that plays an important role in Ricci flow. We will also show that there is a splitting theorem in the case of equality.

We prove a strong Frankel theorem for mean curvature flow shrinkers in all dimensions: Any two shrinkers in a sufficiently large ball must intersect. In particular, the shrinker itself must be connected in all large balls. The key to the proof is a strong Bernstein theorem for incomplete stable Gaussian surfaces.

We will show that if a gradient shrinking Ricci soliton has an approximate symmetry on one scale, this symmetry propagates to larger scales. This is an example of the shrinker principle which roughly states that information radiates outwards for shrinking solitons.