{"id":"fea5ac89-931a-43e1-a687-9885465e5f5a","arxiv_id":"1908.02688","paper_version":2,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":2.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper constructs high-codimension ancient curve shortening flows, but the abstract and Remark 1.3 state the construction was earlier discovered in AAAW13.","lead":"This paper constructs explicit families of ancient curve-shortening flows that live in high-dimensional Euclidean spaces, not in a plane, and shows they connect to bounds on solution complexity. The main construction was previously reported by Altschuler and coauthors, and the paper adds entropy computations and implications for a Colding-Minicozzi codimension bound.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The ODE construction behind Theorem 1.1 is sound; the load-bearing issue is that the entropy computation in Corollary 2.3 contains a measure-factor error that invalidates the printed proof of the lower bound.","rationale":"The paper's main construction is mathematically correct, and the constant-speed parametrization identified by the reader is genuinely safe; the reader's weakest assumption is not the load-bearing problem. The entropy identity is one of the paper's advertised contributions and feeds directly into the sharp-constant discussion, so a measure-factor error in its proof is a real, checkable flaw. Because the error is localized and appears fixable, the appropriate disposition remains a conditional acceptance rather than rejection or unconditional acceptance. The reader's conditional verdict is therefore unchanged, but for a different reason than the one emphasized by the reader.","tokens_in":8355,"tokens_out":19685,"duration_ms":224877,"concrete_test":"Recompute the second displayed equation of Corollary 2.3 using dH^1|_{sγ_t} = s√B dθ. Check whether the printed denominator √B should be a numerator. If it is a numerator, the lower-bound limit is k_m λ(S^1) as claimed; if it is really a denominator, the integral with s̃ tends to 0 and the claim sup_t λ(γ_t) = k_m λ(S^1) is unproved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central existence argument is sound: because |∂θγ_t| is θ-independent, the Laplace–Beltrami reduction to the scalar ODE (2.5) is valid, and Proposition 2.1 stands. The place where the argument is not secure is the entropy computation in Corollary 2.3. For γ_t as in (2.1), with A = Σ r^{2k_j^2} and B = Σ k_j^2 r^{2k_j^2}, the Gaussian integral at translation 0 is (4π)^{-1/2} ∫_{sγ_t} e^{-|x|^2/4} dH^1 = s√π √B e^{-s^2 A/4}, because scaling multiplies the arclength element by s and |∂θγ_t| = √B. The manuscript's displayed formula has the factor √B in the denominator instead of the numerator. With the printed formula, the choice s̃ = √(2/A) gives a lower bound that tends to 0 as t → -∞, not to k_m λ(S^1); the limit written immediately below, λ(S^1) lim √B/√A = k_m λ(S^1), is exactly what follows from the corrected formula. Thus the lower-bound half of the entropy claim is not established as written, though it is likely fixable.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper constructs a family of closed curves in R^{2m} with coordinate functions r(t)^{k_j^2} times trigonometric monomials of frequency k_j, and proves that for an explicitly defined function r(t) these curves solve the curve shortening flow on (-∞,0). The main existence result is Theorem 1.1; the paper also identifies the tangent flows at 0 and -∞, computes the entropy as k_m λ(S^1), constructs analogous noncompact helix solutions in R^{2m+1}, and takes products to obtain higher-dimensional ancient mean curvature flows. The appendix gives an ODE existence proof. The authors state in Remark 1.3 that the main construction was already discovered by Altschuler–Altschuler–Angenent–Wu in [AAAW13].","tokens_in":8576,"tokens_out":11817,"duration_ms":124976,"significance":"The construction itself is clean, self-contained, and machine-checkable in the sense that the Laplacian reduction is valid because |∂_θ γ_t| is θ-independent, and the ODE solution is proved from scratch in the appendix. If the entropy computation is corrected, the paper provides explicit non-planar ancient curve shortening flows with computable entropy and explicit tangent flows, which is a useful addition to the small set of known high-codimension ancient flows and directly motivates the sharp-constant conjecture for a Colding–Minicozzi codimension bound. However, because the central existence result is already present in [AAAW13], the incremental value of the paper rests on the entropy computation, the helix variant, the product construction, and the conjectures; the authors are honest about this priority issue, but it should be reflected in the framing.","major_comments":[{"comment":"The displayed Gaussian integral in the proof of the lower bound has a measure-factor error. For sγ_t, with A = Σ_j r^{2k_j^2} and B = Σ_j k_j^2 r^{2k_j^2}, the correct computation is (4π)^{-1/2} ∫_{sγ_t} e^{-|x|^2/4} dH^1 = s√π √B exp(-s^2 A/4), because the dilation multiplies the arclength element by s and |∂_θ γ_t| = √B. The manuscript writes a factor √B in the denominator instead of the numerator. As printed, the choice s̃ = √2 A^{-1/2} gives an expression that tends to 0 as t → -∞, so the claimed lower bound does not follow. Replacing the denominator by √B makes the displayed limit correct and gives k_m λ(S^1). Since this corollary is the basis for the entropy applications and Conjecture 2.4, the proof must be corrected.","section":"Corollary 2.3"},{"comment":"The proof of Theorem 1.2 is one sentence: 'Take the product of n appropriately chosen torus curves.' This is not sufficient as written, because a product of n torus curves in R^{2m} would lie in R^{2mn}; to obtain a submanifold of R^{2m}, the torus-curve factors must be chosen with ambient dimensions that sum to 2m, and the nondegeneracy of the resulting product must be checked. The statement is plausible and the fix is straightforward, but the proof should be completed explicitly.","section":"Section 2.3, proof of Theorem 1.2"}],"minor_comments":[{"comment":"The Laplacian formula Δ_{γ_t} = |∂_θ γ_t|^{-2} ∂_θ^2 is used because |∂_θ γ_t| is θ-independent; this fact is correct for the given coordinate functions, but it should be stated explicitly as the justification rather than left implicit.","section":"Section 2.1, between (2.2) and (2.3)"},{"comment":"The tangent flows are described as 'the multiplicity k_m circle' and 'the multiplicity k_1 circle' without specifying the radius; the limiting circles have radius √2, as expected for self-shrinking circles, and this should be stated for clarity.","section":"Proposition 2.2"},{"comment":"The upper-bound argument uses monotonicity of entropy to conclude sup_t λ(γ_t) = lim_{t→-∞} λ(γ_t); this is standard for closed mean curvature flows, but a citation or one-line justification would improve rigor.","section":"Corollary 2.3, upper bound"},{"comment":"The add-on remark about [AAAW13] appears twice, once after the abstract and once as Remark 1.3; for a journal submission this should be integrated into the introduction in a single place.","section":"Remark 1.3 and abstract add-on"}],"recommendation":"major_revision","confidential_remarks":"The overlap with [AAAW13] is fully disclosed, and I see no circularity or fabrication concern. The main technical issue is the incorrect entropy integral in Corollary 2.3, which is fixable. The editor may wish to assess whether the incremental results beyond the known [AAAW13] construction—entropy computation, tangent-flow characterization, helix analogue, and product constructions—are substantial enough for the journal's standards."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"What you should know: the main construction—the torus curves—is not new; the authors disclose in the abstract add-on and Remark 1.3 that Altschuler–Altschuler–Angenent–Wu [AAAW13] discovered it. What is actually new here is the entropy computation, the product theorem, the odd-dimensional helix variant, and the discussion of the Colding–Minicozzi constant. The ODE verification is clean and the paper is honestly written, but Corollary 2.3 has a measure-factor error that invalidates the printed proof of the lower bound.\n\nLet me be specific. In the Gaussian integral over sγ_t, the arclength element contributes a factor √B in the numerator, where B = Σ k_j² r^{2k_j²}. The manuscript puts √B in the denominator. With the printed formula, the chosen scale s̃ = √(2/A) makes the lower bound tend to 0 as t→−∞, not k_m λ(S¹). The corrected formula gives exactly the claimed limit, so the entropy statement is almost certainly fixable, but the proof as written does not work. This is a minor-to-moderate bug: it does not undermine the construction itself, but the entropy claim lacks a valid lower-bound proof until corrected.\n\nWhat the paper does well: the reduction of the curve shortening flow to the scalar ODE is rigorous. The claim that |∂θγ| is θ-independent is standard and holds here, so the Laplace–Beltrami computation is sound. Theorem A.1 supplies a complete existence proof. The tangent-flow identifications in Proposition 2.2 are plausible and well argued. The product construction in Theorem 1.2 is straightforward and correct, and the comparison with Choi–Mantoulidis is useful context.\n\nThe bigger weakness is presentational. The abstract and introduction still frame Theorem 1.1 as the main new result, and the line \"As far as we know, this is the first construction...\" is no longer accurate once AAAW13 is known. The authors do disclose the overlap, but the framing should be revised to make the extension, not the base construction, the focus. The entropy upper bound is also sketched rather than fully detailed; it looks correct but should be expanded.\n\nThis paper is for researchers studying ancient mean curvature flows, entropy, and codimension bounds. It deserves a serious referee, not a desk reject. A referee should flag the factor error, ask for a reframing around the new contributions, and request a fuller entropy upper bound. With those changes, it becomes a useful independent exposition and extension of known material.","headline":"Sound explicit construction with a fixable factor error in the entropy proof, but the central example is not new; the paper's value lies in the entropy computation, product construction, and honest discussion of overlap.","tokens_in":9144,"tokens_out":3936,"would_cite":true,"duration_ms":40637,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53C44","35K55"],"pacs":[],"model":"deepseek-v4-flash","headline":"For every positive integer m, a compact ancient curve shortening flow in R^{2m} exists whose image spans the full space.","keywords":["ancient solutions","mean curvature flow","curve shortening flow","high codimension","torus curves","entropy","tangent flows","ODE reduction"],"falsifier":"For $m=2$, $k_1=1$, $k_2=2$, let $r(t)$ solve $\\frac12(r^2+r^8)=-t$ and define $\\gamma_t$ by formula (2.1); evaluate both sides of $\\partial_t\\gamma_t=\\Delta_{\\gamma_t}\\gamma_t$ at $\\theta=\\pi/4$ and $t=-1$. If the two sides differ, the ancient-flow claim for this example is false; if they agree, the reduction is consistent.","tokens_in":8106,"feed_emoji":"🌀","tokens_out":10278,"duration_ms":98294,"temperature":0.7,"pith_summary":"This paper proves that for every positive integer $m$ there is a compact, connected ancient solution of the curve shortening flow (defined for all $t<0$) in $\\mathbb{R}^{2m}$ whose image is not contained in any $(2m-1)$-dimensional subspace. Ancient solutions model the shapes of singularities under the flow, yet previously known ancient curve flows were planar, so the existence of genuinely high-codimension examples was open. The construction is explicit: a ``torus curve'' whose $2m$ coordinate functions are $r(t)^{k_j^2}\\cos(k_j\\theta)$ and $r(t)^{k_j^2}\\sin(k_j\\theta)$, with $r(t)$ solving one scalar ODE. The paper computes the entropy of this flow as $k_m\\lambda(S^1)$, identifies its tangent flows at $0$ and $-\\infty$ as round circles with multiplicity, and uses the example to propose the sharp constant in an existing entropy-versus-codimension bound.","feed_headline":"Ancient curves that use all 2m dimensions exist for every m","feed_subtitle":"Sinusoidal torus curves reduce the flow to one ODE and sharpen the entropy-versus-codimension bound.","key_machinery":"The load-bearing object is the torus curve, a curve on a product of $m$ circles $S^1(r^{k_1^2})\\times\\cdots\\times S^1(r^{k_m^2})$ with coordinate functions $r^{k_j^2}\\cos(k_j\\theta)$ and $r^{k_j^2}\\sin(k_j\\theta)$. Its defining property is that the speed $|\\partial_\\theta\\gamma|$ is independent of $\\theta$, so the vector Laplacian acts as a pure second derivative in $\\theta$; substituting into $\\partial_t\\gamma=\\Delta_\\gamma\\gamma$ collapses the entire high-codimension flow to a single first-order ODE for $r(t)$. The ancient solution of that ODE, obtained by inverting $F(r)=\\frac12\\sum_j r^{2k_j^2}=-t$, is what makes the construction work.","core_discovery":"The central object is the torus curve $\\gamma_t(\\theta) = (r^{k_1^2}\\cos(k_1\\theta), r^{k_1^2}\\sin(k_1\\theta), \\ldots, r^{k_m^2}\\cos(k_m\\theta), r^{k_m^2}\\sin(k_m\\theta))$, which winds $k_j$ times around the $j$-th circle $S^1(r^{k_j^2})$. Because $|\\partial_\\theta\\gamma_t|^2 = \\sum_{j=1}^m k_j^2 r^{2k_j^2}$ is independent of $\\theta$, the curve Laplacian is $|\\partial_\\theta\\gamma_t|^{-2}\\partial_\\theta^2$, so the mean curvature flow equation $\\partial_t\\gamma = \\Delta_\\gamma\\gamma$ reduces to the scalar ODE $r' = -r\\,/\\,\\sum_{j=1}^m k_j^2 r^{2k_j^2}$. The paper proves this ODE has a unique positive solution on $(-\\infty,0)$ with $r\\to\\infty$ as $t\\to-\\infty$ and $r\\to0$ as $t\\to0$, yielding an ancient flow. The coordinate functions are linearly independent, so $\\gamma_t$ spans $\\mathbb{R}^{2m}$; rescaling shows the tangent flow at $-\\infty$ is the multiplicity-$k_m$ circle and at $0$ the multiplicity-$k_1$ circle. Consequently $\\sup_t\\lambda(\\gamma_t) = k_m\\lambda(S^1)$. Adding one linear coordinate to the same Ansatz gives eternal helix-type solutions in odd-dimensional spaces with infinite entropy, and products of torus curves give $n$-dimensional ancient mean curvature flows of high codimension.","pith_inferences":["If the conjectured sharp constant is correct, any ancient curve shortening flow whose image truly spans $\\mathbb{R}^N$ must have entropy at least $(N/2)\\lambda(S^1)$; testing this inequality on multi-frequency or perturbed torus curves would give evidence without classifying all ancient flows.","The identity $F(r)+t=0$ makes the construction essentially algebraic, so these solutions may serve as explicit blow-up models; one could perturb the Ansatz by small non-sinusoidal Fourier modes and check whether the ODE reduction survives to first order.","The same constant-speed idea may generalize to higher-dimensional submanifolds formed from orthogonal products of such curves, with entropy becoming a product of winding numbers; whether the resulting flows remain ancient is a testable extension.","Varying the frequencies $k_j$ gives a family of ancient flows with the same ambient dimension but different entropies, which may help calibrate any future sharp bound."],"forward_implications":["Ancient curve shortening flows in Euclidean space need not be planar: for each $m$ there is one in $\\mathbb{R}^{2m}$ spanning the full space.","Products of these curves produce compact $n$-dimensional ancient mean curvature flows that also span the full space, and further products with Euclidean factors give finite-entropy ancient solutions in every dimension and codimension except curves lying in odd-dimensional Euclidean spaces.","The entropy of the $m$-frequency torus curve is exactly $k_m\\lambda(S^1)$, matching the linear-in-dimension entropy growth that the conjectured sharp constant predicts.","Setting $k_1=1$ makes the tangent flow at time $0$ a single round circle even though the curve has arbitrarily high codimension; the tangent flow at $0$ alone cannot bound codimension.","The example yields the lower bound $\\lambda(S^1)C_1\\ge 2$ for the constant in the known entropy-codimension bound and motivates the conjecture that the sharp value is $C_1=2/\\lambda(S^1)$."],"supporting_citations":[{"why":"Discovered the same torus-curve ancient flow before this preprint; the paper acknowledges priority and contrasts the motivation.","marker":"[AAAW13]"},{"why":"Supplies the entropy-versus-codimension bound that the example tests and for which the sharp constant is conjectured.","marker":"[CM19b]"},{"why":"Defines the entropy functional and the Gaussian integral used in the entropy computation of Corollary 2.3.","marker":"[CM12]"},{"why":"Gives the alternative ancient-flow construction via unstable minimal submanifolds, compared in Remark 2.9; those methods cannot produce ancient curves.","marker":"[CM19a]"},{"why":"Classifies compact ancient curve shortening flows as planar, the background result that the new curves go beyond.","marker":"[DHS10]"},{"why":"Companion paper in preparation that uses the tangent flow at -infinity, motivated by this example, to prove codimension bounds.","marker":"[SS]"}],"fun_headline_variants":["Ancient curve flows spanning all 2m dimensions, for any m","Torus winding gives compact ancient MCF in high codim","Curve shortening reduces to one ODE for ancient tori","High-codimension ancient flows from sinusoidal tori","Constructing ancient MCF: curves that fill 2m space"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The reduction to a single ODE rests on the curve being parametrized with speed independent of $\\theta$, so that the curve Laplacian is a pure second derivative in $\\theta$; if that failed, the coordinate functions would not close under the flow, and the whole construction would collapse.","fun_headline_variants_meta":{"raw":{"variants":["Ancient curve flows spanning all 2m dimensions, for any m","Torus winding gives compact ancient MCF in high codim","Curve shortening reduces to one ODE for ancient tori","High-codimension ancient flows from sinusoidal tori","Constructing ancient MCF: curves that fill 2m space"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000646,"raw_usage":{"total_tokens":2977,"prompt_tokens":964,"completion_tokens":2013,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":580,"completion_tokens_details":{"reasoning_tokens":1927}},"tokens_in":580,"tokens_out":2013,"duration_ms":19362,"temperature":1.0,"reasoning_tokens":1927,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:38:39.902630+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For $m=2$, $k_1=1$, $k_2=2$, let $r(t)$ solve $\\frac12(r^2+r^8)=-t$ and define $\\gamma_t$ by formula (2.1); evaluate both sides of $\\partial_t\\gamma_t=\\Delta_{\\gamma_t}\\gamma_t$ at $\\theta=\\pi/4$ and $t=-1$. If the two sides differ, the ancient-flow claim for this example is false; if they agree, the reduction is consistent.","supporting_citations":[],"review_version":1}