When can minimal hypersurfaces be connected by mean curvature flow?

Jingwen Chen; Pedro Gaspar

REVIEW 3 major objections 3 minor 26 references

When can minimal hypersurfaces be connected by mean curvature flow?

T0 review · 3 major / 3 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper proves that in infinitely many positively curved 3-spheres, the minimal sphere achieving the p-th width cannot be connected by mean curvature flow to any lower-area minimal surface.

desk verdict A clean short note that proves new topological obstructions to connecting mean curvature flows in Berger spheres, but the main theorem leans on an unproved width bound from the authors' companion preprint. read the letter →

arxiv 2507.03837 v1 pith:7KETAYIK submitted 2025-07-04 math.DG

Jingwen Chen , Pedro Gaspar This is my paper · ORCID

classification math.DG MSC 53C4253E1049Q20

keywords meancurvatureflowminimalsurfacesBergerspheresvolumespectrummin-maxwidthsBrakkepositiveRiccitopologicalobstructions

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 asks when one minimal hypersurface can be connected to another by mean curvature flow, the geometric analogue of a gradient flow line between two critical points of the area functional. It proves that in infinitely many positively curved Riemannian metrics on the 3-sphere (Berger metrics), there is a minimal sphere whose area equals the $p$-th min-max width, yet no eternal (weak) mean curvature flow can have that sphere as its backward limit and a lower-area minimal surface as its forward limit. The obstruction is topological: along a mean curvature flow with a multiplicity-one sphere as the backward limit, the topology of the surface cannot get more complicated, so a forward limit of genus zero is impossible; the lower-area minimal surfaces in these examples all have positive genus. The paper also shows that for some Berger spheres several low $p$-widths coincide with the area of the equatorial sphere, and produces indefinitely many such coincidences.

What carries the argument

The machinery is the interaction between min-max widths and the geometry of Berger spheres. The named objects are the $p$-width $\omega_p(S_\tau)$ (the infimum, over $p$-parameter families of cycles sweeping out the manifold, of the maximum mass of a slice) and the equatorial sphere $S^2\subset S_\tau$ with explicit area $A(\tau)$. The proof uses the upper bound $\omega_p(S_\tau)\le 2\pi^2\tau\lfloor\sqrt{p}\rfloor$, the continuity of widths under smooth metric variation, and the explicit values of $A(\tau)$, together with the topological principle that an eternal Brakke flow in a closed 3-manifold whose backward limit is a multiplicity-one sphere must have forward limit of genus zero.

What would settle it

Find a Berger parameter $\tau$ and an eternal Brakke flow in $S_\tau$ whose backward limit is the multiplicity-one equatorial sphere and whose forward limit is a minimal surface of genus at least one; this directly contradicts Theorem A. Alternatively, compute a single width $\omega_p(S_\tau)$ that exceeds $2\pi^2\tau\lfloor\sqrt{p}\rfloor$, breaking the bound (1) on which the argument depends.

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Extended reading notes

Core claim

On the authors' own terms, the central discovery is Theorem A: there exist infinitely many Riemannian 3-spheres $(S^3,g)$ with positive Ricci curvature, each containing a minimal surface $\Sigma_p$ (the equatorial sphere $S^2$) whose area equals the $p$-width $\omega_p(S^3,g)$ for some $p\ge 1$, and with the property that for every minimal surface $\Sigma_q$ of smaller area there is no eternal Brakke flow $\{\Sigma_t\}_{t\in(-\infty,\infty)}$ satisfying $\Sigma_t\to\Sigma_q$ as $t\to+\infty$ and $\Sigma_t\to\Sigma_p$ as $t\to-\infty$ in the varifold sense. The proof selects a Berger parameter $\tau=\tau(p)$ by an intermediate value argument: at small $\tau$ the $p$-width is bounded above by the equatorial area, while at $\tau=1$ (round sphere) the $p$-width exceeds it. Since the only minimal spheres in a Berger sphere are the equatorial ones, any lower-area minimal surface has genus at least one, and the topology-simplification principle for eternal Brakke flows rules out a genus-one forward limit from a multiplicity-one sphere.

Load-bearing premise

The proof rests on the companion upper bound $\omega_p(S_\tau)\le 2\pi^2\tau\lfloor\sqrt{p}\rfloor$; if that bound fails for some $p$, the intermediate-value construction of $\tau(p)$ with $A(\tau(p))=\omega_p(S_{\tau(p)})$ fails, and Theorem A collapses. It also assumes the quoted topological simplification principle holds for eternal Brakke flows in closed 3-manifolds.

Editorial extensions

If this is right

In each of the infinitely many Berger spheres produced by Theorem A, the $p$-width is achieved by the equatorial sphere, which has Morse index 1, so the generic expectation that the $p$-width is realized by a minimal surface of index $p$ fails for these metrics.
Any eternal mean curvature flow in these examples with the equatorial sphere as backward limit cannot settle on a lower-area minimal surface; the only possible forward limits of lower area are excluded by genus.
Proposition 2 exhibits a single Berger sphere for which $\operatorname{area}_{S_\tau}(S^2)=\omega_p(S_\tau)>\omega_1(S_\tau)$ for some $p>1$, and Corollary 1 iterates this to produce an increasing sequence $q_i$ and decreasing $\tau_{q_i}$ with the first $q_i$ widths all equal to the equatorial area.
Remark 1 keeps open the possibility that a different minimal surface realizing the same $p$-width (constructed by approximation arguments) could be connected to a lower-index surface; the conjecture is not disproved.

Reading between the lines

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

The obstruction is likely not special to Berger spheres: any closed 3-manifold containing a multiplicity-one minimal sphere that realizes a width, with all lower-area minimal surfaces having positive genus, would exhibit the same no-flow conclusion.
Because the equality $A(\tau(p))=\omega_p(S_{\tau(p)})$ is obtained by a continuity and intermediate-value argument rather than by explicit computation, the set of $p$ for which such a coincidence happens could be much larger than the constructed sequence; numerical computation of low widths of Berger spheres could test this.
The collapse of several consecutive widths to the same value resembles a degenerate Morse function with repeated critical values, suggesting the min-max spectrum of Berger spheres is far from the generic simple spectrum; small metric perturbations might split these widths and restore connecting flows.
If the companion upper bound is sharpened, the same framework would pin down exactly which $p$ admit $\tau(p)$ and could yield estimates on how close $\tau(p)$ is to $1/(\pi\lfloor\sqrt{p}\rfloor)$.

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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

Dear Colleague,

Read this one if you care about the connecting-flows program for minimal hypersurfaces. The paper shows there are infinitely many Berger 3-spheres where the equatorial sphere area equals a p-width, and then uses a topological theorem to rule out any eternal Brakke flow connecting that sphere to a smaller-area minimal surface. The new ingredient is the observation that a width upper bound from the authors' companion paper [CG25b] can be fed into a continuity argument to force these coincidences.

The paper does well what it sets out to do. The proof of Proposition 1 is a clean IVT: at small tau the width bound forces omega_p <= A(tau), at tau=1 the Willmore result gives omega_p > A(1). The topological part is standard but the application is new. I also appreciate that the authors state explicitly that Theorem A is not a counterexample to the Morse-theoretic conjecture, since other min-max surfaces might still connect. That is honest.

The soft spot is exactly what the reader flagged: the whole construction rests on the external upper bound (1), omega_p(S_tau) <= 2 pi^2 tau floor(sqrt(p)), proved only in the unreviewed companion preprint. If that bound has a hidden failure, Proposition 1 collapses and Theorem A loses its examples. The topological Proposition 3 from [MS24] is a second dependency, but that one sits on published work and is less fragile. This is a genuine citation dependency, not a disguised circularity. The authors are upfront about it, and in the current culture of geometric analysis this is normal. But a referee must check [CG25b] carefully.

There are a couple of minor points. The proof of Proposition 2 is a bit dense, and the boundedness of f relies on unboundedness of widths from the Weyl law, which is fine but would benefit from a sentence saying why the contradiction is uniform. The second proof is cleaner. Corollary 1's iteration is okay.

For the right reader--a specialist in min-max theory or mean curvature flow--this is a worthwhile short note. It is not groundbreaking, but it fills a real gap in the program of connecting critical points. I would send it to peer review: the result is conditional on a companion preprint, but that is the kind of dependency referees can check. My verdict matches yours: conditional, with the caveat being the [CG25b] bound.

Warmly, [your name]

Referee Report

3 major / 3 minor

Summary. The paper studies whether minimal hypersurfaces in a closed 3-manifold can be connected by an eternal mean curvature flow. Building on the volume spectrum of Berger spheres, the authors prove (Proposition 1) that for every integer p there exists a Berger metric S_{τ(p)} on S^3 for which the equatorial sphere has area equal to the p-width. They then use a topological obstruction for eternal Brakke flows (Proposition 3, quoted from [MS24]) to show (Theorem A) that for such a metric, no eternal Brakke flow can connect a lower-area minimal surface to the equatorial sphere. This yields infinitely many positive-Ricci metrics on S^3 in which the Morse-theoretic expectation of connecting flows fails. The paper also proves (Proposition 2 and Corollary 1) that there are Berger spheres with prescribed coincidences of the first several min-max widths. The argument is short and mostly an application of known results; the main novelty is the explicit construction of the metrics via an intermediate value argument.

Significance. If the external inputs are accepted, the main theorem is a clean and explicit illustration of how topology can obstruct the existence of minimizing mean curvature flow trajectories between minimal hypersurfaces. The IVT construction is elegant, and the paper correctly identifies the role of White's topology theorem and the classification of minimal spheres in Berger spheres. The paper gives concrete examples rather than a new general theorem, and it is a useful contribution to the Morse-theoretic study of the area functional. The main caveat is that the central construction is conditional on two external sources: the upper bound (1) from the authors' companion preprint [CG25b] and Proposition 3 from the preprint [MS24]. Both are load-bearing and are not proved in the manuscript. The authors should be credited for a clear reduction of the problem to known min-max and flow results, but the current presentation leaves the reader unable to fully verify the main claim from the text alone.

major comments (3)

[Section 2.3, Eq. (1); proof of Proposition 1] The upper bound ω_p(S_τ) ≤ 2π^2 τ⌊√p⌋ is stated without proof and attributed to the companion preprint [CG25b]. This bound is load-bearing: the intermediate value argument in Proposition 1 requires the inequality at the left endpoint τ_0 = 1/(π⌊√p⌋), and if the bound were false or had a weaker form (e.g., with a different constant or a ceiling instead of the floor), the root τ(p) might not exist and Theorem A would lose its examples. The authors should either include a proof of (1) in the present paper or reproduce the precise statement from [CG25b] and clarify its publication status. As written, the main theorem is conditional on an unreviewed companion paper.
[Section 2.2, Proposition 3; proof of Theorem A] The topological obstruction used to conclude nonexistence of eternal Brakke flows is quoted from the preprint [MS24] (Proposition 3.7 therein) and is not proved in this paper. Since Theorem A is essentially the contrapositive of this proposition combined with the classification of minimal spheres in Berger spheres, the main nonexistence result is not self-contained. Please provide a proof or a detailed derivation from the cited results of White, Brendle, Choi–Haslhofer–Hershkovits–White, and Bamler–Kleiner, or at minimum state the exact version used as a lemma with a proof sketch. The current presentation makes the central claim dependent on an unpublished source.
[Section 3, Theorem A] The statement that there are 'infinitely many' Riemannian spheres with the stated property is not explicitly justified in the proof. Since a τ(p) is produced for each p, one must argue that the metrics S_{τ(p)} are distinct. This follows because for any fixed τ>0 the Weyl law gives ω_p(S_τ) ~ c p^{1/3}, so eventually ω_p(S_τ) > A(τ); hence the root τ(p) must tend to 0 as p→∞. The authors should add a sentence making this point clear.

minor comments (3)

[Section 2.3, metric definition] In the displayed definition of the Berger metric, the expression '⟨u, v⟩' should presumably be '⟨v, w⟩' to be symmetric in the two tangent vectors; as printed, the formula is not symmetric.
[Title and running header] The running header contains typographical artifacts such as 'HYPERSURF ACES' and 'CUR V ATURE'; these should be cleaned up in the final version.
[Proof of Proposition 1] The equality ω_5(S_1) = 2π^2 is used without an explicit reference to the precise statement in [MN14] or [CG25a]; since this is not entirely immediate from the Willmore conjecture alone, a citation or a brief explanation would help the reader.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Proposition 1 is an intermediate-value construction from independent width bounds, and Theorem A is an external topological consequence; no fitted parameter or definitional identity is passed off as a prediction.

full rationale

I walked the derivation chain of Proposition 1, Theorem A, Proposition 2, and Corollary 1. Proposition 1 finds tau(p) by the intermediate value theorem: at tau = 1 the p-width exceeds A(1), and for tau <= 1/(pi * floor(sqrt(p))) the quoted upper bound (1) forces omega_p <= 2pi <= A(tau); continuity of both functions then gives a root. The inputs are the companion estimate omega_p(S_tau) <= 2pi^2 tau floor(sqrt(p)) from [CG25b], the explicit area A(tau) from [Tor10], the Willmore-based lower bound from [MN14], and continuity from [MNS19]. None of these assumes the equality being concluded; the equality is an output of a sign change, not a fitted parameter or a definitional identity. Theorem A uses only this equality together with Proposition 3 from [MS24] (itself based on [Whi95, Bre16, CHHW22, BK23]) and the classification of minimal spheres in Berger spheres from [TU09]; both are external inputs. The repeated use of [CG25b] in Proposition 2 and Corollary 1 is a citation dependency: if that bound were false the constructions would fail. That is a correctness risk, not circularity, because the cited bound is a parameter-free estimate whose statement and assumptions lie outside the target result; there is no exhibited reduction of Theorem A to its own assumptions. No self-definitional step, renamed known result, or imported uniqueness theorem occurs. Hence the circularity score is 0, with the caveat that the companion-preprint dependency is the main fragility of the paper.

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

The paper introduces no free parameters or new entities. The central claims are deduced from standard min-max theorems plus a small number of external results, several from preprints by the same authors or collaborators, which are not reproduced here.

assumptions (7)

standard math Weyl law for the volume spectrum (Theorem 1, [LMN18])
Used to show that ω_p(S_τ) is unbounded in p, necessary for the contradiction in Proposition 2 and for Corollary 1.
standard math Continuity of p-widths under smooth changes of the Riemannian metric ([MNS19])
Used in the IVT argument in Proposition 1 to find τ(p) with ω_p(S_τ(p)) = A(τ(p)).
domain assumption Upper bound ω_p(S_τ) ≤ 2π^2 τ⌊√p⌋ ([CG25b], Eq. (1))
Load-bearing for Proposition 1: ensures ω_p drops below A(τ) at small τ. This bound is from the authors' own companion preprint and is not proved here.
domain assumption Topological simplification for eternal Brakke flows (Proposition 3, after [MS24])
Underpins Theorem A: a backward sphere of multiplicity 1 forces the forward limit to have genus 0. Proof is in [MS24] and ingredients include [Whi95, Bre16, CHHW22, BK23].
domain assumption Uniqueness of embedded minimal spheres in Berger spheres ([TU09])
Ensures any minimal surface with area < A(τ) has genus ≥1, so Proposition 3 obstructs connecting flows. Attributed to [TU09], a preprint.
standard math Explicit area A(τ) and minimality of equatorial spheres ([Tor10])
Provides the target area A(τ) = area_{S_τ}(S^2) and its continuity; used throughout the paper.
standard math Known round sphere widths: ω_1..ω_4 = 4π and ω_5 = 2π^2 ([MN14, CM23])
Sets the endpoint comparison at τ=1: ω_p(S_1) > A(1) for p≥5.

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Cite this review

Pith. "Pith review of When can minimal hypersurfaces be connected by mean curvature flow?." pith.science (2026). https://pith.science/paper/7KETAYIK

@misc{pith2026250703837,
  author       = {Pith},
  title        = {Pith review of: When can minimal hypersurfaces be connected by mean curvature flow?},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7KETAYIK}},
  note         = {Machine review of arXiv:2507.03837}
}

read the original abstract

From the perspective of Morse theory, it is natural to investigate gradient flow trajectories between critical points. In this short note, we explore the minimal hypersurface analogue of this phenomenon and present examples that suggest additional topological and variational obstructions to the existence of connecting mean curvature flows.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

26 extracted references · 25 canonical work pages

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

Richard H Bamler and Bruce Kleiner, On the multiplicity one conjecture for mean curvature flows of surfaces, arXiv preprint arXiv:2312.02106 (2023)

arXiv 2023

[2] [2]

Brakke, The motion of a surface by its mean curvature, Mathematical Notes, vol

Kenneth A. Brakke, The motion of a surface by its mean curvature, Mathematical Notes, vol. 20, Princeton University Press, Princeton, N.J., 1978

work page 1978

[3] [3]

Simon Brendle, Embedded self-similar shrinkers of genus 0, Ann. of Math. (2) 183 (2016), no. 2, 715--728. 3450486

work page 2016

[4] [4]

Jingwen Chen and Pedro Gaspar, Mean curvature flow and low energy solutions of the parabolic A llen- C ahn equation on the three-sphere , J. Geom. Anal. 33 (2023), no. 9, Paper No. 283, 30. 4605575

work page 2023

[5] [5]

, Morse theory for the A llen- C ahn functional , J. Funct. Anal. 288 (2025), no. 9, Paper No. 110876, 30. 4864882

work page 2025

[6] [6]

, Volume spectrum of fiber bundles and the widths of B erger spheres , arXiv preprint arXiv:2505.09548 (2025)

work page Pith review arXiv 2025

[7] [7]

239 (2024), Paper No

Jingwen Chen, Symmetric mean curvature flow on the n -sphere , Nonlinear Anal. 239 (2024), Paper No. 113437, 13. 4663465

work page 2024

[8] [8]

Kyeongsu Choi, Robert Haslhofer, Or Hershkovits, and Brian White, Ancient asymptotically cylindrical flows and applications, Invent. Math. 229 (2022), no. 1, 139--241. 4438354

work page 2022