Construction of High Codimension Ancient Mean Curvature Flows

Ao Sun; Douglas Stryker

REVIEW 2 major objections 4 minor 19 references

Construction of High Codimension Ancient Mean Curvature Flows

T0 review · 2 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read For every positive integer m, a compact ancient curve shortening flow in R^{2m} exists whose image spans the full space.

desk verdict Sound explicit construction with a fixable factor error in the entropy proof, but the central example is not new; the paper's value lies in the entropy computation, product construction, and honest discussion of overlap. read the letter →

arxiv 1908.02688 v2 pith:IZCJSHWZ submitted 2019-08-07 math.DG

Douglas Stryker , Ao Sun This is my paper · ORCID

classification math.DG MSC 53C4435K55

keywords ancientsolutionsmeancurvatureflowcurveshorteninghighcodimensiontoruscurvesentropytangentflowsODEreduction

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper proves that for every positive integer $m$ there is a compact, connected ancient solution of the curve shortening flow (defined for all $t<0$) in $\mathbb{R}^{2m}$ whose image is not contained in any $(2m-1)$-dimensional subspace. Ancient solutions model the shapes of singularities under the flow, yet previously known ancient curve flows were planar, so the existence of genuinely high-codimension examples was open. The construction is explicit: a ``torus curve'' whose $2m$ coordinate functions are $r(t)^{k_j^2}\cos(k_j\theta)$ and $r(t)^{k_j^2}\sin(k_j\theta)$, with $r(t)$ solving one scalar ODE. The paper computes the entropy of this flow as $k_m\lambda(S^1)$, identifies its tangent flows at $0$ and $-\infty$ as round circles with multiplicity, and uses the example to propose the sharp constant in an existing entropy-versus-codimension bound.

What carries the argument

The load-bearing object is the torus curve, a curve on a product of $m$ circles $S^1(r^{k_1^2})\times\cdots\times S^1(r^{k_m^2})$ with coordinate functions $r^{k_j^2}\cos(k_j\theta)$ and $r^{k_j^2}\sin(k_j\theta)$. Its defining property is that the speed $|\partial_\theta\gamma|$ is independent of $\theta$, so the vector Laplacian acts as a pure second derivative in $\theta$; substituting into $\partial_t\gamma=\Delta_\gamma\gamma$ collapses the entire high-codimension flow to a single first-order ODE for $r(t)$. The ancient solution of that ODE, obtained by inverting $F(r)=\frac12\sum_j r^{2k_j^2}=-t$, is what makes the construction work.

What would settle it

For $m=2$, $k_1=1$, $k_2=2$, let $r(t)$ solve $\frac12(r^2+r^8)=-t$ and define $\gamma_t$ by formula (2.1); evaluate both sides of $\partial_t\gamma_t=\Delta_{\gamma_t}\gamma_t$ at $\theta=\pi/4$ and $t=-1$. If the two sides differ, the ancient-flow claim for this example is false; if they agree, the reduction is consistent.

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Extended reading notes

Core claim

The central object is the torus curve $\gamma_t(\theta) = (r^{k_1^2}\cos(k_1\theta), r^{k_1^2}\sin(k_1\theta), \ldots, r^{k_m^2}\cos(k_m\theta), r^{k_m^2}\sin(k_m\theta))$, which winds $k_j$ times around the $j$-th circle $S^1(r^{k_j^2})$. Because $|\partial_\theta\gamma_t|^2 = \sum_{j=1}^m k_j^2 r^{2k_j^2}$ is independent of $\theta$, the curve Laplacian is $|\partial_\theta\gamma_t|^{-2}\partial_\theta^2$, so the mean curvature flow equation $\partial_t\gamma = \Delta_\gamma\gamma$ reduces to the scalar ODE $r' = -r\,/\,\sum_{j=1}^m k_j^2 r^{2k_j^2}$. The paper proves this ODE has a unique positive solution on $(-\infty,0)$ with $r\to\infty$ as $t\to-\infty$ and $r\to0$ as $t\to0$, yielding an ancient flow. The coordinate functions are linearly independent, so $\gamma_t$ spans $\mathbb{R}^{2m}$; rescaling shows the tangent flow at $-\infty$ is the multiplicity-$k_m$ circle and at $0$ the multiplicity-$k_1$ circle. Consequently $\sup_t\lambda(\gamma_t) = k_m\lambda(S^1)$. Adding one linear coordinate to the same Ansatz gives eternal helix-type solutions in odd-dimensional spaces with infinite entropy, and products of torus curves give $n$-dimensional ancient mean curvature flows of high codimension.

Load-bearing premise

The reduction to a single ODE rests on the curve being parametrized with speed independent of $\theta$, so that the curve Laplacian is a pure second derivative in $\theta$; if that failed, the coordinate functions would not close under the flow, and the whole construction would collapse.

Editorial extensions

If this is right

Ancient curve shortening flows in Euclidean space need not be planar: for each $m$ there is one in $\mathbb{R}^{2m}$ spanning the full space.
Products of these curves produce compact $n$-dimensional ancient mean curvature flows that also span the full space, and further products with Euclidean factors give finite-entropy ancient solutions in every dimension and codimension except curves lying in odd-dimensional Euclidean spaces.
The entropy of the $m$-frequency torus curve is exactly $k_m\lambda(S^1)$, matching the linear-in-dimension entropy growth that the conjectured sharp constant predicts.
Setting $k_1=1$ makes the tangent flow at time $0$ a single round circle even though the curve has arbitrarily high codimension; the tangent flow at $0$ alone cannot bound codimension.
The example yields the lower bound $\lambda(S^1)C_1\ge 2$ for the constant in the known entropy-codimension bound and motivates the conjecture that the sharp value is $C_1=2/\lambda(S^1)$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the conjectured sharp constant is correct, any ancient curve shortening flow whose image truly spans $\mathbb{R}^N$ must have entropy at least $(N/2)\lambda(S^1)$; testing this inequality on multi-frequency or perturbed torus curves would give evidence without classifying all ancient flows.
The identity $F(r)+t=0$ makes the construction essentially algebraic, so these solutions may serve as explicit blow-up models; one could perturb the Ansatz by small non-sinusoidal Fourier modes and check whether the ODE reduction survives to first order.
The same constant-speed idea may generalize to higher-dimensional submanifolds formed from orthogonal products of such curves, with entropy becoming a product of winding numbers; whether the resulting flows remain ancient is a testable extension.
Varying the frequencies $k_j$ gives a family of ancient flows with the same ambient dimension but different entropies, which may help calibrate any future sharp bound.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

What you should know: the main construction—the torus curves—is not new; the authors disclose in the abstract add-on and Remark 1.3 that Altschuler–Altschuler–Angenent–Wu [AAAW13] discovered it. What is actually new here is the entropy computation, the product theorem, the odd-dimensional helix variant, and the discussion of the Colding–Minicozzi constant. The ODE verification is clean and the paper is honestly written, but Corollary 2.3 has a measure-factor error that invalidates the printed proof of the lower bound.

Let me be specific. In the Gaussian integral over sγ_t, the arclength element contributes a factor √B in the numerator, where B = Σ k_j² r^{2k_j²}. The manuscript puts √B in the denominator. With the printed formula, the chosen scale s̃ = √(2/A) makes the lower bound tend to 0 as t→−∞, not k_m λ(S¹). The corrected formula gives exactly the claimed limit, so the entropy statement is almost certainly fixable, but the proof as written does not work. This is a minor-to-moderate bug: it does not undermine the construction itself, but the entropy claim lacks a valid lower-bound proof until corrected.

What the paper does well: the reduction of the curve shortening flow to the scalar ODE is rigorous. The claim that |∂θγ| is θ-independent is standard and holds here, so the Laplace–Beltrami computation is sound. Theorem A.1 supplies a complete existence proof. The tangent-flow identifications in Proposition 2.2 are plausible and well argued. The product construction in Theorem 1.2 is straightforward and correct, and the comparison with Choi–Mantoulidis is useful context.

The bigger weakness is presentational. The abstract and introduction still frame Theorem 1.1 as the main new result, and the line "As far as we know, this is the first construction..." is no longer accurate once AAAW13 is known. The authors do disclose the overlap, but the framing should be revised to make the extension, not the base construction, the focus. The entropy upper bound is also sketched rather than fully detailed; it looks correct but should be expanded.

This paper is for researchers studying ancient mean curvature flows, entropy, and codimension bounds. It deserves a serious referee, not a desk reject. A referee should flag the factor error, ask for a reframing around the new contributions, and request a fuller entropy upper bound. With those changes, it becomes a useful independent exposition and extension of known material.

Referee Report

2 major / 4 minor

Summary. The paper constructs a family of closed curves in R^{2m} with coordinate functions r(t)^{k_j^2} times trigonometric monomials of frequency k_j, and proves that for an explicitly defined function r(t) these curves solve the curve shortening flow on (-∞,0). The main existence result is Theorem 1.1; the paper also identifies the tangent flows at 0 and -∞, computes the entropy as k_m λ(S^1), constructs analogous noncompact helix solutions in R^{2m+1}, and takes products to obtain higher-dimensional ancient mean curvature flows. The appendix gives an ODE existence proof. The authors state in Remark 1.3 that the main construction was already discovered by Altschuler–Altschuler–Angenent–Wu in [AAAW13].

Significance. The construction itself is clean, self-contained, and machine-checkable in the sense that the Laplacian reduction is valid because |∂_θ γ_t| is θ-independent, and the ODE solution is proved from scratch in the appendix. If the entropy computation is corrected, the paper provides explicit non-planar ancient curve shortening flows with computable entropy and explicit tangent flows, which is a useful addition to the small set of known high-codimension ancient flows and directly motivates the sharp-constant conjecture for a Colding–Minicozzi codimension bound. However, because the central existence result is already present in [AAAW13], the incremental value of the paper rests on the entropy computation, the helix variant, the product construction, and the conjectures; the authors are honest about this priority issue, but it should be reflected in the framing.

major comments (2)

[Corollary 2.3] The displayed Gaussian integral in the proof of the lower bound has a measure-factor error. For sγ_t, with A = Σ_j r^{2k_j^2} and B = Σ_j k_j^2 r^{2k_j^2}, the correct computation is (4π)^{-1/2} ∫_{sγ_t} e^{-|x|^2/4} dH^1 = s√π √B exp(-s^2 A/4), because the dilation multiplies the arclength element by s and |∂_θ γ_t| = √B. The manuscript writes a factor √B in the denominator instead of the numerator. As printed, the choice s̃ = √2 A^{-1/2} gives an expression that tends to 0 as t → -∞, so the claimed lower bound does not follow. Replacing the denominator by √B makes the displayed limit correct and gives k_m λ(S^1). Since this corollary is the basis for the entropy applications and Conjecture 2.4, the proof must be corrected.
[Section 2.3, proof of Theorem 1.2] The proof of Theorem 1.2 is one sentence: 'Take the product of n appropriately chosen torus curves.' This is not sufficient as written, because a product of n torus curves in R^{2m} would lie in R^{2mn}; to obtain a submanifold of R^{2m}, the torus-curve factors must be chosen with ambient dimensions that sum to 2m, and the nondegeneracy of the resulting product must be checked. The statement is plausible and the fix is straightforward, but the proof should be completed explicitly.

minor comments (4)

[Section 2.1, between (2.2) and (2.3)] The Laplacian formula Δ_{γ_t} = |∂_θ γ_t|^{-2} ∂_θ^2 is used because |∂_θ γ_t| is θ-independent; this fact is correct for the given coordinate functions, but it should be stated explicitly as the justification rather than left implicit.
[Proposition 2.2] The tangent flows are described as 'the multiplicity k_m circle' and 'the multiplicity k_1 circle' without specifying the radius; the limiting circles have radius √2, as expected for self-shrinking circles, and this should be stated for clarity.
[Corollary 2.3, upper bound] The upper-bound argument uses monotonicity of entropy to conclude sup_t λ(γ_t) = lim_{t→-∞} λ(γ_t); this is standard for closed mean curvature flows, but a citation or one-line justification would improve rigor.
[Remark 1.3 and abstract add-on] The add-on remark about [AAAW13] appears twice, once after the abstract and once as Remark 1.3; for a journal submission this should be integrated into the introduction in a single place.

Circularity Check

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No circularity: the construction is verified from scratch by direct Laplacian computation and an explicit ODE solution; the acknowledged overlap with AAAW13 is a priority issue, not a circular dependency.

full rationale

The derivation is self-contained. Equation (2.1) defines the torus curves γ_t, and Proposition 2.1 verifies the curve shortening flow equation by computing both sides: since |∂_θγ_t|^2 = Σ k_j^2 r^{2k_j^2} is θ-independent, the curve Laplacian is |∂_θγ_t|^{-2}∂_θ^2, and ∂_tγ_t matches Δ_{γ_t}γ_t exactly when r satisfies the ODE (2.5). The ODE is solved explicitly in Theorem A.1 via the conserved quantity F(r)=½Σ r^{2k_j^2}=-t, which also yields the required asymptotics r→∞ as t→−∞ and r→0 as t→0. No parameter is fitted to the target theorem, and no conclusion is assumed as an input. The prior discovery by Altschuler–Altschuler–Angenent–Wu is disclosed in the abstract and in Remark 1.3, but it is not used as a proof ingredient, so the overlap is a novelty/priority concern, not circularity. The entropy computation in Corollary 2.3 appears to have a measure-factor typo in the displayed Gaussian integral (√B in the denominator instead of the numerator), but the stated limit k_mλ(S^1) follows from the corrected formula; this is a correctness/typo issue, not a circular reduction. Overall, the paper's central existence claim does not reduce by construction to its own inputs.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The construction is explicit and uses no fitted constants or hidden postulates. The only inputs are the chosen positive integers k_1,...,k_m, which are not fitted but are arbitrary inputs that define the curve. The axioms are standard background facts in differential geometry and ODE theory, none of which are ad hoc to this paper.

assumptions (6)

standard math The Laplacian on a curve with θ-independent speed equals |∂_θγ|^{-2}∂_θ^2.
Used to derive equation (2.3); holds because |∂_θγ| is independent of θ for the trigonometric parametrization.
domain assumption Mean curvature flow equation ∂_t x = Δ x for immersed submanifolds.
Section 1; this is the defining equation that the construction verifies.
standard math Entropy is monotone nonincreasing along mean curvature flow (Huisken monotonicity).
Used in Corollary 2.3 to pass from the estimate for t ≤ T_ε to the supremum over all t.
standard math The product formula Δ_{M1×M2} = Δ_{M1} + Δ_{M2}.
Used in Section 2.3 to build n-dimensional ancient solutions from products of torus curves.
domain assumption Colding-Minicozzi Theorem 0.6/0.9 as cited external results on dimension bounds.
Used in the applications in Section 2.1; the sharp value is only conjectured and is not needed for the construction.
standard math The implicit equation F(r) = -t has a unique positive solution on t ∈ (-∞,0) because F is strictly monotone.
Appendix A, Theorem A.1; this is the main ODE existence step.

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Cite this review

Pith. "Pith review of Construction of High Codimension Ancient Mean Curvature Flows." pith.science (2026). https://pith.science/paper/IZCJSHWZ

@misc{pith2026190802688,
  author       = {Pith},
  title        = {Pith review of: Construction of High Codimension Ancient Mean Curvature Flows},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IZCJSHWZ}},
  note         = {Machine review of arXiv:1908.02688}
}

read the original abstract

We construct a class of compact ancient solutions to the mean curvature flow in Euclidean space with high codimension. In particular, we construct higher codimensional ancient curve shortening flows. Moreover, we characterize the asymptotic behavior of these solutions. Add on remark: the construction in this paper has been discovered by Altschuler D.-Altschuler S.-Angenent-Wu in [AAAW13].

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

19 extracted references · 3 canonical work pages

[1]

Altschuler, Steven J

Dylan J. Altschuler, Steven J. Altschuler, Sigurd B. Angenent, and Lani F. Wu. The zoo of solitons for curve shortening in R^n . Nonlinearity , 26(5):1189--1226, 2013

work page 2013
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Unique asymptotics of ancient convex mean curvature flow solutions

Sigurd Angenent, Panagiota Daskalopoulos, and Natasa Sesum. Unique asymptotics of ancient convex mean curvature flow solutions. J. Differential Geom. , 111(3):381--455, 2019

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Uniqueness of convex ancient solutions to mean curvature flow in R^3

Simon Brendle and Kyeongsu Choi. Uniqueness of convex ancient solutions to mean curvature flow in R^3 . Invent. Math. , 217(1):35--76, 2019

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Ancient low entropy flows, mean convex neighborhoods, and uniqueness

Kyeongsu Choi, Robert Haslhofer, and Or Hershkovits. Ancient low entropy flows, mean convex neighborhoods, and uniqueness. arXiv preprint arXiv:1810.08467 , 2018

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Ildefonso Castro and Ana M. Lerma. The C lifford torus as a self-shrinker for the L agrangian mean curvature flow. Int. Math. Res. Not. IMRN , (6):1515--1527, 2014

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Colding and William P

Tobias H. Colding and William P. Minicozzi, II. Generic mean curvature flow I : generic singularities. Ann. of Math. (2) , 175(2):755--833, 2012

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Minicozzi, II

Tobias Holck Colding and William P. Minicozzi, II. Uniqueness of blowups and ojasiewicz inequalities. Ann. of Math. (2) , 182(1):221--285, 2015

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Ancient gradient flows of elliptic functionals

Kyeongsu Choi and Christos Mantoulidis. Ancient gradient flows of elliptic functionals. arXiv preprint arXiv:1902.07697 , 2019

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Panagiota Daskalopoulos, Richard Hamilton, and Natasa Sesum. Classification of compact ancient solutions to the curve shortening flow. J. Differential Geom. , 84(3):455--464, 2010

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Altschuler, Steven J

Dylan J. Altschuler, Steven J. Altschuler, Sigurd B. Angenent, and Lani F. Wu. The zoo of solitons for curve shortening in R^n . Nonlinearity , 26(5):1189--1226, 2013

work page 2013

[2] [2]

Unique asymptotics of ancient convex mean curvature flow solutions

Sigurd Angenent, Panagiota Daskalopoulos, and Natasa Sesum. Unique asymptotics of ancient convex mean curvature flow solutions. J. Differential Geom. , 111(3):381--455, 2019

2019

[3] [3]

Uniqueness of convex ancient solutions to mean curvature flow in R^3

Simon Brendle and Kyeongsu Choi. Uniqueness of convex ancient solutions to mean curvature flow in R^3 . Invent. Math. , 217(1):35--76, 2019

2019

[4] [4]

Ancient low entropy flows, mean convex neighborhoods, and uniqueness

Kyeongsu Choi, Robert Haslhofer, and Or Hershkovits. Ancient low entropy flows, mean convex neighborhoods, and uniqueness. arXiv preprint arXiv:1810.08467 , 2018

arXiv 2018

[5] [5]

Ildefonso Castro and Ana M. Lerma. The C lifford torus as a self-shrinker for the L agrangian mean curvature flow. Int. Math. Res. Not. IMRN , (6):1515--1527, 2014

2014

[6] [6]

Colding and William P

Tobias H. Colding and William P. Minicozzi, II. Generic mean curvature flow I : generic singularities. Ann. of Math. (2) , 175(2):755--833, 2012

2012

[7] [7]

Minicozzi, II

Tobias Holck Colding and William P. Minicozzi, II. Uniqueness of blowups and ojasiewicz inequalities. Ann. of Math. (2) , 182(1):221--285, 2015

2015

[8] [8]

Ancient gradient flows of elliptic functionals

Kyeongsu Choi and Christos Mantoulidis. Ancient gradient flows of elliptic functionals. arXiv preprint arXiv:1902.07697 , 2019

arXiv 1902