Paper · Oct 16, 2025
Singularities of Curve Shortening Flow with Convex Projections
As on arXiv
Abstract
We show that any smooth closed immersed curve in with a one-to-one convex projection onto some -plane develops a Type I singularity and becomes asymptotically circular under Curve Shortening flow in . As an application, we prove an analog of Huisken's conjecture for Curve Shortening flow in , showing that any smooth closed immersed curve in can be smoothly perturbed to a closed immersed curve in which shrinks to a round point under Curve Shortening flow. Our proof relies on a novel contradiction argument in which Type II singularities are excluded by proving both the uniqueness and non-uniqueness of the tangent flows at the singular point.
Comments: 59 pages, 11 figures, minor updates
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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
- 6A.10Curve Shortening and the First Geometric Flows
Shrinking circles, grim reapers, neckpinches and Huisken’s monotonicity, seen in the plane.
- 12C.4Loops, Disks and Curve Shrinking
Perelman III §§2–3 and Morgan–Tian §§18.3–18.7: π₃ as spheres of loops, minimal disks and the −2π estimate, curve shrinking, ramps in M × S¹ and the estimates that make the argument rigorous.
- 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.