Codimension Bounds and Rigidity of Ancient Mean Curvature Flows by the Tangent Flow at $-\infty$

Ao Sun; Douglas Stryker

REVIEW 2 major objections 3 minor 22 references

Codimension Bounds and Rigidity of Ancient Mean Curvature Flows by the Tangent Flow at $-\infty$

T0 review · 2 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read An ancient mean curvature flow that converges quickly enough to its tangent flow at $-\infty$ is forced to have the same codimension as that tangent flow, and with sufficiently fast convergence it must be exactly equal to it.

desk verdict A serious extension of the Colding-Minicozzi codimension program to general compact shrinkers, with one underived but likely repairable estimate in the proof of Theorem 1.3. read the letter →

arxiv 1909.02535 v2 pith:UHDNUJWJ submitted 2019-09-05 math.DG math.AP

Douglas Stryker , Ao Sun This is my paper · ORCID

classification math.DGmath.AP MSC 53C44

keywords ancientmeancurvatureflowcodimensionself-shrinkertangentat-∞driftLaplaciancaloricfunctionscurveshorteningrigidity

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 establishes that an ancient mean curvature flow that, after rescaling by $\sqrt{-t}$, converges to a compact shrinker fast enough is constrained in two ways: it cannot live in a higher-dimensional ambient space than the shrinker does, and with sufficiently rapid convergence it must be exactly the shrinker. The threshold is quantitative: convergence at rate $(-t)^{1/2-\delta}$ for some $\delta$ below the first nonzero eigenvalue $\lambda_1$ of the drift Laplacian forces equal codimension, while convergence at merely the borderline rate permits higher codimension, as the paper's example of torus curves shows. A separate theorem shows that if the graph function decays faster than $e^{-\alpha t}$ in an exponential sense, the flow is rigidly identical to the shrinker. These results matter because ancient solutions model singularity formation, and knowing that their geometry is controlled by their behavior at $-\infty$ gives a practical way to classify them.

What carries the argument

The central objects are the drift Laplacian on the limiting shrinker and the caloric functions on the moving flow. The first eigenvalue $\lambda_1$ of the drift Laplacian sets the critical convergence rate: the codimension bound holds only when the graph error decays faster than $(-t)^{-1/2+\lambda_1}$. The proof controls the dimension of spaces of ancient solutions to the heat equation along the flow (caloric functions) with polynomial growth, using Gaussian-weighted norms; coordinate functions of the immersion are caloric, so bounding that dimension bounds the ambient dimension. Two inequalities do the work: a Poincar\'e-type upper bound comparing Gaussian norms of caloric functions to gradients via the Rayleigh quotient on the shrinker, and a lower bound from orthonormalizing functions at times $-\Omega^m$. For the rigidity theorem, a Carleman inequality for functions on $\Sigma\times(-\infty,0)$ converts the assumed exponential decay of the graph function into vanishing of the graph function.

What would settle it

Build or find an ancient mean curvature flow whose rescaled flow is a $C^1$ graph over a compact shrinker with error decaying at the rate $(-t)^{1/2-\delta}\varepsilon(t)\to0$ for some $\delta<\lambda_1$, but whose codimension is strictly larger than that of the shrinker; the theorem predicts no such flow exists. At the borderline rate the torus-curve example already shows the conclusion is false, so the decay gap is the only place to attack.

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Extended reading notes

Core claim

The central discovery is a sharp codimension rigidity theorem for ancient mean curvature flows. If an ancient flow $M_t^n \subset \mathbb{R}^N$ has rescaled flow $M_t/\sqrt{-t}$ forming a $C^1$ graph over a fixed compact shrinker $\Sigma$ with error $\varepsilon(t)$ satisfying $\lim_{t\to-\infty}(-t)^{1/2-\delta}\varepsilon(t)=0$ for some $\delta<\lambda_1$, then $\operatorname{codim}(M_t)=\operatorname{codim}(\Sigma)$. The proof works by showing that the space of slowly growing caloric functions on the flow has the same dimension as on the shrinker: constants plus coordinate functions, so no extra ambient directions can appear. The same machinery, with eigenfunction-transplantation assumptions and a spectral gap, yields a weaker bound $\operatorname{codim}(M_t)\le\operatorname{codim}(\Sigma)+r_1$ when convergence is only at the borderline rate. Finally, for graphs over a closed shrinker, a Carleman inequality proof shows that if $\limsup_{t\to-\infty}\varphi^2e^{-\alpha t}=0$, then $\varphi\equiv0$, so the rescaled flow is exactly the shrinker.

Load-bearing premise

The whole codimension result rests on the assumption that the rescaled flow converges to the shrinker at a rate faster than the borderline rate set by the first nonzero eigenvalue: $(-t)^{1/2-\delta}\varepsilon(t)\to0$ for some $\delta<\lambda_1$, and the paper's own torus-curve example shows that at the borderline rate $(-t)^{1/2-\lambda_1}\varepsilon(t)\le C$, higher codimension does occur.

Editorial extensions

If this is right

An ancient flow that converges to a compact shrinker faster than the $\lambda_1$ threshold cannot escape into higher codimension; its entire history lies in the same Euclidean subspace as the shrinker.
For ancient curve shortening flows, if the rescaled flow is a $C^1$ graph over the multiplicity-$m$ circle with the stated rate and only type I singularities, the flow is forced to be the shrinking circle with multiplicity $m$.
The torus-curve solutions show the rate is sharp: at the borderline rate $(-t)^{1/2-\lambda_1}\varepsilon(t)\le C$, one can have tangent flow a multiplicity-$k_m$ circle yet live in arbitrarily high codimension.
Under faster-than-exponential convergence in the graph sense, an ancient mean curvature flow is identical to its tangent flow at $-\infty$, not merely close to it.
The type I assumption in the circle rigidity result is necessary: there exist nontrivial rescaled ancient flows converging to $mS^1$ exponentially fast but with type II singularities.

Reading between the lines

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

The same caloric-function machinery likely extends to noncompact shrinkers, since the proofs work at the level of spectral gaps and Gaussian weights; the paper explicitly leaves that setting open.
The borderline torus-curve examples suggest a testable dichotomy: any ancient flow converging exactly at the borderline rate should either have extra codimension or be accounted for by eigenfunctions below $1/2$; one could search for other solutions realizing the bound $\operatorname{codim}(\Sigma)+r_1$ in Theorem 1.3.
The Carleman rigidity argument is local in the graph function and likely extends to higher codimension by tracking normal-bundle components, replacing the scalar $\varphi$ by a section of the normal bundle.
The sharp constant conjecture for the entropy bound could be probed computationally by enumerating torus-curve-like solutions with different winding numbers and checking whether any ancient curve shortening flow in high codimension beats $\sup_t\lambda(M_t)\ge (N/2)\lambda(S^1)$.

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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

Good paper, worth refereeing carefully. The main results are real: Theorem 1.1 extends the Colding-Minicozzi cylinder codimension bound to arbitrary compact shrinkers, with a sharp rate assumption; Theorem 1.3 gives a more flexible bound using the modified space P-tilde_d of linear caloric functions; and the rigidity theorem (Theorem 4.1) is a clean Carleman argument under super-exponential convergence. The torus curve discussion is also useful, and the authors are honest that the motivating example comes from AAAW13.

The proof strategy is coherent and mostly careful. The genuine soft spot is in the proof of Theorem 1.3: after orthonormalizing the extra functions at t = -Omega^{m_q+1}, the authors assert |V_{r1+1}| <= C Omega^{m_q(-lambda1+rho)} with no derivation. This bound is load-bearing: it is exactly what makes the kappa term in Lemma 3.7 go to zero against the decay hypothesis. A generic orthonormal linear function only gives |U| ~ (-t)^{-1/2} = Omega^{-m_q/2}, which is weaker when lambda1 > 1/2. The text says the bound follows from the proof of Theorem 1.1, but that isn't immediate; the normalization is done at a different time, and the orthogonal decomposition into eigenfunction directions is not written out. I don't think it's a fatal gap; it's the kind of spectral and normalization bookkeeping a referee can ask to be filled in. The analogous step in Theorem 1.1 is easier and likely fine.

Two other remarks. First, the assumptions of Theorem 1.3 are heavy (psi_i, monotone scaling, second-order Phi decay), and as the authors admit, the torus curve doesn't satisfy the Phi decay condition, so the motivating example is not covered by the stronger theorem. That is a limitation but not a flaw. Second, the citation pattern looks fine: Corollary 1.2 uses Baldauf-Sun as an external entropy bound, and the AAAW13 example is properly credited.

Net: the paper is a solid contribution, clearly written, and the soft spots are local. I would send it to a serious referee. The referee should push for a full derivation of the coefficient-vector estimate in Theorem 1.3 and a cleaner statement of the spectral assumption.

Referee Report

2 major / 3 minor

Summary. The paper studies ancient mean curvature flows in arbitrary codimension whose rescaled flow at -∞ is a C^1-graph over a compact shrinker Σ. Theorem 1.1 proves that if the graph error decays as (-t)^{1/2-δ}ε(t)→0 with δ<λ_1, where λ_1 is the first nonzero drift-Laplacian eigenvalue on Σ, then the ambient codimension of the flow equals that of Σ. Corollary 1.2 applies this to the m-covered circle, giving rigidity under a type I assumption. Theorem 1.3 weakens the convergence-rate assumption at the cost of extra spectral assumptions and proves codim(M_t) ≤ codim(Σ)+r_1. Theorem 1.4 uses a Carleman inequality to show that sufficiently fast exponential convergence forces the rescaled flow to equal the shrinker exactly. The paper is motivated by the explicit torus-curve examples from [AAAW13], which are used to test sharpness of the rates.

Significance. If the proofs are completed, the paper gives a meaningful extension of Colding-Minicozzi's codimension-complexity paradigm from round cylinders to arbitrary compact shrinkers, with a sharp rate in Theorem 1.1 and an explicit sharpness example via torus curves. The entropy computation for torus curves and the discussion around Conjecture 2.4 are valuable contributions, and the Carleman-based rigidity in Theorem 1.4 is a clean and potentially useful tool. However, the proof of Theorem 1.3 contains an underived coefficient-vector estimate that is load-bearing for the main weaker-convergence result, and the statement of Theorem 1.4 needs a uniformity clarification. These are local but necessary repairs rather than refutations of the central strategy.

major comments (2)

[§3.3, proof of Theorem 1.3, after (3.33)] The estimate |V_{r_1+1}| ≤ C Ω^{m_q(-λ_1+ρ)} is asserted without derivation, and it is load-bearing: it is what makes the κ-term in Lemma 3.7 vanish in (3.34), and the final contradiction in Theorem 1.3 depends on that. The preceding facts do not imply it automatically. Orthonormality at t=-Ω^{m_q+1} with v_{r_1+1}=⟨x,V_{r_1+1}⟩ gives |V| of order Ω^{-(m_q+1)/2} on the natural scaling, while orthogonality to the transplanted eigenfunctions ψ_i does not by itself improve this to a power with exponent -λ_1, especially when λ_1>1/2. The authors need to supply the missing spectral/normalization argument, including the exact scaling convention for V_{r_1+1}; without it, the upper bound (3.34) and the contradiction in Theorem 1.3 are not established.
[§4, Theorem 4.1 and its proof] The hypothesis is written as limsup_{t→-∞} φ^2 e^{-αt}=0 for a function on Σ×(-∞,0), but the proof requires that ∫_Σ φ^2(·,T_1)e^{-αT_1}→0 as T_1→-∞. Pointwise convergence in x on the compact manifold Σ does not imply the integral of the boundary term converges to zero without uniformity in x. The statement should clarify that the limsup is uniform in x, e.g. limsup_{t→-∞} sup_Σ φ^2 e^{-αt}=0, or the proof should establish the needed uniform decay from the graph equation. As written, the passage from the Carleman inequality to φ≡0 has a gap.

minor comments (3)

[§3.1, proof of Theorem 1.1] The assertion that an additional coordinate function lies in P_{2δ'}(M_t) is stated without derivation; a short justification using |u|≤C√(-t)ε(t)=o((-t)^δ) and the graph closeness to Σ would make the proof self-contained.
[§3.3, proof of Lemma 3.7] The proof says 'Applying Lemma 3.1 to v', but the correct reference is Lemma 3.6, since v is constructed to be orthogonal to 1 and to the ψ_i and the bound uses λ_{l+1} rather than λ_1.
[§3.3, Lemma 3.7] The notation in the assumption ||ψ_i/s_i(t)-φ_i||_{C^1} is ambiguous because φ_i is originally an eigenfunction on Σ while the norm is taken on M_t; the text should specify that φ_i denotes the transplanted function, and likewise in Theorem 1.3.

Circularity Check

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No circular dependence: the central theorems are proved from explicit convergence hypotheses and external geometric lemmas; the one coauthor citation is an independent entropy bound.

full rationale

I walked the derivation chain of Theorems 1.1, 1.3, and 4.1 and found no equation that reduces to an input by construction. Theorem 1.1 translates the graph-convergence rate into a polynomial-growth space P_{2δ'} and then combines caloric-function norm lower bounds with Colding-Minicozzi upper bounds; the contradiction is genuine and not a restatement of the assumption. Theorem 1.3 has a proof gap at the unstated coefficient-vector estimate |V_{r1+1}| ≤ C Ω^{m_q(-λ1+ρ)}, but that is a missing derivation, not circularity: the asserted bound is not used as an input nor defined in terms of the conclusion. Corollary 1.2 invokes the coauthor paper [BS18], but that theorem is an independent sharp entropy bound for plane curves, not a restatement of the present rigidity claim, so the self-citation is not load-bearing circularity. Theorem 4.1 uses a Carleman inequality to convert the exponential decay hypothesis into an integral estimate; the conclusion φ ≡ 0 follows from the resulting inequality over arbitrary T1 → -∞, not from assuming the conclusion. There are no fitted parameters renamed as predictions, no uniqueness theorem imported from the authors' prior work, and no ansatz smuggled in via citation. The omitted coefficient-vector estimate should be addressed as a correctness or completeness issue, but it does not raise the circularity score.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

No free parameters are fitted; the proofs use the shrinker's spectral gap lambda_1 and the given convergence rate epsilon(t) as inputs. The axiomatic inputs are standard results in geometric analysis, the most important being the caloric-function estimates from [CM19b], the entropy bound from [BS18], and the graph estimate from [Wan14]. The spectral convergence assumption in Theorem 1.3 is a theorem hypothesis, not an axiom.

assumptions (7)

domain assumption Weighted monotonicity formula for mean curvature flow (Ecker, cited as [CM19b, Eq. 3.1])
Used in Lemmas 3.2, 3.7 and 3.8 to turn the flow equation into differential inequalities for Gaussian norms; the paper cites this external theorem rather than proving it.
standard math Rayleigh inequality for the drift Laplacian L_Sigma on a compact shrinker
Invoked in Lemmas 3.1 and 3.6 to extract the spectral gap from orthogonality to low eigenfunctions; this is a standard fact in spectral theory for the self-adjoint drift Laplacian.
domain assumption Colding-Minicozzi [CM19b, Lemma 3.9] lower bound for caloric functions
Used in the contradiction proofs to obtain the lower bounds (3.11) and (3.32). The paper states that the proof is the same as in Colding-Minicozzi and relies on that result.
domain assumption Colding-Minicozzi [CM19b, Lemma 7.1] and C^1-graph estimates (7.21)-(7.23)
Used to transplant inner products and gradient norms between M_t/sqrt(-t) and Sigma; the paper describes the calculations as straightforward and similar to Colding-Minicozzi.
domain assumption Baldauf-Sun [BS18, Theorem A] sharp entropy bound for plane curves with turning number m
Used in Corollary 1.2 to conclude entropy equality from the type I assumption. One author is a coauthor of [BS18], but the cited theorem is an independent sharp entropy statement.
domain assumption Wang [Wan14, Lemma 2.4] estimate for normal graphs over a shrinker
Used in the proof of Theorem 4.1 to convert the rescaled mean curvature flow equation into the semilinear inequality |partial_t phi - Delta phi| <= C(|phi|^2 + |grad phi|^2).
domain assumption Compactness and finite entropy bound lambda(M_t) <= lambda_0 for the flow
Assumed in Lemmas 3.1, 3.2, 3.6, 3.7 to apply the Colding-Minicozzi estimates. For compact ancient flows, entropy is monotone and finite.

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Cite this review

Pith. "Pith review of Codimension Bounds and Rigidity of Ancient Mean Curvature Flows by the Tangent Flow at $-\infty$." pith.science (2026). https://pith.science/paper/UHDNUJWJ

@misc{pith2026190902535,
  author       = {Pith},
  title        = {Pith review of: Codimension Bounds and Rigidity of Ancient Mean Curvature Flows by the Tangent Flow at $-\infty$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UHDNUJWJ}},
  note         = {Machine review of arXiv:1909.02535}
}

abstract

Motivated by the limiting behavior of an explicit class of compact ancient curve shortening flows, we prove codimension bounds for ancient mean curvature flows by their tangent flow at $-\infty$, generalizing a theorem for cylinders in [CM19b]. In the case of the $m$-covered circle, we apply this bound to prove a strong rigidity theorem. Furthermore, we extend this paradigm by showing that under the assumption of sufficiently rapid convergence, a compact ancient mean curvature flow is identical to its tangent flow at $-\infty$.

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Reference graph

Works this paper leans on

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Pith tools

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

[1] [1]

The zoo of solitons for curve shortening in ^n

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

work page 2019

[4] [4]

Sharp Entropy Bounds for Plane Curves and Dynamics of the Curve Shortening Flow

Julius Baldauf and Ao Sun. Sharp entropy bounds for plane curves and dynamics of the curve shortening flow. arXiv preprint arXiv:1808.03936 , 2018

work page Pith review arXiv 2018

[5] [5]

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

work page Pith review arXiv 2018

[6] [6]

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

work page 2014

[7] [7]

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

[8] [8]

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