{"id":"e19ed8ed-f79b-4a60-8abe-d5a6c44c163c","arxiv_id":"2507.03837","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Min-max minimal spheres in infinitely many Berger 3-spheres are never connected by eternal mean curvature flow to lower-area minimal surfaces, due to a genus obstruction.","lead":"The authors show that on certain Berger 3-spheres, a minimal sphere whose area equals a min-max width cannot be connected by an eternal mean curvature flow to any smaller-area minimal surface. This reveals a topological obstruction to gradient-flow connections between minimal surfaces.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem A's construction of the p-width Berger metrics rests entirely on the external upper bound (1) from [CG25b]; if that companion estimate has a hidden failure, Proposition 1 and hence the main theorem collapse.","rationale":"After reading the proof, I find no internal inconsistency in the IVT argument itself; the signs at the endpoints are correct assuming (1). Theorem A is a short consequence of Proposition 1, the [TU09] classification of minimal spheres in Berger spheres, and the [MS24] genus obstruction. The paper is honest that (1) comes from the companion [CG25b]. The main risk is therefore external validation of that bound. The bound is quite strong in the collapsed regime, asserting ω_p(S_τ) ≤ 2π at τ = 1/(π⌊√p⌋), so it deserves independent checking before the theorem is fully accepted. Proposition 3 is also external, but it is a qualitative topological statement that is consistent with White's theorem and less likely to hide a quantitative error. Because the reader already marked the paper CONDITIONAL, my stress-test confirms that conditional status without escalating to reject or accept.","tokens_in":6610,"tokens_out":27928,"duration_ms":341635,"concrete_test":"Verify (1) at the critical left endpoint for small p by directly computing the mass of the p-sweepout of S_τ constructed in [CG25b]: take p=5, τ=1/(2π), and p=9, τ=1/(3π). Pull back the p-sweepout of the base S^2 via the Hopf fibration and compute the supremum mass; if it exceeds 2π^2τ⌊√p⌋ (i.e., 2π in these cases), the bound is false and the IVT root is not guaranteed. Also confirm from [Tor10] that A(τ) ≥ 2π at these parameter values.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The single most load-bearing step is the quoted upper bound (1), ω_p(S_τ) ≤ 2π^2 τ⌊√p⌋, taken from the companion preprint [CG25b]. The IVT argument in Proposition 1 needs, at the left endpoint τ_0 = 1/(π⌊√p⌋), the inequalities ω_p(S_{τ_0}) ≤ 2π ≤ A(τ_0), so that the continuous function ω_p - A has opposite signs at the two endpoints. Thus (1) is not a decoration; it is the only mechanism guaranteeing that the p-width can drop to the value A(τ). If the companion bound is mis-stated, or if the sweepout construction behind it only gives a weaker estimate (for instance with ⌈√p⌉ instead of ⌊√p⌋, or with a larger constant at small τ), then for some p there may be no root τ(p), and Theorem A loses its examples. Since [CG25b] is an unreviewed companion paper, this is a genuine citation dependency rather than a verified internal lemma. The topological obstruction Proposition 3, quoted from [MS24], is a second external input, but the width bound is the more quantitative and more fragile one.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":6868,"tokens_out":17266,"duration_ms":187277,"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":[{"comment":"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":"Section 2.3, Eq. (1); proof of Proposition 1"},{"comment":"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":"Section 2.2, Proposition 3; proof of Theorem A"},{"comment":"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.","section":"Section 3, Theorem A"}],"minor_comments":[{"comment":"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.","section":"Section 2.3, metric definition"},{"comment":"The running header contains typographical artifacts such as 'HYPERSURF ACES' and 'CUR V ATURE'; these should be cleaned up in the final version.","section":"Title and running header"},{"comment":"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.","section":"Proof of Proposition 1"}],"recommendation":"major_revision","confidential_remarks":"The paper is plausible and the mathematical argument is coherent once the two external inputs are accepted. However, the central result depends on the authors' own unpublished companion [CG25b] for the width bound (1), and also on [MS24] for Proposition 3. I would advise the editor to ask the authors to either include proofs of these inputs or confirm that both references have been accepted for publication, and to make the exact statements available in the present paper."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear Colleague,\n\nRead 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.\n\nThe 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.\n\nThe 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.\n\nThere 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.\n\nFor 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.\n\nWarmly, [your name]","headline":"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.","tokens_in":7353,"tokens_out":2946,"would_cite":true,"duration_ms":33097,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53C42","53E10","49Q20"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["mean curvature flow","minimal surfaces","Berger spheres","volume spectrum","min-max widths","Brakke flow","positive Ricci curvature","topological obstructions"],"falsifier":"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.","tokens_in":6426,"feed_emoji":"🌊","tokens_out":11806,"duration_ms":120324,"temperature":0.7,"pith_summary":"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.","feed_headline":"Some minimal spheres cannot flow to any smaller minimal surface","feed_subtitle":"In these positively curved 3-spheres, a width-minimizing sphere can never flow to a lower-area minimal surface.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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 $1/(\\pi\\lfloor\\sqrt{p}\\rfloor)$."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the upper bound $\\omega_p(S_\\tau)\\le 2\\pi^2\\tau\\lfloor\\sqrt{p}\\rfloor$ that seeds the intermediate-value argument.","marker":"[CG25b]"},{"why":"Provides the round-sphere lower bound $\\omega_p(S^3)\\ge 2\\pi^2$, the other endpoint of the intermediate value step.","marker":"[MN14]"},{"why":"Gives continuity of widths under smooth metric variation, making the intermediate value argument legitimate.","marker":"[MNS19]"},{"why":"Supplies the explicit area formula $A(\\tau)$ for the equatorial sphere in Berger metrics.","marker":"[Tor10]"},{"why":"Shows the equatorial spheres are the only embedded minimal spheres in $S_\\tau$, so lower-area minimal surfaces have positive genus.","marker":"[TU09]"},{"why":"Establishes the topology-simplification principle for hypersurfaces moving by mean curvature that underlies the genus obstruction.","marker":"[Whi95]"},{"why":"States the closed-3-manifold version: an eternal Brakke flow with a multiplicity-one sphere backward limit has genus-zero forward limit.","marker":"[MS24]"}],"fun_headline_variants":["Minimal spheres that never flow downward","Infinite family of minimal surfaces with no connecting flow","Positive Ricci spheres: no mean curvature bridges between widths","No eternal flow links these minimal surfaces"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Minimal spheres that never flow downward","Infinite family of minimal surfaces with no connecting flow","Positive Ricci spheres: no mean curvature bridges between widths","No eternal flow links these minimal surfaces"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000189,"raw_usage":{"total_tokens":1277,"prompt_tokens":826,"completion_tokens":451,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":442,"completion_tokens_details":{"reasoning_tokens":394}},"tokens_in":442,"tokens_out":451,"duration_ms":5943,"temperature":1.0,"reasoning_tokens":394,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T20:03:09.646698+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}