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arxiv: 2510.14863 · v2 · pith:ZN4WYU7Onew · submitted 2025-10-16 · 🧮 math.DG

Singularities of Curve Shortening Flow with Convex Projections

Qi Sun This is my paper

Pith reviewed 2026-05-22 12:03 UTC · model grok-4.3

classification 🧮 math.DG
keywords curve shortening flowsingularitiesconvex projectionType I singularityimmersed curvesHuisken conjectureR^n
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The pith

Smooth closed immersed curves in R^n with a one-to-one convex projection onto a 2-plane develop Type I singularities and become asymptotically circular under curve shortening flow.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that any smooth closed immersed curve in R^n, if it admits a one-to-one convex projection onto some 2-plane, evolves by curve shortening flow to a Type I singularity while approaching a circle. This matters because the result lifts the classical rounding behavior of planar curves to arbitrary dimensions, where singularities are typically harder to classify. The authors then apply the statement to obtain an analog of Huisken's conjecture: every closed curve in R^n can be smoothly perturbed into R^{n+2} so that the flow shrinks it to a round point. The central proof device is a contradiction that rules out Type II singularities by simultaneously establishing uniqueness and non-uniqueness of tangent flows at the blow-up point.

Core claim

We show that any smooth closed immersed curve in R^n with a one-to-one convex projection onto some 2-plane develops a Type I singularity and becomes asymptotically circular under Curve Shortening flow in R^n. As an application, we prove an analog of Huisken's conjecture for Curve Shortening flow in R^n, showing that any smooth closed immersed curve in R^n can be smoothly perturbed to a closed immersed curve in R^{n+2} 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.

What carries the argument

A contradiction argument that excludes Type II singularities by establishing both the uniqueness and non-uniqueness of tangent flows at the singular point.

Load-bearing premise

The contradiction ruling out Type II singularities rests on being able to prove both uniqueness and non-uniqueness of the tangent flows at the blow-up point.

What would settle it

A concrete counter-example would be a smooth closed immersed curve satisfying the one-to-one convex projection condition whose curve shortening flow develops a Type II singularity or fails to become asymptotically circular.

Figures

Figures reproduced from arXiv: 2510.14863 by Qi Sun.

Figure 1
Figure 1. Figure 1: Examples on CSF with a one-to-one convex projection 1.4. Application. Huisken’s generic singularities conjecture [Ilm03, # 8] for em￾bedded MCF of surfaces has been settled recently by remarkable works, particularly [CM12,CCMS24a,CCS23,BK24]; see also [CMI19,CM21,CCMS24b,SX21,SX25]. It was pointed out by Altschuler [Alt91] that embedded space curves can evolve to have self-intersections under space CSF and… view at source ↗
Figure 2
Figure 2. Figure 2: Snapshots of the evolution of a perturbation of the pla￾nar figure eight curve from different angles. Previously appeared in [Sun24]. 1.5. Strategy of our proof. The principal part of the proof is devoted to ruling out Type II singularities; the argument proceeds by contradiction. For CSF with convex projections, developing Type II singularities, we first improve the known blow-up results in the literature… view at source ↗
Figure 4
Figure 4. Figure 4: Points p(τ ), q(τ ) on S bis τ Γ(·, τ ). Sketch of the proof of Proposition 7.5. In §7.2 we derive gradient estimates for the upper branch of the rotated curve S 1 τΓ(·, τ ). Based on the estimates for S 1 τΓ(·, τ ) in §7.2, we derive gradient estimates in §7.3 for the upper branch of the rotated curve S bis τ Γ(·, τ ). Because of the choice of the horizontal rotation S bis τ (Definition 7.16), the estimat… view at source ↗
read the original abstract

We show that any smooth closed immersed curve in $\mathbb R^n$ with a one-to-one convex projection onto some $2$-plane develops a Type~I singularity and becomes asymptotically circular under Curve Shortening flow in $\mathbb R^n$. As an application, we prove an analog of Huisken's conjecture for Curve Shortening flow in $\mathbb R^n$, showing that any smooth closed immersed curve in $\mathbb R^n$ can be smoothly perturbed to a closed immersed curve in $\mathbb R^{n+2}$ 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript proves that any smooth closed immersed curve in R^n with a one-to-one convex projection onto some 2-plane develops a Type I singularity and becomes asymptotically circular under curve shortening flow. As an application, it shows that any such curve admits a smooth perturbation in R^{n+2} that shrinks to a round point, yielding an analog of Huisken's conjecture for CSF.

Significance. If the central claims hold, the work supplies a geometric criterion guaranteeing Type I singularities for CSF in arbitrary codimension and introduces a contradiction argument that simultaneously invokes uniqueness (via monotonicity) and non-uniqueness (via the projection) of tangent flows. The application to a perturbed round-point shrinking result is of clear interest to geometric analysts working on singularity classification.

major comments (2)
  1. [§4] §4, proof of Theorem 1.1: the non-uniqueness of tangent flows is asserted to follow from the one-to-one convex projection, yet the construction of two distinct blow-up limits is only sketched; it is unclear whether the projection condition alone produces limits with different curvatures or whether an additional rescaling argument is required. This step is load-bearing for the contradiction that rules out Type II singularities.
  2. [§3.2] §3.2, monotonicity formula (3.7): uniqueness of the tangent flow under the Type II assumption is invoked via a standard entropy functional, but the precise adaptation of Huisken's monotonicity to immersed curves in R^n (rather than hypersurfaces) is not written out; a short derivation or reference to the exact statement used would strengthen the argument.
minor comments (2)
  1. [Introduction] Notation: the symbol for the convex projection is introduced in the abstract but first defined only in §2; an earlier sentence in the introduction would improve readability.
  2. [Figure 1] Figure 1: the caption does not indicate the time at which the curve is shown or whether the projection plane is fixed; adding this information would clarify the illustration of the singularity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment point by point below.

read point-by-point responses

  1. Referee: [§4] §4, proof of Theorem 1.1: the non-uniqueness of tangent flows is asserted to follow from the one-to-one convex projection, yet the construction of two distinct blow-up limits is only sketched; it is unclear whether the projection condition alone produces limits with different curvatures or whether an additional rescaling argument is required. This step is load-bearing for the contradiction that rules out Type II singularities.

    Authors: We agree that the sketch of the two distinct blow-up limits in the proof of Theorem 1.1 would benefit from additional detail. The one-to-one convex projection directly implies non-uniqueness: the projected curve evolves to a round circle under CSF, yielding one tangent flow, while the embedding in R^n permits a second limit with nontrivial curvature in the orthogonal complement, contradicting uniqueness from the monotonicity formula. To clarify whether rescaling is needed and to make the construction explicit, we will expand this argument in the revised §4 with a step-by-step construction of the distinct limits. revision: yes

  2. Referee: [§3.2] §3.2, monotonicity formula (3.7): uniqueness of the tangent flow under the Type II assumption is invoked via a standard entropy functional, but the precise adaptation of Huisken's monotonicity to immersed curves in R^n (rather than hypersurfaces) is not written out; a short derivation or reference to the exact statement used would strengthen the argument.

    Authors: We acknowledge that while the monotonicity formula (3.7) is a standard adaptation of Huisken's entropy functional to the curve shortening flow setting, an explicit short derivation or precise reference for immersed curves in R^n was omitted. We will add a brief derivation of the formula adapted to immersed curves in the revised §3.2 to make the uniqueness invocation fully self-contained. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation proceeds by contradiction: assume a Type II singularity occurs, then invoke standard monotonicity/entropy functionals for CSF to obtain uniqueness of the tangent flow, while the one-to-one convex projection supplies an independent geometric reason for the existence of multiple distinct blow-up limits, yielding non-uniqueness. This pair of properties rules out Type II without any reduction to self-definition, fitted parameters, or load-bearing self-citation. The subsequent application to a perturbed Huisken-type result follows directly once Type I is secured. The argument is self-contained against external benchmarks (standard CSF monotonicity formulas and the stated projection hypothesis) and exhibits no circular step.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard assumptions from the theory of curve shortening flow and singularity analysis in Euclidean space, with no free parameters or invented entities; the novelty is in the application of a contradiction argument.

axioms (2)
  • standard math Curve shortening flow is well-posed for smooth closed immersed curves in R^n and develops singularities in finite time
    This is a foundational property of the flow used throughout the analysis.
  • domain assumption Tangent flows exist at singular points and can be analyzed for uniqueness properties
    Directly invoked in the contradiction argument to derive both uniqueness and non-uniqueness.

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Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages · 1 internal anchor

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