{"id":"51f609cf-a599-4f5c-8218-714cc5d37d5e","arxiv_id":"2608.13531","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"A stationary minimal surface with a mildly flat singular point admits a non-trivial Brakke flow, so Brakke flow starting from it is non-unique.","lead":"This paper proves that certain singular minimal surfaces can start a genuinely moving mean-curvature flow, not just remain static. It shows Brakke flows are non-unique from such initial data under a mild flatness condition at a singular point.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 4.1's slab bound (4.2) is asserted from Allard's estimate although the density at the origin exceeds 1; no quantitative height bound is supplied for this regime.","rationale":"The reader's weakest_assumption is exactly the step I find most load-bearing: the slab estimate (4.2). The rest of the paper is a careful adaptation of [ST25], and I could not locate a separate fatal flaw; the Tε=∞ issue in Lemma 7.1 does not damage the proof because Lemma 7.3 uses equality at Tε only after assuming Tε→0, and if Tε=∞ the desired limsup conclusion is trivial. The abstract's omission of Assumption 3.1 is a presentation issue, not a correctness issue. The real problem is that (4.2) is asserted rather than proved: the cited Allard estimate normally requires density close to 1, which fails at the singular point. A quantitative height estimate may be obtainable from the monotonicity formula and the standard density lower bound Θ≥1 at points of the support, so the claim is plausible; but the paper does not provide it. Since the entire hole construction rests on (4.2), the paper should be accepted only after this step is justified. I therefore keep the reader's CONDITIONAL verdict.","tokens_in":15465,"tokens_out":31921,"duration_ms":366667,"concrete_test":"Read the exact statement of [All72, 7.5(2),(6)] and check whether it assumes the density ratio ||V||(U_r)/(ω_n r^n) is within 1+ε of unity. If it does, the derivation of (4.2) is invalid at the singular point. Then attempt a direct proof of (4.2): for x∈Γ0∩U_{2ε}, use stationarity to obtain ||V0||(B_{|x^⊥|/2}(x)) ≥ ω_n(|x^⊥|/2)^n, combine this with (3.1) over U_{4ε} to get |x^⊥| ≤ C(n) μ^{2/(n+2)} ε, and check whether μ0 can be chosen so this is ≤ ε/20. If this direct estimate succeeds, the gap is cosmetic; if it fails, Lemma 4.1 is unproven and the conditional acceptance should be withdrawn.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The hole construction in Lemma 4.1 depends on the pointwise bound Γ0∩U_{2ε}⊂{|x_{n+1}|≤ε/20}, claimed from [All72, 7.5(2),(6)] and Assumption 3.2. Assumption 3.2 only provides the scale-invariant L2 closeness (3.1) and Θ_n(||V0||,0)>1. Allard's height/regularity estimate, in the form usually quoted, requires the mass ratio in the ball to be close to 1; the origin has density >1, so the theorem does not apply there, and no argument is given for nearby points. L2 closeness alone does not imply Hausdorff or pointwise closeness: e.g., two n-planes meeting at angle θ have L2 excess c_n sin^2θ but maximum height ε sinθ at radius ε. For any θ>1/20 the slab (4.2) fails; whether such a θ is compatible with μ<μ1 depends on the unstated constants. Even if the two-plane example is ruled out by a small Allard constant, the proof must still show a quantitative inequality |x^⊥|≤C μ |x| for all x near 0. This is not derived; without it, the modified surfaces lose the uniform bounds (4.1) and (4.4), and the rest of the non-uniqueness proof has no starting point.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves that a stationary multiplicity-one varifold associated to a closed countably n-rectifiable set Γ0 is dynamically unstable (and hence has non-unique Brakke flows) provided that the density at a singular point exceeds 1 and that the scale-invariant L2 distance of Γ0 from some n-plane has sufficiently small limsup at the origin. The proof follows the framework of Stuvard and Tonegawa: one constructs modified initial surfaces with a hole, obtains Brakke flows starting from them, uses Brakke's expanding holes lemma and Huisken's monotonicity formula to get uniform density and L2-excess estimates, and then runs a blow-up argument. The new ingredient is an integrality-based density-gap lemma that prevents the density ratio from crossing a threshold near the initial time, removing the logarithmic decay condition from [ST25] and avoiding any uniqueness assumption on tangent cones.","tokens_in":15702,"tokens_out":22451,"duration_ms":223681,"significance":"If the gap described below is repaired, the result constitutes a substantial advance: it broadens the class of singularities known to be dynamically unstable, removes the logarithmic decay assumption of [ST25], and applies to several well-studied classes (two-valued Lipschitz graphs, stable codimension-one integral varifolds, area-minimizing mod p hypersurfaces). The integrality argument in Lemma 7.2 is elegant and the overall structure of the proof is transparent. The paper is generally well written and clearly exposes its reliance on external results. However, the present version contains a load-bearing unproved assertion in Lemma 4.1, namely the pointwise slab estimate (4.2), and this must be addressed before the theorem can be considered established.","major_comments":[{"comment":"The pointwise slab bound Γ0∩U_{2ε}⊂{|x_{n+1}|≤ε/20} is asserted directly from [All72, 7.5(2),(6)] and Assumption 3.2. Allard's height estimate in that form is a regularity theorem that applies when the mass ratio is close to 1, whereas Assumption 3.2 permits the density at the origin to be any value in (1,Θ0]. The L2 excess condition (3.1) alone does not imply a pointwise or Hausdorff closeness statement of the type (4.2); for example, two unit-density planes crossing at angle θ have L2 excess comparable to sin^2 θ while their maximum height at radius ε is ε sin θ. The author needs to prove a quantitative bound of the form |x^⊥| ≤ C μ |x| near 0, or supply a precise citation that covers the density >1 case, because (4.2) is used to obtain the uniform estimates (4.1), (4.4), and property (5) of Lemma 4.1, on which all subsequent sections rely.","section":"Lemma 4.1, Eq. (4.2)"}],"minor_comments":[{"comment":"In the statement of Lemma 7.2, equation (7.6) is written with d∥V_t∥ but should be d∥V∥; the lemma concerns a single varifold V, not a flow.","section":"Lemma 7.2, Eq. (7.6)"},{"comment":"In the displayed Brakke inequality (2.4), the integrand should be parenthesized as (−ϕ(x,t)h(V_t,x)+∇ϕ(x,t))·h(V_t,x)+∂_tϕ(x,t) for readability and to avoid ambiguity in the sign of the ∂_t term.","section":"Definition 2.1(4)"},{"comment":"Proposition 4.3 is stated before the uniform mass estimates of Section 6 are available; the compactness argument is only cited from [ST25, Proposition 4.3]. The author should either move this proposition after Section 6 or explicitly indicate that the needed uniform mass bounds follow from Proposition 4.2 together with Lemma 4.1.","section":"Proposition 4.3"},{"comment":"The approximation argument that produces a compactly supported test function ψ from 1−Φ_0(·,s_0) is only sketched. A short justification using monotone convergence with càdlàg cutoffs would make the argument complete.","section":"Proof of Theorem 3.4, final paragraph"},{"comment":"When the rescaled test function Φ_{T_ε^{-1/2} ε_j} first appears, it has not been defined; the author should state explicitly that it is the function Φ_λ from Section 5 evaluated with λ = T_ε^{-1/2} ε_j.","section":"Lemma 7.3"}],"recommendation":"major_revision","confidential_remarks":"The essential issue is the justification of (4.2) in Lemma 4.1. The rest of the proof is coherent and the new integrality argument is a genuine contribution. If a correct height bound for stationary varifolds with density >1 can be supplied, or if the assumptions are modified to include a pointwise flatness condition, the paper would be suitable for publication. The current gap is serious enough that the theorem is not yet established."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Let me tell you what I think of Motegi's paper.\n\nThe new thing here is real. The paper takes Stuvard–Tonegawa's hole-nucleation strategy for dynamical instability and pushes it past two restrictions that were clearly artificial: the logarithmic convergence rate to a flat tangent cone, and tangent-cone uniqueness. The key novelty is the integrality argument in Lemmas 7.2–7.3. The idea is to use the fact that a stationary integral varifold with small L2 distance to a plane has density close to an integer, so it cannot cross the threshold 1+δ at arbitrarily small times. That is a genuinely nice piece of reasoning, and it is what removes the log-decay assumption. The paper is also well organized, and the estimates in Sections 5 and 6 are adapted carefully from the standard toolkit.\n\nThe main soft spot is the pointwise slab estimate (4.2) in Lemma 4.1. The proof just asserts that Allard's [All72, 7.5] gives Γ0 ∩ U_2ε ⊂ {|x_{n+1}| ≤ ε/20} under Assumption 3.2. The trouble is that Allard's height/regularity estimate normally requires the mass ratio in the ball to be close to 1, and here the density at the origin is >1. L2 closeness does not by itself give pointwise closeness; a small set of high points could still have small L2 excess. This might be fixable, e.g., via a monotonicity argument: a point at height h would carry mass at least c h^n, which would violate the L2 bound for μ small. But the author does not supply that argument, and (4.2) is load-bearing for the hole construction. As it stands, this is a genuine gap that a referee should close before acceptance.\n\nSmaller issues: Lemma 7.1 states ||V^ε_{T_ε}||(Φ_ε(·,T_ε)) = αc, but if T_ε = ∞ the value at T_ε is undefined. Not a big deal, because the contradiction in Lemma 7.3 only runs when T_ε is finite, but the lemma as written overreaches. The abstract also omits Assumption 3.1 (the Caccioppoli partition and boundary conditions), so the advertised statement is slightly stronger than Theorem 3.4.\n\nOverall, the proof is coherent once you grant (4.2), and the gap looks bridgeable rather than fatal. This is a significant paper for the GMT community and deserves a careful referee. I would send it to review and ask for a justification of (4.2), either a proof or a precise citation covering the density >1 case.","headline":"A substantial generalization of Stuvard–Tonegawa with a fresh integrality argument, but the proof of the key slab estimate (4.2) is asserted rather than derived.","tokens_in":16283,"tokens_out":11466,"would_cite":true,"duration_ms":114833,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["49Q15","53E10","35K55"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that a stationary minimal hypersurface with a singular point of density greater than 1 and sufficiently small scale-invariant $L^2$ distance to a plane admits a genuinely time-dependent Brakke flow.","keywords":["Brakke flow","mean curvature flow","dynamical instability","stationary varifold","non-uniqueness","minimal surface","singularity","tangent cone"],"falsifier":"Check whether the slab estimate (4.2) holds for a concrete stationary integral cone with density $3/2$ that satisfies the $L^2$-excess condition (3.1), such as three half-planes meeting along a common axis at small angles: compute the Hausdorff distance from the plane $T$ over $U_{2\\varepsilon}$. If for arbitrarily small $\\varepsilon$ the cone is not contained in $\\{|x_{n+1}|\\le \\varepsilon/20\\}$ despite having small $L^2$ distance, then Lemma 4.1 cannot supply the uniform excess and density bounds (4.1) and (4.4), and the proof of Theorem 3.4 collapses.","tokens_in":2052,"feed_emoji":"🌀","tokens_out":5567,"duration_ms":113602,"temperature":0.7,"pith_summary":"The paper establishes that a stationary minimal hypersurface $\\Gamma_0$ with a singular point where its density exceeds 1 and its scale-invariant $L^2$ distance to some plane has sufficiently small limsup is dynamically unstable in Brakke's weak sense. Dynamical instability means there exists a non-trivial Brakke flow—one whose mass is strictly less than the initial mass for every $t>0$—starting from $\\Gamma_0$. Since $\\Gamma_0$ is stationary, the constant flow is already a Brakke flow, so this gives non-uniqueness of weak mean curvature flows from the same initial datum. The proof removes a logarithmic-decay assumption on tangent cones used in earlier work and does not require the tangent cone at the singularity to be unique.","feed_headline":"One singular point makes mean curvature flow non-unique","feed_subtitle":"Stationary minimal surfaces with density above 1 admit a genuinely moving Brakke flow, so the static flow is not unique.","key_machinery":"The argument runs through a hole-nucleation construction: a Lipschitz map $g_\\varepsilon$ modifies $\\Gamma_0$ inside a tiny ball, producing a new surface $\\Gamma^\\varepsilon_0$ that agrees with $\\Gamma_0$ outside $U_{2\\varepsilon}$ and has a hole at the origin. Brakke's expanding holes lemma, with a Gaussian test function $\\Phi_\\lambda$, bounds the weighted area of the flow at later times by its initial value plus an error term controlled by the scale-invariant $L^2$ excess from $T$. Huisken's monotonicity formula supplies uniform-in-time density and excess estimates for the approximating flows. The decisive new ingredient is integrality: in a blow-up limit, a varifold supported on the plane $T$ with bounded energy must have constant integer density, so its density ratio is forced to be $0$ or $1$; this contradicts the level-crossing at $1+\\delta$ that would occur if the first hitting time $T_\\varepsilon$ collapsed to zero. The integrality of varifolds is what replaces the logarithmic-decay assumption of earlier work.","core_discovery":"The central claim is Theorem 3.4: for any $\\Theta_0>1$ there is a constant $\\mu_0=\\mu_0(n,\\Theta_0)\\in(0,1)$ such that if $\\Gamma_0$ is a closed countably $n$-rectifiable set whose associated multiplicity-one varifold is stationary, whose density at the origin belongs to $(1,\\Theta_0]$, and which satisfies the small-$L^2$-excess condition $\\limsup_{R\\to 0} R^{-n-2}\\int_{\\Gamma_0\\cap U_R} |T^\\perp x|^2\\,d\\mathcal{H}^n(x)\\le \\mu_0^2$ for some $n$-plane $T$, then there exists a Brakke flow $\\{V_t\\}_{t\\ge 0}$ with fixed boundary $\\partial\\Gamma_0$ such that $\\lim_{t\\downarrow 0}\\|V_t\\|=\\|V_0\\|=\\mathcal{H}^n\\lfloor \\Gamma_0$ and $\\|V_t\\|(U)<\\|V_0\\|(U)$ for all $t>0$. The genuinely time-dependent flow is obtained as a limit of Brakke flows starting from modified surfaces with a small hole cut at the singular point, and the strict mass loss shows that the static flow is not the only weak evolution.","pith_inferences":["Beyond the paper: the proof's waiting-time estimate $T_\\varepsilon\\ge \\delta_0\\varepsilon^2$ and the blow-up contradiction suggest a quantitative lower bound on the rate of mass loss near $t=0$; one could try to extract an explicit rate from the constants in Lemmas 7.1 and 7.3.","Beyond the paper: the level-crossing argument uses integrality to rule out density ratios strictly between $0$ and $1$; a similar argument with thresholds $kc$ for integer $k$ might probe singularities of higher density where the limit density could jump across integer levels.","Beyond the paper: the slab estimate (4.2) is the only step that imports pointwise closeness from $L^2$ flatness; an $L^2$-only hole construction would remove the most delicate assumption and likely cover unions of half-planes at small angles without invoking a classical regularity theorem."],"forward_implications":["If the theorem is correct, uniqueness of Brakke flows fails for every stationary hypersurface satisfying Assumption 3.2: the static flow is never the only weak evolution from such an initial datum.","The logarithmic decay condition on tangent cones from previous work becomes unnecessary; only a small limsup of the $L^2$ excess at the singular point is needed.","The result applies to broad natural classes—stationary two-valued Lipschitz graphs, stable codimension-one integral varifolds, and area-minimizing mod-$p$ hypersurfaces—whenever existing tangent-cone uniqueness results place them in Assumption 3.2.","The constructed flow loses mass immediately: $\\|V_t\\|(U)<\\|V_0\\|(U)$ for every $t>0$, so the singularity does not merely delay motion but initiates it right away."],"supporting_citations":[{"why":"Supplies the overall hole-nucleation strategy, the expanding-holes framework, and the earlier dynamical-instability result that this paper improves.","marker":"[ST25]"},{"why":"Provides the existence theorem for Brakke flows with fixed boundary from the modified surfaces $\\Gamma^\\varepsilon_0$ and the initial-time continuity used in Proposition 4.2.","marker":"[ST21]"},{"why":"Defines Brakke flow and contains the original expanding holes lemma that Lemma 5.1 modifies.","marker":"[Bra78]"},{"why":"Gives Allard's compactness theorem and the regularity/height estimates used for the slab estimate (4.2) in Lemma 4.1.","marker":"[All72]"},{"why":"Supplies the monotonicity formula and density existence for stationary varifolds used throughout the paper.","marker":"[Sim83]"},{"why":"Provides the Huisken-type monotonicity formula propositions adapted in Proposition 6.1 to control density and $L^2$ excess uniformly in time.","marker":"[KT14]"},{"why":"Supplies the compactness theorem for Brakke flows used to pass from the approximating flows to a limit flow as $\\varepsilon\\to 0$.","marker":"[Ton19]"}],"fun_headline_variants":["Singular point yields non-unique Brakke flows","Stationary minimal surface with singularity has moving flow","Non-uniqueness from a single singular point","Minimal surface singularity forces multiple Brakke flows","Brakke flow non-unique without tangent cone uniqueness"],"cache_read_input_tokens":18304,"weakest_assumption_plain":"The load-bearing premise is that a regularity estimate from Allard's theory holds for the singular point—namely that near the origin the entire surface lies in a slab of width proportional to the scale around the planar direction $T$—even though the density there exceeds 1, so Allard's classical regularity theorem does not directly apply.","fun_headline_variants_meta":{"raw":{"variants":["Singular point yields non-unique Brakke flows","Stationary minimal surface with singularity has moving flow","Non-uniqueness from a single singular point","Minimal surface singularity forces multiple Brakke flows","Brakke flow non-unique without tangent cone uniqueness"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000202,"raw_usage":{"total_tokens":1385,"prompt_tokens":953,"completion_tokens":432,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":569,"completion_tokens_details":{"reasoning_tokens":356}},"tokens_in":569,"tokens_out":432,"duration_ms":4461,"temperature":1.0,"reasoning_tokens":356,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T04:52:07.400887+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Check whether the slab estimate (4.2) holds for a concrete stationary integral cone with density $3/2$ that satisfies the $L^2$-excess condition (3.1), such as three half-planes meeting along a common axis at small angles: compute the Hausdorff distance from the plane $T$ over $U_{2\\varepsilon}$. If for arbitrarily small $\\varepsilon$ the cone is not contained in $\\{|x_{n+1}|\\le \\varepsilon/20\\}$ despite having small $L^2$ distance, then Lemma 4.1 cannot supply the uniform excess and density bounds (4.1) and (4.4), and the proof of Theorem 3.4 collapses.","supporting_citations":[],"review_version":1}