Unique tangent flows at the first surface singularity

The explanation

Start with a round sphere of radius 1. Under mean curvature flow every point moves inward at a speed set by how much the surface bends there. For this sphere the radius squared falls by 4 per unit time, so it equals 1 − 4t. The sphere vanishes to a point at time 1/4. Now zoom in on that point. At time 0.24 the radius is 0.2, and magnifying by 10 gives radius 2. At time 0.2499 the radius is 0.02, and magnifying by 100 gives radius 2 again. Every zoom sequence sees the same sphere of radius 2, shrinking in the same way.

This limit is a tangent flow: what the flow looks like at a singularity after magnifying space and time. A singularity is where curvature becomes infinite. The limiting shapes are self-shrinkers, surfaces that shrink without changing shapeSource 1. Compactness gives a limit only along some zoom sequences. The question is whether all sequences give the same limit, in the same place and facing the same waySource 2. For a neck pinch the limit is a cylinder, and its axis could in principle turn as the zoom changes.

Several kinds of limit already had uniqueness theorems. Schulze handled compact limitsSource 3. Colding and Minicozzi handled round cylinders, including their axesSource 4. Chodosh and Schulze handled limits that open out like conesSource 5. A limit can have several ends, some like cones and some like cylinders, and it need not be an exact round cylinderSource 6. That mixed case was open.

The paper claims uniqueness at every singular point at the first singular time, for every smooth closed embedded surface in three-dimensional spaceSource 7. All zooms converge smoothly, with multiplicity one, to a single self-shrinker in the original coordinates, including its axes and positionSource 8. Multiplicity one means the limit is covered once, not several stacked times. It needs no convexity assumption and no prescribed modelSource 9. The main obstacle is a small change that grows along a cylindrical end and changes the type of that endSource 10. The paper shows that no curve of critical points can open up such an endSource 11. It uses exact algebraic certificates for this stepSource 12.

How sure is this? There is no Lean check. A computer proof checker has not looked at any part of it. The proof is very long and relies on delicate estimates for complex families of surfaces. No specialist review of the paper has been published.

What it doesn't show. This does not cover later singular times, flows continued through a singularity, or zooms whose center moves. It has no Lean check.