{"id":"ac441715-3f8b-48b7-9cc9-ef901c03b200","arxiv_id":"2608.11011","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Ancient mean curvature flows asymptotic to the Simons cone from one side have unique asymptotics; adding mean convexity forces them to be stationary Hardt-Simon leaves.","lead":"Ancient mean curvature flows that approach the Simons cone from one side must eventually match a single predicted asymptotic profile. With an extra mean-convexity assumption, the flow is forced to be stationary and equal to one of the Hardt-Simon foliation leaves.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Equation (25)'s entropy bound is not justified and is likely false for the Hardt-Simon leaf flow, which is needed for the compactness and graphical-radius arguments.","rationale":"The reader's weakest-assumption analysis focused on one-sidedness (A2). That is a reasonable concern, but the more immediate gap I find is the entropy bound in equation (25). The proof of the main theorems repeatedly uses a uniform entropy bound strictly below 2 to pass to Brakke limits, to apply the strong maximum principle, and to rule out excess varifolds in proposition 3.3. The paper justifies this bound by local smooth convergence to \\Sigma, but that convergence does not control the entropy because entropy is sensitive to all scales and centers. The Hardt-Simon leaf flow is a concrete member of the theorem's own class and has constant entropy equal to the entropy of \\Sigma_1^+, not of \\Sigma, under rescaling. Thus the stated inequality is not a consequence of (A1) and appears to be false. If the entropy bound is only needed in the weaker form \\lambda < 2, the proof may be repairable, but that weaker bound is not established. I therefore recommend conditional acceptance: the central argument requires an additional entropy estimate or a corrected version of proposition 3.3 before the theorem can be considered proved.","tokens_in":53787,"tokens_out":42732,"duration_ms":418001,"concrete_test":"For n=5, solve the Hardt-Simon profile ODE (26) numerically, construct \\Sigma_1^+, and compute its Gaussian entropy \\lambda(\\Sigma_1^+) by maximizing F_{x_0,r}(\\Sigma_1^+) over a grid of centers x_0 and scales r; compare with the explicit \\lambda(\\Sigma) from (22). If \\lambda(\\Sigma_1^+) > \\lambda(\\Sigma), then equation (25) is false and the paper must supply a different entropy bound before proposition 3.3 can be applied to the flows it claims to classify; if the two entropies agree, this objection collapses.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Section 2.2, equation (25), asserts that if the rescaled flow \\tilde M_\\tau converges to the Simons cone \\Sigma locally smoothly away from 0, then \\lambda(\\tilde M_\\tau) \\le \\lambda(\\Sigma) < 2. This does not follow: entropy is scale invariant and samples all centers and scales, whereas C^\\infty_loc convergence away from 0 only controls compact annuli. The stationary Hardt-Simon leaf flow M_t \\equiv \\Sigma_1^+ satisfies (A1), (A2), and (A3); its rescaled slices are e^{\\tau/2}\\Sigma_1^+, so \\lambda(\\tilde M_\\tau)=\\lambda(\\Sigma_1^+) for all \\tau. The leaf is not scale invariant, and like the catenoid relative to two planes, it has strictly larger Gaussian entropy than its asymptotic cone; hence the displayed inequality in (25) is false. This is load-bearing because proposition 3.3 has as a hypothesis exactly \\lambda(M_t) \\le \\lambda(\\Sigma), and its proof uses that sharp bound to exclude excess varifolds in the limit. Lemma 3.8 invokes proposition 3.3 via (25), and proposition 3.6, lemma 5.2, and lemma 6.1 all inherit the compactness from these steps. As written, the proof gives no valid uniform entropy bound \\Lambda < 2 for flows satisfying (A1), (A2), so the graphical-radius estimate and the mode isolation in Section 5 are unsupported.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":54064,"tokens_out":26006,"duration_ms":280744,"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":[{"comment":"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.","section":"§2.2, Eq. (25)"},{"comment":"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.","section":"§4, Corollary 4.2"}],"minor_comments":[{"comment":"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.","section":"§2.2"},{"comment":"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.","section":"§5, Lemma 5.2"},{"comment":"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.","section":"§3, Proposition 3.1"},{"comment":"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.","section":"§5, Lemma 5.8"},{"comment":"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.","section":"§2.2, Proposition 2.3"}],"recommendation":"major_revision","confidential_remarks":"The main theorem is plausible and the manuscript is carefully organized, but the entropy bound (25) is the fulcrum of the compactness arguments and is currently asserted without proof. If the author can supply a valid proof of (25) (or a replacement entropy bound) and fix the corresponding issue for the Huisken functional in Corollary 4.2, the paper would be acceptable. If the entropy bound turns out to be false for one-sided flows asymptotic to the cone, the compactness argument in Proposition 3.3 would need substantial restructuring, and the current proof would not be salvageable in its present form. I would also suggest checking the Hardt-Simon leaf flow explicitly against (25) and including the computation or a reference."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nThis 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.\n\nI 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.\n\nThis 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.","headline":"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.","tokens_in":54606,"tokens_out":8451,"would_cite":false,"duration_ms":107855,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53E10","53C42"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["mean curvature flow","ancient solutions","Simons cone","Hardt-Simon foliation","self-shrinkers","eigenmode decomposition","unique asymptotics","parabolic region"],"falsifier":"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).","tokens_in":46,"feed_emoji":"📐","tokens_out":6028,"duration_ms":153398,"temperature":0.7,"pith_summary":"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.","feed_headline":"Ancient flows near the Simons cone have one forced shape","feed_subtitle":"With mean convexity, the flow must be a stationary Hardt-Simon leaf; without it, unique asymptotics still hold.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the inverse operator for the linearized minimal surface equation and the Liouville theorems used to prove stationarity of the mean-convex flow.","marker":"[41]"},{"why":"Provides the template for graphical radius estimates, normal-graph evolution, and Merle–Zaag mode isolation for asymptotically conical flows.","marker":"[10]"},{"why":"Supplies the ODE lemma that forces the dynamics onto the first eigenmode once the nonlinear error is controlled.","marker":"[38]"},{"why":"Introduces the inner–outer inverse Poincaré inequality that controls gradient and weighted $L^2$ terms by the $L^2$ mass in an outer annulus.","marker":"[1]"},{"why":"Provides Huisken's monotonicity formula, used for the entropy bound and for the conclusion that blowdowns are self-shrinkers.","marker":"[30]"},{"why":"Defines the Hardt–Simon foliation leaves and their profile functions, the stationary solutions appearing in the uniqueness theorem.","marker":"[24]"},{"why":"Classifies minimal hypersurfaces asymptotic to quadratic cones, used to identify the backward limit of the mean-convex flow as a Hardt–Simon leaf.","marker":"[40]"},{"why":"Gives the spectral decomposition of the linearized rescaled mean curvature operator at the Simons cone, including the first four eigenvalues and eigenfunctions.","marker":"[29]"}],"fun_headline_variants":["Ancient flow near Simons cone has unique asymptotics","Mean convex flow asymptotic to Simons cone is stationary","Forced leading term for flows hugging Simons cone","One-sided flow asymptotic to Simons cone has pinning"],"cache_read_input_tokens":56704,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Ancient flow near Simons cone has unique asymptotics","Mean convex flow asymptotic to Simons cone is stationary","Forced leading term for flows hugging Simons cone","One-sided flow asymptotic to Simons cone has pinning"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000234,"raw_usage":{"total_tokens":1417,"prompt_tokens":786,"completion_tokens":631,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":402,"completion_tokens_details":{"reasoning_tokens":571}},"tokens_in":402,"tokens_out":631,"duration_ms":6147,"temperature":1.0,"reasoning_tokens":571,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T12:06:53.618002+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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).","supporting_citations":[{"cited_title":"Existence of mean curvature flow singularities with bounded mean curvature","cited_arxiv_id":null,"evidence_quote":"Supplies the inverse operator for the linearized minimal surface equation and the Liouville theorems used to prove stationarity of the mean-convex flow."},{"cited_title":"Mean curvature flow with generic initial data","cited_arxiv_id":null,"evidence_quote":"Provides the template for graphical radius estimates, normal-graph evolution, and Merle–Zaag mode isolation for asymptotically conical flows."},{"cited_title":"Optimal estimates for blowup rate and behavior for nonlinear heat equations","cited_arxiv_id":null,"evidence_quote":"Supplies the ODE lemma that forces the dynamics onto the first eigenmode once the nonlinear error is controlled."},{"cited_title":"Unique asymptotics of ancient convex mean curvature flow solutions","cited_arxiv_id":null,"evidence_quote":"Introduces the inner–outer inverse Poincaré inequality that controls gradient and weighted $L^2$ terms by the $L^2$ mass in an outer annulus."},{"cited_title":"Asymptotic behavior for singularities of the mean curvature flow","cited_arxiv_id":null,"evidence_quote":"Provides Huisken's monotonicity formula, used for the entropy bound and for the conclusion that blowdowns are self-shrinkers."},{"cited_title":"Area minimizing hypersurfaces with isolated singularities","cited_arxiv_id":null,"evidence_quote":"Defines the Hardt–Simon foliation leaves and their profile functions, the stationary solutions appearing in the uniqueness theorem."},{"cited_title":"Minimal hypersurfaces asymptotic to quadratic cones in R n+ 1","cited_arxiv_id":null,"evidence_quote":"Classifies minimal hypersurfaces asymptotic to quadratic cones, used to identify the backward limit of the mean-convex flow as a Hardt–Simon leaf."},{"cited_title":"Mean curvature flow converging to an minimizing cone and its Hardt-Simon foliation","cited_arxiv_id":null,"evidence_quote":"Gives the spectral decomposition of the linearized rescaled mean curvature operator at the Simons cone, including the first four eigenvalues and eigenfunctions."}],"review_version":1}