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An Ancient Stacked Pancake Solution to Mean Curvature Flow

T0 review · 4 major / 4 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read For every n≥3 there is a closed, non-convex ancient mean curvature flow trapped in a slab, shaped like two pancakes joined by a neck.

desk verdict A novel and significant construction whose main theorem is plausible, but the key uniform entropy bound in Lemma 3.3 is asserted without proof, so the current version is incomplete. read the letter →

arxiv 2509.26515 v3 pith:45W34OUI submitted 2025-09-30 math.DG

Alexander Mramor , Louis Yudowitz This is my paper · ORCID
classification math.DG MSC 53E10
keywords ancientsolutionmeancurvatureflownon-convexslabrotationalsymmetrypancakecatenoidBrakke
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

The paper constructs the first compact, embedded ancient solution to mean curvature flow that is non-convex and confined to a slab. Ancient solutions exist for all negative times and serve as models for singularity formation; previously, the only slab-trapped compact examples were convex 'ancient pancakes.' The new solution looks like two such pancakes stacked and joined by a neck, and its existence shows that convexity is not a hidden requirement for a compact ancient flow to stay inside a slab. The construction works in every dimension n≥3.

What carries the argument

The construction is carried out on the profile curve in a plane, where rotational symmetry reduces mean curvature flow to a forced curve-shortening flow. The initial profiles are two copies of the ancient pancake profile joined by a catenoidal neck, parametrized by the neck width m. A continuity argument selects a neck width that sits at the boundary between eventual neck-pinch and quick convexification, producing a family of 'old-but-not-ancient' flows whose Brakke limit is the ancient solution. The Sturmian theorem on intersection numbers is the main tool that controls singularities and upgrades the weak limit to a smooth flow.

What would settle it

Compute the entropy of the approximating flows M^i_t on a fixed parabolic cylinder; if the bound ever exceeds 5 for some i, the compactness step fails. More directly, if the catenoid used in Claim 1 of Lemma 3.4 intersects some approximating profile more than twice, the singularity-free argument collapses.

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Extended reading notes

Core claim

For every n≥3 there exists an ancient solution M^n_t ⊂ R^{n+1}, t∈(−∞,0], that is rotationally symmetric, embedded, closed, non-convex, and contained in a slab; its profile curve has two strict local maxima (the pancake tips) and one strict local minimum (the neck). The flow is obtained as a limit of long-lived smooth rotationally symmetric flows, and the paper proves the weak limit is actually smooth by combining Sturmian intersection-counting with catenoid and pancake barriers.

Load-bearing premise

The asserted but unproved uniform entropy bound λ(M^i_t) ≤ 5 in Lemma 3.3 is load-bearing: without it, Brakke compactness cannot produce the ancient limit flow B_t.

Editorial extensions

If this is right

  • The convexity assumption in the earlier classification of compact slab-trapped ancient flows is genuinely necessary for n≥3.
  • At every time the profile has exactly two tips and one neck, so the flow is a stack of two pancakes, not a single pancake.
  • The flow is rotationally and reflection symmetric, and stays inside a fixed slab for all t≤0.
  • The resulting ancient flow is smoothly embedded for all t≤0, not merely a weak Brakke flow.
  • The construction suggests that similar stacks of finitely many pancakes can be built, and possibly an infinite stack whose convex hull is a halfspace.

Reading between the lines

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

  • One could test the continuity argument numerically: evolve the two-pancake profile for large negative times and track the neck width; the paper's proof predicts the neck remains bounded and no pinch occurs before time 0.
  • The failure at n=2 is tied to the catenoid being entire; a similar construction using a different entire self-shrinker might produce an analogous stack in R^3 with different topology.
  • If the asserted entropy bound in Lemma 3.3 could be verified with an explicit constant, the compactness step would be placed on firmer ground and might extend to k>2 pancakes without new ideas.
  • The halfspace-convex-hull scenario, if realized from an infinite stack, would give an embedded counterpart to the known immersed example and sharpen the Chini–Møller trichotomy.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The paper constructs, for every n≥3, an ancient solution of mean curvature flow in R^{n+1} that is O(n)×O(1)-invariant, embedded, non-convex, closed, contained in a slab, and whose profile curve has two strict local maxima and one strict local minimum. The construction proceeds by building a sequence of rotationally symmetric 'old-but-not-ancient' flows from two copies of the Bourni–Langford–Tinaglia ancient pancake joined by a catenoidal neck, extracting a weak ancient Brakke limit via compactness, then upgrading the limit to a smooth ancient flow through a sequence of regularity claims involving Sturmian intersection arguments and barrier constructions.

Significance. If correct, Theorem 1.1 provides the first compact non-convex ancient solution of MCF trapped in a slab, thereby showing that the convexity hypothesis in Bourni–Langford–Tinaglia's Theorem 1.2 is necessary for n≥3. The paper's ambitious strategy—using Brakke compactness to obtain a weak ancient limit and then proving regularity—is a natural and potentially influential approach. The authors are explicit about the limitations of their method, including the obstruction in n=2 and the heuristic nature of parts of the regularity argument. The paper also credits and builds on the non-fattening results for rotationally symmetric flows, which is appropriate.

major comments (4)
  1. [§3.1, Lemma 3.3] The proof asserts 'One can check that we have the (crude) uniform entropy bound λ(M^i_t)≤5' with no derivation. This bound is the only quantitative input for Brakke compactness, and without it the existence of the ancient Brakke flow B_t is not established. The bound is not obvious: each initial surface Σ_{s_i}^{m_i} consists of two pancake components (each contributing entropy near 2) joined by a catenoidal neck, whose entropy at its own scale must be controlled. Please provide a proof or a precise reference for this estimate; otherwise Lemma 3.3 and all subsequent claims are unsupported.
  2. [§3.2, Lemma 3.4, Claim 1] The proof that the singular set of B_t avoids the axis of rotation relies on the assertion that the partial profile functions f_m have 'uniformly controlled geometry' (a uniform interval of definition, uniform C^1 bounds, and uniformly bounded minima). These are not proved, and they are load-bearing: they justify the existence of a fixed catenoid W whose profile intersects each approximating profile in exactly two points. Without a rigorous verification of these three conditions, the Sturmian intersection argument for excluding axis singularities fails.
  3. [§3.2, Lemma 3.4, Claims 3–4] The regularity upgrade via 'wide but thin pancakes' and 'thin grim reaper' barriers is described largely in qualitative terms ('must quickly push in', 'would lead to a contradiction', 'one can see'). These claims are the core of the proof that the weak limit B_t is smooth; they require precise barrier constructions and quantitative estimates. As written, the argument is a sketch rather than a proof. This is a central gap because the final theorem requires a smooth ancient flow, not merely a Brakke flow.
  4. [§3.3, Proof of Theorem 1.1] The final step invokes Theorem 2.4 from [1] to conclude that the flow starting from a smooth slice B_{t_i} is non-fattening and agrees with B_t, but the applicability to the weak limit is not fully justified. In particular, the graphicality of B_{t_i} is asserted from Theorem 2.1 and the graphicality of the approximating flows; this inference for a weak limit needs a proof. The phrase 'as long as m>0' also needs to be tied to the evolution of the neck parameter m(t) and to the fact that B_t has no neck singularity after t_i.
minor comments (4)
  1. [General] There are several typographical and formatting issues: 'particulary' in the acknowledgements, 'Its easy' should be 'It is easy', and the reference to 'nvent. Math' is missing 'I' in 'Invent.'.
  2. [Captions] The captions for Figures 1 and 2 contain ambiguous notation such as 'a = 2.05, h = 1.3t = -19.5'; please clarify the parameters being displayed.
  3. [§2.1] The definition of Brakke flow is standard but the statement of Brakke compactness is informal; since the paper depends heavily on this theorem, a precise statement with the exact hypotheses (including the entropy bound) would help the reader.
  4. [References] Several references are to preprints or in-press items; please update where possible, especially [2], [8], [14], and [16].

Circularity Check

0 steps flagged · score 0.0 of 10

No circular reduction: the construction is a genuine gluing/continuity existence proof; the unproved entropy bound in Lemma 3.3 is a gap, not a circularity.

full rationale

The derivation is not circular. The old-but-not-ancient surfaces Sigma_{s_i}^{m(i)} are built from the known ancient pancakes of [3] (published, with one overlapping author) by explicit circular-arc necks and a continuity parameter m; the ancient flow B_t is obtained by Brakke compactness and then upgraded by Sturmian intersection arguments, barrier estimates, and entropy monotonicity. No target quantity is defined in terms of a fitted parameter, and the two-maxima/one-minimum profile is constructed geometrically and shown to persist, not derived from an input that already contains Theorem 1.1. The self-citations ([3], [5], [17], [18]) are used as published ingredients (for example, Theorem A.2 in [18] is invoked in Lemma 3.3 to identify the convex regime), not as a uniqueness theorem that forces the outcome, so they do not create a loop. The weakest point is Lemma 3.3's assertion 'One can check that we have the (crude) uniform entropy bound lambda(M^i_t) <= 5': this bound is the gatekeeper for Brakke compactness and is not derived. That is an omitted verification and a correctness risk, not an equivalence of input and output by construction; similarly, Lemma 3.4's regularity claims are sketched. Such gaps do not make the paper circular. Score 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The construction is a proof of existence, not an empirical fit; no free parameters are fitted. The geometric parameters like the neck width m are part of the construction and are varied to take a limit. The paper relies on prior results from the theory of rotationally symmetric curvature flows, ancient pancakes, and weak flows, listed as axioms above. No new particles or entities are invented.

assumptions (5)
  • domain assumption Existence and stated asymptotics of ancient pancake solutions in [3] (width w(t)=2π−o(1), girth g(t)=−t+(n−1)log(−t)+c_n+o(1))
    The ansatz copies two ancient pancakes at distance 1 apart (Section 3.1). If these solutions or their asymptotics did not exist, the initial data for the approximating flows would not have the required form.
  • standard math Sturmian intersection principle for rotationally symmetric flows (Theorem 2.1 in [1]), including monotonicity of critical points (Theorem 2.2) and singularity structure (Theorem 2.3)
    Used throughout to control profile intersections, to ensure the number of critical points does not increase, and to locate singularities on the axis of rotation.
  • standard math Non-fattening and properties of level-set flows for compact rotationally symmetric graphical initial data (Theorem 2.4 in [1])
    Needed to guarantee a unique well-behaved weak flow through singularities, so that profile curves of the weak flow are well-defined (Section 2.1).
  • standard math Brakke compactness and regularity; uniqueness of unit-regular cyclic mod 2 Brakke flows from a given initial surface (Brakke's theorems, Corollary G.5 in [9], Theorem 4.37 in [14])
    Used in Lemma 3.2 and Lemma 3.3 to pass from the approximating smooth flows to the ancient limit B_t and to give it structure.
  • standard math Catenoids in R^{n+1} for n≥3 are contained in a slab and have finite width
    Used in Lemma 3.4, Claim 1, to construct a catenoid barrier that intersects the approximating profile curves only twice; this is the step requiring n≥3.

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Cite this review

Pith. "Pith review of An Ancient Stacked Pancake Solution to Mean Curvature Flow." pith.science (2026). https://pith.science/paper/45W34OUI

@misc{pith2026250926515,
  author       = {Pith},
  title        = {Pith review of: An Ancient Stacked Pancake Solution to Mean Curvature Flow},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/45W34OUI}},
  note         = {Machine review of arXiv:2509.26515}
}
read the original abstract

We construct an embedded ancient solution to the mean curvature flow which is qualitatively given by a ``stack'' of two ancient pancakes joined by a neck. Our solution is closed, non-convex, and contained in a slab.

Figures

Figures reproduced from arXiv: 2509.26515 by the authors.

Figure 1
Figure 1. A “stack” of two ancient pancakes. Informally, the strict local maxima mentioned above correspond to “tips” of the pancakes and the strict local minimum corresponds to a “neck” joining them. To be more precise, the construction will be reflection symmetric across the hyper￾plane perpendicular to the rotation axis which passes through the local minimum. Throughout the article we use these symmetries, particularly the… view at source ↗
Figure 2
Figure 2. The profile curve Γ m s . Now we turn to actually obtaining a non-trivial ancient solution, worthy of the moniker “stacked pancakes”. This is by a continuity argument and towards that end we first show the following [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Behavior of (Σm s )Tm(s) for m ≪ 1 on the left and for m ≫ 1 on the right. Now we consider the smooth flows Mi t := (Σm(i) si )t with si := −10i+100, and then recenter time so that Tm(i) = 0. Note that the Mi 0 are each compact, connected, and do not bound a convex domain (in fact, are a uniform distance away from any such set in Hausdorff distance). By the choice of Ti := Tm(i) and that si → −∞ as i → ∞, the flows … view at source ↗

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Reference graph

Works this paper leans on

23 extracted references · 4 linked inside Pith

  1. [1]

    Angenent, and Yoshikazu Giga

    Steven Altschuler, Sigurd B. Angenent, and Yoshikazu Giga. Mean curvature flow through singularities for surfaces of rotation.J. Geom. Anal., 5(3):293–358, 1995

    1995

  2. [2]

    On the multiplicity one conjecture for mean curvature flows of surfaces

    Richard Bamler and Bruce Kleiner. On the multiplicity one conjecture for mean curvature flows of surfaces. Preprint, arXiv:2312.02106

    arXiv

  3. [3]

    Collapsing ancient solutions of mean curvature flow.J

    Theodora Bourni, Mat Langford, and Giuseppe Tinaglia. Collapsing ancient solutions of mean curvature flow.J. Differential Geom., 119(2):187–219, 2021

    2021

  4. [4]

    Ancient mean curva- ture flows out of polytopes.Geom

    Theodora Bourni, Mat Langford, and Giuseppe Tinaglia. Ancient mean curva- ture flows out of polytopes.Geom. Topol., 26(4):1849–1905, 2022

    1905

  5. [5]

    On the construc- tion of closed nonconvex nonsoliton ancient mean curvature flows.Int

    Theodora Bourni, Mathew Langford, and Alexander Mramor. On the construc- tion of closed nonconvex nonsoliton ancient mean curvature flows.Int. Math. Res. Not. IMRN, (1):757–768, 2021

    2021

  6. [6]

    Compact curve shortening flow solutions out of non-compact curves.Int

    Theodora Bourni and Martin Reiris. Compact curve shortening flow solutions out of non-compact curves.Int. Math. Res. Not. IMRN, (1):804–816, 2024

    2024

  7. [7]

    Gluing Eguchi-Hanson metrics and a question of Page.Comm

    Simon Brendle and Nikolaos Kapouleas. Gluing Eguchi-Hanson metrics and a question of Page.Comm. Pure Appl. Math., 70(7):1366–1401, 2017

    2017

  8. [8]

    Ancient mean curvature flows and their spacetime tracks

    Francesco Chini and Niels Martin Møller. Ancient mean curvature flows and their spacetime tracks. Preprint, arXiv:1901.05481v2

    arXiv 1901

Show all 23 references
  1. [9]

    Mean curvature flow with generic initial data.nvent

    Kyeongsu Choi, Otis Chodosh, Christos Mantoulidis, and Felix Schulze. Mean curvature flow with generic initial data.nvent. Math, 237(1):121–220, 2024. AN ANCIENT STACKED PANCAKE SOLUTION TO MEAN CUR V ATURE FLOW 21

    2024

  2. [10]

    Interior estimates for hypersurfaces moving by mean curvature.Invent

    Klaus Ecker and Gerhard Huisken. Interior estimates for hypersurfaces moving by mean curvature.Invent. Math., 105(3):547–569, 1991

    1991

  3. [11]

    Nonfattening of mean curvature flow of mean convex type.Comm

    Or Hershkovits and Brian White. Nonfattening of mean curvature flow of mean convex type.Comm. Pure Appl. Math., 73(3):558–580, 2020

    2020

  4. [12]

    Generalized flow of sets by mean curvature on a manifold.Indiana Univ

    Tom Ilmanen. Generalized flow of sets by mean curvature on a manifold.Indiana Univ. Math. J., 41(3):671–705, 1992

    1992

  5. [13]

    Elliptic regularization and partial regularity for motion by mean curvature.Mem

    Tom Ilmanen. Elliptic regularization and partial regularity for motion by mean curvature.Mem. Amer. Math. Soc., 108(520):x+90, 1994

    1994

  6. [14]

    An intersection principle for mean curvature flow

    Tang-Kai Lee and Alec Payne. An intersection principle for mean curvature flow. Preprint, arXiv:2505.11600

    arXiv

  7. [15]

    Ancient solutions to the curve shortening flow spanning the halfplane.Trans

    John Man Shun Ma. Ancient solutions to the curve shortening flow spanning the halfplane.Trans. Amer. Math. Soc., 374(6):4207–4226, 2021

    2021

  8. [16]

    A classification result for eternal mean convex flows of finite total curvature type

    Alexander Mramor. A classification result for eternal mean convex flows of finite total curvature type. Preprint, arXiv:2403.12020

  9. [17]

    Ancient and eternal solutions to mean curvature flow from minimal surfaces.Math

    Alexander Mramor and Alec Payne. Ancient and eternal solutions to mean curvature flow from minimal surfaces.Math. Ann., 380(1-2):569–591, 2021

    2021

  10. [18]

    Nonconvex surfaces which flow to round points.Comm

    Alexander Mramor and Alec Payne. Nonconvex surfaces which flow to round points.Comm. Anal. Geom., 32(3):837–887, 2024

    2024

  11. [19]

    An introduction to brakke flows

    Felix Schulze. An introduction to brakke flows. Lecture Notes. Summer School 2021: Curvature Constraints and Spaces of Metrics. Institut Fourier, Grenoble

    2021

  12. [20]

    SpringerBriefs in Mathe- matics

    Yoshihiro Tonegawa.Brakke’s mean curvature flow. SpringerBriefs in Mathe- matics. Springer, Singapore, 2019. An introduction

    2019

  13. [21]

    An embedded minimal surface with no symmetries.J

    Martin Traizet. An embedded minimal surface with no symmetries.J. Differ- ential Geom., 60(1):103–153, 2002

    2002

  14. [22]

    Asymptotic structure of self-shrinkers

    Lu Wang. Asymptotic structure of self-shrinkers. Preprint, arXiv:1610.04904

    arXiv

  15. [23]

    The avoidance principle for noncompact hypersurfaces moving by mean curvature flow.Calc

    Brian White. The avoidance principle for noncompact hypersurfaces moving by mean curvature flow.Calc. Var. Partial Differential Equations, 63(5):Paper No. 111, 20, 2024. Mathematical Sciences Institute, Australian National University, Canberra, ACT 2601, Australia Department o...

    2024

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