Ancient mean curvature flow asymptotic to Simons cone

Junyoung Park

REVIEW 2 major objections 5 minor 42 references

Ancient mean curvature flow asymptotic to Simons cone

T0 review · 2 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read For n≥5, every one-sided ancient mean curvature flow asymptotic to the Simons cone has a single forced asymptotic profile; with mean convexity it must be a stationary Hardt–Simon leaf.

desk verdict A genuinely novel attack on ancient flows asymptotic to the singular Simons cone, but the proof has a load-bearing gap: the entropy bound (25) is false for the leaf flows the paper classifies. read the letter →

arxiv 2608.11011 v1 pith:VYPBBC5U submitted 2026-08-11 math.DG math.AP

Junyoung Park This is my paper · ORCID

classification math.DGmath.AP MSC 53E1053C42

keywords meancurvatureflowancientsolutionsSimonsconeHardt-Simonfoliationself-shrinkerseigenmodedecompositionuniqueasymptoticsparabolicregion

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 claims that one-sidedness plus a prescribed blowdown determines ancient mean curvature flow asymptotically. For flows asymptotic to the $O(n)\times O(n)$ Simons cone with $n\geq 5$ and lying entirely on one side of it, the rescaled flow must approach the cone in a specific way: its graph function behaves like $c\,e^{(1-\alpha)\tau/2}|x|^\alpha$ for a positive constant $c$, with a power-law error. This matters because ancient flows are the blowup models for singularities, and uniqueness of asymptotics is the first step toward classifying them. When the flow is additionally mean convex, the asymptotics upgrade to full uniqueness: the flow is stationary and equals one leaf of the Hardt–Simon foliation up to dilation and reflection.

What carries the argument

The load-bearing object is the linearized rescaled mean curvature operator $L = \Delta_\Sigma - \tfrac12\langle x,\nabla_\Sigma\rangle + (\tfrac12 + \tfrac{2n-2}{|x|^2})$ on the Simons cone, whose top eigenvalue is $\lambda_1=(1-\alpha)/2$ with a one-dimensional, sign-definite eigenfunction $|x|^\alpha$. The proof localizes the graph function with a cutoff, projects onto eigenmodes of $L$, and uses a Merle–Zaag ODE lemma to show that the first mode dominates; one-sidedness kills the sign-changing zero and third eigenmodes. Two geometric estimates—a graphical radius estimate and an inner–outer inverse Poincaré estimate—control the nonlinear and cutoff errors in terms of higher powers of the weighted $L^2$ norm. The barrier family consists of $O(n)\times O(n)$ symmetric self-shrinkers (the paper's 'inner self shrinkers' and 'trumpets') constructed by ODE analysis.

What would settle it

Construct, or numerically find, a smooth properly embedded ancient rescaled mean curvature flow asymptotic to the $O(n)\times O(n)$ Simons cone for some $n\geq 5$ that crosses the cone, and measure the graph function $u_\tau$ on a fixed annulus $\Sigma_{r,R}$. If $\|u_\tau - c\,e^{(1-\alpha)\tau/2}|x|^\alpha\|_{C^k}$ does not decay like $e^{(1-\alpha)(1+p)\tau/2}$ for any positive $c$, or if the projection onto the sign-changing zero eigenmode grows at the same rate as the first mode, Theorem 1.1 fails without (A2).

Watch

Extended reading notes

Core claim

The discovery, on the paper's own terms, is that the parabolic asymptotics of a smooth, properly embedded ancient rescaled mean curvature flow asymptotic to the Simons cone are not arbitrary: under one-sidedness, the dynamics of the localized graph function collapse onto the first eigenmode of the linearized operator. The leading term is $u_\tau(x) \approx c\,e^{(1-\alpha)\tau/2}|x|^\alpha$, the same profile exhibited by the type-I rescaling of a Hardt–Simon leaf, and the error is smaller by a fixed power. The constant $c$ is positive. With mean convexity, Theorem 6.5 upgrades agreement at the asymptotics level to exact agreement: the unrescaled flow converges backward to a Hardt–Simon leaf and, by a Liouville argument, is stationary, hence is that leaf.

Load-bearing premise

The load-bearing premise is one-sidedness (A2): the flow must lie entirely on one side of the Simons cone, which is what makes the graph function nonnegative and rules out the sign-changing eigenmodes; if a smooth ancient flow asymptotic to the cone crosses it, the proof's conclusion is expected to fail.

Editorial extensions

If this is right

In the parabolic region, every one-sided flow asymptotic to the Simons cone has the same leading asymptotics as the stationary Hardt–Simon leaf, so no other ancient blowup profile can occur on one side of the cone.
For the unrescaled flow, uniform graphicality over the cone holds for all sufficiently negative times, and the flow is trapped between two nearby Hardt–Simon leaves (Corollaries 5.10 and 5.11).
With mean convexity, the flow is stationary; its time slices are exactly a Hardt–Simon leaf, giving a complete classification in this class (Theorem 6.5).
The paper's Remark 1.4 indicates that Theorem 1.3 should continue to hold without mean convexity, because the unique asymptotics of Theorem 1.1 do not use that assumption.

Reading between the lines

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

If one-sidedness is dropped, I would expect the sign-changing fourth eigenmode (the degree-two homogeneous mode) to dominate or compete with the first mode; a counterexample with oscillatory or different parabolic asymptotics would confirm the author's prediction that (A2) is optimal.
The same mode-isolation scheme should apply to other self-shrinking cones whose linearized operator has a simple positive eigenvalue with one sign-definite eigenfunction, making one-sided asymptotics a general phenomenon rather than a special property of the Simons cone.
A quantitative consequence not drawn in the paper is that the exponent $p(n)$ in the error estimate is controlled by spectral gaps such as $\lambda_1-\lambda_2$ and $|\lambda_5|$, so explicit lower bounds on $p(n)$ could be extracted from the stated eigenvalues.

Share X Bluesky LinkedIn Reddit HN

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

This is the first serious attack on ancient mean curvature flow asymptotic to the singular Simons cone, and the main idea is genuinely new: isolate the first eigenmode of the linearized operator on the cone, and use the singular model to build inner self-shrinkers and trumpets. The one-sidedness assumption is flagged explicitly, and the paper is honest about expecting a counterexample without it. Sections 7 and 8 contain real, explicit constructions. If the argument worked, this would be a major step.

The problem is that a load-bearing estimate is wrong. Equation (25) claims that locally smooth convergence to Sigma as tau -> -infty implies lambda(tilde M_tau) <= lambda(Sigma) < 2. It does not. Entropy is scale-invariant and global; the Hardt-Simon leaf flow M_t = Sigma_1^+ satisfies (A1)-(A3), and its rescaled slices are e^{tau/2} Sigma_1^+, which converge to Sigma locally away from 0. But lambda(e^{tau/2} Sigma_1^+) = lambda(Sigma_1^+) > lambda(Sigma), just as the catenoid has larger entropy than its tangent plane at infinity. So the inequality in (25) is false for the very flows Theorem 1.3 claims to classify.

This is not cosmetic. Proposition 3.3 has lambda(M_t) <= lambda(Sigma) as a hypothesis, and its proof uses the sharp bound to exclude excess varifolds in the limit. Lemma 3.8 and Proposition 3.6 inherit that. Without a valid uniform entropy bound, the graphical-radius estimate and the mode isolation in Section 5 have no foundation as written. A weaker bound lambda < 2 would not fix the proof, because the sharpness is doing the work when the limit measure is asserted to be exactly the cone.

I did not check every estimate in Sections 7 and 8; those constructions look plausible and the ODE analysis is careful. The reliance on cited spectral and Liouville results is acceptable. The writing has typos but the structure is clear. The flaw may be repairable, possibly by using one-sidedness plus the strong maximum principle instead of the sharp entropy bound, but as it stands the proof of both main theorems is incomplete.

This paper deserves referee time because the setup and the tools are important, and the author is working on a real problem. But I would not cite it yet; wait for a corrected version.

Referee Report

2 major / 5 minor

Summary. The paper studies smooth, properly embedded ancient mean curvature flows in R^{2n}, n≥5, that are asymptotic to the O(n)×O(n) Simons cone and lie on one side of it. The main theorem (Theorem 1.1 / 5.9) asserts a unique parabolic-region asymptotics: the rescaled flow is an exponentially small normal graph over the cone, with leading term c e^{(1-α)τ/2}|x|^α, where α is the exponent of the first eigenfunction of the linearized rescaled operator. Under an additional mean-convexity assumption, Theorem 1.3 upgrades this to full uniqueness: the flow is stationary and is a Hardt-Simon leaf. The proof combines a graphical radius estimate, an inner-outer (inverse Poincaré) estimate, spectral mode isolation in the Gaussian weighted L^2 space, and two families of auxiliary self-shrinkers (inner self-shrinkers and trumpets) constructed by ODE analysis.

Significance. If the proof is correct, this is a substantial step: it is among the first classification results for ancient MCF asymptotic to a singular cone, and it shows that one-sidedness plus prescribed asymptotic behavior at -∞ forces a unique asymptotics in the parabolic region. The paper has several genuine strengths: the exponent α is derived from the first eigenvalue of the linearized operator rather than fitted; the auxiliary objects (inner self-shrinkers and trumpets) are constructed explicitly by ODE methods; and the proof is structured so that the main claims are falsifiable by checking the displayed asymptotics. No numerical fitting or ad hoc parameters enter the argument. The main issue is that a nonlocal entropy bound used throughout the compactness argument is asserted rather than proved.

major comments (2)

[§2.2, Eq. (25)] The implication in Eq. (25) is not justified and is load-bearing. From C^∞_loc convergence of \tilde M_τ to Σ away from 0 one cannot conclude λ(\tilde M_τ) ≤ λ(Σ): entropy is invariant under independent translations and dilations and is defined as a supremum over all centers and scales, while local smooth convergence on compact annuli only controls the Gaussian integrands on those annuli. This inequality is exactly the hypothesis used in Proposition 3.3 (the bound λ(M_t)≤λ(Σ)<2 in (73)), and the proof of Proposition 3.3 uses it to rule out excess varifolds in the limit. Lemma 3.8 invokes Proposition 3.3 via (25), and Proposition 3.6, Lemma 5.2, Proposition 5.3, and Lemma 6.1 all inherit the compactness from these steps. The paper should either prove (25) from assumptions (A1)-(A2) plus properness, or replace it by a directly established entropy bound for the class of flows considered. The Hardt-Simon leaf flow e^{τ/2}Σ_1^+ mentioned in Remark 3.7 is a natural test case: it satisfies (A1)-(A2), and no argument is given in the paper that its slice entropy is at most λ(Σ). As written, this is a gap in the central compactness argument, not a presentation issue.
[§4, Corollary 4.2] The same type of nonlocal input occurs in Corollary 4.2: the assertion H(\tilde M_τ) ≤ H(Σ) is credited to Huisken's monotonicity formula, but monotonicity alone does not yield this inequality unless one also proves that the Huisken functional of the rescaled slices converges to H(Σ) as τ→-∞ under assumption (A1). Huisken's formula gives monotonicity in τ for the Gaussian density at the spacetime origin, and identifying its τ=-∞ limit with the cone requires a separate argument controlling the noncompact tails and the region near the origin. Since Proposition 4.1 and the subsequent weighted-L^2 estimates in Section 5 depend on this bound, the missing argument should be supplied or replaced by a directly verified bound for the graphical hypersurfaces under consideration.

minor comments (5)

[§2.2] The spectral facts in subsection 2.2 are stated for n≥4 in some places and for n≥5 in others; the explicit first four eigenvalues and eigenfunctions are needed only for n≥5, and the statement should make this convention uniform.
[§5, Lemma 5.2] In the proof of Lemma 5.2, the text refers to “(420)” when it presumably means the estimate in Corollary 8.3 (or the display preceding (170)); the equation number should be corrected.
[§3, Proposition 3.1] The pasting argument in the proof of Proposition 3.1 is only sketched: after obtaining graphicality on annuli of the form B(0,2e^{(τ-τ_0)/2})\B(0,e^{(τ-τ_0)/2}) for varying τ_0, the passage to graphicality on \tilde M_τ\B(0,r) should be written with an explicit dyadic covering or with a precise choice of τ_0 depending on τ.
[§5, Lemma 5.8] In the proof of Lemma 5.8, the final exclusion of c=0 relies on the strong maximum principle after the flow has been identified with the cone on an open set; this is correct in spirit, but the sentence “˜uτ ≡0, which means that ˜Mτ has to agree with Σ” should be expanded, since ˜uτ is only the localized graph function and the identification of the flow with the cone requires connectedness and one-sidedness together with the maximum principle.
[§2.2, Proposition 2.3] The proofs of Propositions 2.2 and 2.3 reduce to the averaged function and then cite Stolarski's theorems; it would help the reader to state the precise hypotheses of theorem 5.2 and corollary 5.5 of [41] being used, rather than only referring to them.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main asymptotics are derived from stated geometric assumptions with no fitted parameters, and no load-bearing self-citation chain is present.

full rationale

The paper's central claim, Theorem 1.1/5.9, is derived by writing the rescaled graph function as a solution to an evolution equation, projecting onto eigenmodes of the linearized rescaled mean curvature operator, and applying the Merle-Zaag ODE lemma. Nothing is fitted to the target conclusion: the exponent alpha is fixed independently by the spectrum of the linearized operator around the Simons cone, and the constant c arises as an ODE integration constant whose positivity is forced by the one-sidedness assumption. The auxiliary inner self-shrinkers and trumpets used as barriers are constructed explicitly in Sections 7 and 8 by ODE existence arguments rather than assumed or imported from the main theorem. The Liouville-type propositions 2.2 and 2.3 are cited from Stolarski's independent work, not from the present author's own papers, and the Hardt-Simon foliation and its properties are external standard results. The assertion in equation (25), that local smooth convergence to the Simons cone implies the entropy bound lambda(M_tau) <= lambda(Simons cone) < 2, is a delicate input estimate and may be a correctness risk, but it is not circular: it is not a case of a quantity being defined in terms of the object it is used to predict, nor does it reduce the theorem to a fitted parameter or a self-citation. Overall, no step in the derivation chain is equivalent by construction to its own input.

Assumptions & free parameters 0 free parameters · 6 assumptions · 2 invented entities

The central claim rests on the paper's explicit hypotheses (A1)-(A3) plus standard analytic machinery imported from the cited literature. No free parameters are fitted to data. The auxiliary surfaces are proven to exist, not postulated.

assumptions (6)

domain assumption (A1) Smooth, properly embedded ancient MCF with rescaled flow converging to Simons cone Sigma locally smoothly away from 0.
This is the main hypothesis of Theorem 1.1; it supplies the blowdown model and all a priori graphicality far from the origin.
domain assumption (A2) Flow lies on one side of Sigma (after reflection).
Used to select a global normal and to guarantee positivity of the graph function; rules out sign-changing eigenmodes in lemmas 5.5 and 5.7.
domain assumption (A3) Mean convexity, H = div(nu) >= 0 with nu pointing away from the component containing 0.
Needed only for Theorem 1.3; yields monotonicity of enclosed domains and quasistatic backward limits.
standard math Spectral decomposition of L on Sigma: first four eigenvalues (1-alpha)/2, 1/2, -(1+alpha)/2, 0, and lambda_5 < 0, with stated eigenfunctions.
Imported from Huang [29, Section 2]; the entire mode-isolation proof depends on this gap structure.
standard math Liouville rigidity theorems for nonnegative ancient solutions of linear heat equations on Sigma and Sigma^+_1 (propositions 2.2 and 2.3).
Consequences of theorems 5.2 and 5.5 in Stolarski [41]; used to force partial_t h = 0 in section 6.
standard math Entropy bound lambda(Sigma) < 2 for n >= 4 (from Bernstein-Wang [5]) and Huisken monotonicity.
Gives entropy control, compactness, and the Gaussian density bound used in pseudo-locality and maximum principle arguments.

invented entities (2)

Inner self-shrinkers Gamma_theta, Lambda_theta independent evidence
purpose: Barriers that control the graph function near the cone singularity and provide the foliation used in the inner-outer estimate.
Constructed as ODE solutions in proposition 7.1 with explicit initial conditions; their profile is analyzed in propositions 7.4 and theorem 7.5, so an independent reader can verify them.
Trumpets Sigma_sigma independent evidence
purpose: Exterior barriers controlling the C0 growth of the graph function in the outer region (corollary 8.3).
Constructed via ODE in proposition 8.1 with explicit asymptotics u_sigma(x) ~ sigma x; independently checkable.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Ancient mean curvature flow asymptotic to Simons cone." pith.science (2026). https://pith.science/paper/VYPBBC5U

@misc{pith2026260811011,
  author       = {Pith},
  title        = {Pith review of: Ancient mean curvature flow asymptotic to Simons cone},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VYPBBC5U}},
  note         = {Machine review of arXiv:2608.11011}
}

abstract

In this paper, we prove that a smooth, properly embedded ancient mean curvature flow that is asymptotic to $O(n)\times O(n)$ symmetric Simons cone for $n \geq 5$, and lies on one side of the cone has to have a unique asymptotics in the parabolic region. When additionally assuming mean convexity, we upgrade unique asymptotics to full uniqueness, and show that such flow has to be a stationary flow given by one of the leaves of the Hardt-Simon foliation.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

42 extracted references · 36 canonical work pages

[1]

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”. In:Journal of Differential Geometry111.3 (2019), pp. 381–455

work page 2019
[2]

Uniqueness of two-convex closed ancient solutions to the mean curvature flow

Sigurd Angenent, Panagiota Daskalopoulos, and Natasa Sesum. “Uniqueness of two-convex closed ancient solutions to the mean curvature flow”. In:Annals of mathematics192.2 (2020), pp. 353–436

work page 2020
[3]

Bamler and Yi Lai.Classification of ancient cylindrical mean curvature flows and the Mean Convex Neighborhood Conjecture

Richard H. Bamler and Yi Lai.Classification of ancient cylindrical mean curvature flows and the Mean Convex Neighborhood Conjecture. 2025. arXiv: 2512.24524[math.DG]

arXiv 2025
[4]

The PDE-ODI principle and cylindrical mean curvature flows

Richard H. Bamler and Yi Lai. “The PDE-ODI principle and cylindrical mean curvature flows”. In: arXiv preprint arXiv:2512.25050(2025)

arXiv 2025
[5]

Lower bounds on density for topologically nontrivial minimal cones up to dimension six

Jacob Bernstein and Lu Wang. “Lower bounds on density for topologically nontrivial minimal cones up to dimension six”. In:Forum of Mathematics, Sigma. Vol. 13. Cambridge University Press. 2025, e122

work page 2025
[6]

Collapsing ancient solutions of mean curvature flow

T Bourni, M Langford, and G Tinaglia. “Collapsing ancient solutions of mean curvature flow”. In: Journal of Differential Geometry119.2 (2021), pp. 187–219

work page 2021
[7]

Convex ancient solutions to curve shortening flow

Theodora Bourni, Mat Langford, and Giuseppe Tinaglia. “Convex ancient solutions to curve shortening flow”. In:Calculus of Variations and Partial Differential Equations58.4 (2019), p. 133

work page 2019
[8]

Uniqueness of convex ancient solutions to mean curvature flow in higher dimensions

Simon Brendle and Kwok-Kun Choi. “Uniqueness of convex ancient solutions to mean curvature flow in higher dimensions”. In:Geometry & Topology25.5 (2021), pp. 2195–2234.doi: 10.2140/gt.2021.25. 2195.url: https://doi.org

doi:10.2140/gt.2021.25 2021

Show all 42 references

[9]

Mean curvature flow with generic initial data II

Otis Chodosh, Kyeongsu Choi, and Felix Schulze. “Mean curvature flow with generic initial data II”. In: (Feb. 2023). arXiv: 2302.08409[math.DG]

2023 arXiv
[10]

Mean curvature flow with generic initial data

Otis Chodosh et al. “Mean curvature flow with generic initial data”. en. In:Invent. Math.237.1 (July 2024), pp. 121–220

2024
[11]

Classification of bubble-sheet ovals inR 4

Beomjun Choi et al. “Classification of bubble-sheet ovals inR 4”. In:Geometry & Topology29 (2025), pp. 931–964.doi: 10.2140/gt.2025.29.931.url: https://msp.org/gt/2025/29-2/gt-v29-n2-p08-p.pdf

2025 doi
[12]

Classification of ancient noncollapsed flows inR 4

Kyeongsu Choi and Robert Haslhofer. “Classification of ancient noncollapsed flows inR 4”. In:arXiv preprint arXiv:2412.10581(2024)

2024 arXiv
[13]

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”. In:Acta Mathematica228.2 (2022), pp. 217–301

2022
[14]

Classification of noncollapsed translators in R4

Kyeongsu Choi, Or Hershkovits, and Robert Haslhofer. “Classification of noncollapsed translators in R4”. In:arXiv preprint arXiv:2208.14280(2022)

2022 arXiv
[15]

Ancient asymptotically cylindrical flows and applications

Kyeongsu Choi et al. “Ancient asymptotically cylindrical flows and applications”. In:Inventiones mathematicae229.1 (2022), pp. 139–241

2022
[16]

Classification of ancient finite-entropy curve shortening flows

Kyeongsu Choi et al. “Classification of ancient finite-entropy curve shortening flows”. In:arXiv preprint arXiv:2603.09112(2026)

2026
[17]

Stability of translating solutions to mean curvature flow

Julie Clutterbuck, Oliver C. Schn¨ urer, and Felix Schulze. “Stability of translating solutions to mean curvature flow”. In:Calculus of Variations and Partial Differential Equations29.3 (2007), pp. 281– 293.doi: 10.1007/s00526-006-0033-1

2007 doi
[18]

Generic mean curvature flow I; Generic singularities

Tobias Colding and William Minicozzi. “Generic mean curvature flow I; Generic singularities”. In: Annals of Mathematics. Second Series2 (Mar. 2012).doi: 10.4007/annals.2012.175.2.7

2012 doi
[19]

Classification of compact ancient solutions to the curve shortening flow

Panagiota Daskalopoulos, Richard Hamilton, and Nataˇ sa ˇSeˇ sum. “Classification of compact ancient solutions to the curve shortening flow”. In:Journal of Differential Geometry84.3 (2010), pp. 455–464. 59

2010
[20]

Spectral quantization for ancient asymptotically cylindrical flows

Wenkui Du and Jingze Zhu. “Spectral quantization for ancient asymptotically cylindrical flows”. In: Advances in Mathematics479 (2025), p. 110422.issn: 0001-8708.doi: 10.1016/j.aim.2025.110422

2025
[21]

Interior estimates for hypersurfaces moving by mean curva- ture

Klaus Ecker and Gerhard Huisken. “Interior estimates for hypersurfaces moving by mean curva- ture”. In:Inventiones mathematicae105 (1991), pp. 547–569.url: https://api.semanticscholar.org/ CorpusID:122642136

1991
[22]

Regularity of minimal surfaces near quadratic cones

Nick Edelen and Luca Spolaor. “Regularity of minimal surfaces near quadratic cones”. In:Annals of Mathematics198.3 (2023), pp. 1013–1046

2023
[23]

Analysis of Vel´ azquez’s solution to the mean curvature flow with a type II singularity

Siao-Hao Guo and Natasa Sesum. “Analysis of Vel´ azquez’s solution to the mean curvature flow with a type II singularity”. In:Communications in Partial Differential Equations43.2 (2018), pp. 185–285

2018
[24]

Area minimizing hypersurfaces with isolated singularities

Robert Hardt and Leon Simon. “Area minimizing hypersurfaces with isolated singularities.” In: (1985)

1985
[25]

Mean curvature flow through singularities

Robert Haslhofer. “Mean curvature flow through singularities”. In:Proceedings of the International Congress of Mathematicians (ICM)(2026). arXiv:2510.01355

2026
[26]

Uniqueness of the bowl soliton

Robert Haslhofer. “Uniqueness of the bowl soliton”. In:Geometry & Topology19.4 (2015), pp. 2393– 2406

2015
[27]

Ancient solutions of the mean curvature flow

Robert Haslhofer and Or Hershkovits. “Ancient solutions of the mean curvature flow”. In:Communi- cations in Analysis and Geometry24.3 (2016), pp. 593–604

2016
[28]

Translators asymptotic to cylinders

Or Hershkovits. “Translators asymptotic to cylinders”. In:Journal f¨ ur die reine und angewandte Mathematik (Crelle’s Journal)(2021)

2021
[29]

Mean curvature flow converging to an minimizing cone and its Hardt-Simon foliation

Jiuzhou Huang. “Mean curvature flow converging to an minimizing cone and its Hardt-Simon foliation”. In:arXiv preprint arXiv:2510.11430(2025)

2025
[30]

Asymptotic behavior for singularities of the mean curvature flow

Gerhard Huisken. “Asymptotic behavior for singularities of the mean curvature flow”. In:Journal of Differential Geometry31.1 (1990), pp. 285–299

1990
[31]

Convex ancient solutions of the mean curvature flow

Gerhard Huisken and Carlo Sinestrari. “Convex ancient solutions of the mean curvature flow”. In: Journal of differential geometry101.2 (2015), pp. 267–287

2015
[32]

On short time existence for the planar network flow

Tom Ilmanen, Andr´ e Neves, and Felix Schulze. “On short time existence for the planar network flow”. In:Journal of Differential Geometry111.1 (2019), pp. 39–89

2019
[33]

Self-shrinkers with a rotational symmetry

Stephen Kleene and Niels Møller. “Self-shrinkers with a rotational symmetry”. In:Transactions of the American Mathematical Society366.8 (2014), pp. 3943–3963

2014
[34]

Ancient solutions in Lagrangian mean curvature flow

Ben Lambert, Jason D Lotay, and Felix Schulze. “Ancient solutions in Lagrangian mean curvature flow”. In:Annali della Scuola Normale Superiore di Pisa, Classe di Scienze22.3 (2021), pp. 1169– 1205.doi: 10.2422/2036-2145.201901 016.url: https://doi.org

2021
[35]

Sharp one-sided curvature estimates for fully nonlinear curvature flows and applications to ancient solutions

Mat Langford and Stephen Lynch. “Sharp one-sided curvature estimates for fully nonlinear curvature flows and applications to ancient solutions”. In:Journal f¨ ur die reine und angewandte Mathematik (Crelle’s Journal)765 (2020), pp. 1–33

2020
[36]

Ancient solutions and translators of Lagrangian mean curvature flow

Jason D. Lotay, Felix Schulze, and G´ abor Sz´ ekelyhidi. “Ancient solutions and translators of Lagrangian mean curvature flow”. In:Publications Math´ ematiques de l’IH ´ES139 (2024), pp. 259–294.doi: 10. 1007/s10240-023-00143-5.url: https://springer.com

2024
[37]

Minimal hypersurfaces asymptotic to Simons cones

Laurent Mazet. “Minimal hypersurfaces asymptotic to Simons cones”. In:Journal of the Institute of Mathematics of Jussieu16.4 (2017), pp. 821–845.url: https://arxiv.org/abs/1407.2474

2017 arXiv
[38]

Optimal estimates for blowup rate and behavior for nonlinear heat equations

Frank Merle and Hatem Zaag. “Optimal estimates for blowup rate and behavior for nonlinear heat equations”. In:Communications on pure and applied mathematics51.2 (1998), pp. 139–196

1998
[39]

An introduction to Brakke flows

Felix Schulze. “An introduction to Brakke flows”. In:Lecture Notes. Summer School(2021)

2021
[40]

Minimal hypersurfaces asymptotic to quadratic cones in R n+ 1

Leon Simon and Bruce Solomon. “Minimal hypersurfaces asymptotic to quadratic cones in R n+ 1”. In:Inventiones mathematicae86.3 (1986), pp. 535–551

1986
[41]

Existence of mean curvature flow singularities with bounded mean curvature

Maxwell Stolarski. “Existence of mean curvature flow singularities with bounded mean curvature”. In: Duke Mathematical Journal172.7 (2023), pp. 1235–1292

2023
[42]

A local regularity theorem for mean curvature flow

Brian White. “A local regularity theorem for mean curvature flow”. en. In:Ann. Math.161.3 (May 2005), pp. 1487–1519. 60

2005

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

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”. In:Journal of Differential Geometry111.3 (2019), pp. 381–455

work page 2019

[2] [2]

Uniqueness of two-convex closed ancient solutions to the mean curvature flow

Sigurd Angenent, Panagiota Daskalopoulos, and Natasa Sesum. “Uniqueness of two-convex closed ancient solutions to the mean curvature flow”. In:Annals of mathematics192.2 (2020), pp. 353–436

work page 2020

[3] [3]

Bamler and Yi Lai.Classification of ancient cylindrical mean curvature flows and the Mean Convex Neighborhood Conjecture

Richard H. Bamler and Yi Lai.Classification of ancient cylindrical mean curvature flows and the Mean Convex Neighborhood Conjecture. 2025. arXiv: 2512.24524[math.DG]

arXiv 2025

[4] [4]

The PDE-ODI principle and cylindrical mean curvature flows

Richard H. Bamler and Yi Lai. “The PDE-ODI principle and cylindrical mean curvature flows”. In: arXiv preprint arXiv:2512.25050(2025)

arXiv 2025

[5] [5]

Lower bounds on density for topologically nontrivial minimal cones up to dimension six

Jacob Bernstein and Lu Wang. “Lower bounds on density for topologically nontrivial minimal cones up to dimension six”. In:Forum of Mathematics, Sigma. Vol. 13. Cambridge University Press. 2025, e122

work page 2025

[6] [6]

Collapsing ancient solutions of mean curvature flow

T Bourni, M Langford, and G Tinaglia. “Collapsing ancient solutions of mean curvature flow”. In: Journal of Differential Geometry119.2 (2021), pp. 187–219

work page 2021

[7] [7]

Convex ancient solutions to curve shortening flow

Theodora Bourni, Mat Langford, and Giuseppe Tinaglia. “Convex ancient solutions to curve shortening flow”. In:Calculus of Variations and Partial Differential Equations58.4 (2019), p. 133

work page 2019

[8] [8]

Uniqueness of convex ancient solutions to mean curvature flow in higher dimensions

Simon Brendle and Kwok-Kun Choi. “Uniqueness of convex ancient solutions to mean curvature flow in higher dimensions”. In:Geometry & Topology25.5 (2021), pp. 2195–2234.doi: 10.2140/gt.2021.25. 2195.url: https://doi.org

doi:10.2140/gt.2021.25 2021