As on arXiv
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.
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Connected papers
The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2021, plus the canon.
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Singularities and diffeomorphisms
Topics Boundaries & non-compact, Singularities · Same authors Tobias Holck Colding, William P. Minicozzi · Shared terms “additional structure”, “diffeomorphism”, “great generality”
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Optimal growth bounds for eigenfunctions
Topics Boundaries & non-compact, Ricci solitons · Same authors Tobias Holck Colding, William P. Minicozzi · Shared terms “growth bounds”, “compact spaces”, “almost splitting”
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Propagation of symmetries for Ricci shrinkers
Topics Ricci solitons · Same authors Tobias Holck Colding, William P. Minicozzi · Shared terms “shrinker principle”, “propagation”
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A strong Frankel Theorem for shrinkers
Topics Ricci solitons · Same authors Tobias Holck Colding, William P. Minicozzi
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Quantitative uniqueness for mean curvature flow
Topics Singularities, Stability & uniqueness · Same authors Tobias Holck Colding, William P. Minicozzi
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Eigenvalue lower bounds and splitting for modified Ricci flow
Topics Ricci flow · Same authors Tobias Holck Colding, William P. Minicozzi · Shared terms “splitting”
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Distance between minimal surfaces and flows
Same authors Tobias Holck Colding, William P. Minicozzi · Shared terms “PDE”
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On the geometry of asymptotically conical Ricci shrinkers (II): strong uniqueness
Topics Boundaries & non-compact, Singularities, Stability & uniqueness · Shared terms “tangent flow”
Read around it
Background reading
The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.
In the guidebook
- 11B.4Blow-Ups, Blow-Downs and Splitting at Infinity
Limits of rescaled manifolds and flows: why blow-up limits are nonnegatively curved, what blow-downs see, and how lines appear at infinity.
- 11B.6Rotationally Symmetric Flows and the Standard Solution
The neckpinch in equations: warped products under the flow, what the maximum principle forces, and the standard solution that surgery glues in.
- 6A.13Self-Similar Variables and Monotonicity
Changing variables so that self-similar solutions stand still: convergence to the Gaussian, Giga–Kohn blow-up, Gaussian-weighted monotonicity formulas, frequency functions, and the blow-up logic of Huisken, Struwe and Perelman.