{"id":"0ffbf4b0-10db-45bf-addf-796ee357b0d5","arxiv_id":"1908.04699","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"Closed strictly convex self-similar hypersurfaces for quotient curvature flows (σ_k/σ_l)^α = ⟨X,ν⟩ are spheres whenever α > 1/(k-l).","lead":"A geometry paper proves that the only closed convex hypersurfaces that shrink self-similarly with speed given by a quotient of elementary curvature functions are round spheres. The result covers a broad family of fully nonlinear curvature flows and completes a uniqueness picture left open for exponents above the threshold 1/(k-l).","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 4.1(i), the key algebraic estimate for the Z-function maximum principle, is quoted from a self-cited preprint without proof; if Lemma 2.1 of [11] is misstated or has hidden hypotheses, the proof of Theorem 2.1 collapses.","rationale":"The reader's weakest-assumption analysis identifies the same load-bearing point: Lemma 4.1(i) is essential and is not proved in the present paper. I checked the surrounding structure of the proof as far as possible: Proposition 2.3 is consistent with direct computation, the maximum-principle step for W is executed in detail, and the final application of the strong maximum principle to Z is sound provided Lemma 4.1 holds. The remaining soft spot is genuinely the algebraic inequality (15), together with the fact that the written derivation in Lemma 4.1(i) divides by l and therefore does not cover l=0 as stated. Since (15) is an equality for l=0, the missing l=0 discussion is repairable, but it should be stated explicitly. The range condition '1 > p > 1 - k - l' in Theorem 2.1 also appears to be a typo for '1 > p > 1 - (k-l)' (the proof needs p0 = (p-1)/(k-l) in (-1,0)), though this does not affect Theorem 1.1, where the alpha condition is equivalent to p0 in (-1,0). Overall the central argument appears credible conditional on the quoted algebraic lemma; I therefore do not change the reader's CONDITIONAL verdict.","tokens_in":6898,"tokens_out":40417,"duration_ms":349958,"concrete_test":"Independently verify Lemma 2.1 of [11]: for arbitrary n and 2 <= k <= n, prove from Newton's inequalities that sigma_1/(k(k-1)) - k sigma_k/((k-1) sigma_{k-1}) + (k+1) sigma_{k+1}/(k sigma_k) >= 0 on positive definite (b_ij), and check the telescoping step in (15) explicitly, splitting off the l=0 case where the displayed formula divides by zero. A numeric spot-check on random positive eigenvalue tuples, for example n=4, k=2, would catch transcription errors, but the decisive test is either a derivation from Newton's inequalities or a counterexample to the quoted lemma.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central maximum-principle argument for Z requires the first group of terms in Proposition 2.3 to be nonnegative. Lemma 4.1(i) reduces this to inequality (15), and inequality (15) is derived in the paper solely from Lemma 2.1 of [11], an algebraic inequality about elementary symmetric polynomials that is quoted without proof. The derivation also divides by l, so as written it excludes the l=0 case (where (15) is actually an equality, but the proof does not say this). If Lemma 2.1 of [11] has a missing hypothesis (e.g., k >= 2, or positivity requirements on the eigenvalues) or the transcription is inexact, then Lemma 4.1(i) fails, the nonnegativity of the first two terms in Proposition 2.3 fails, and the strong maximum principle step establishing that Z is constant has no foundation. This is a genuine load-bearing external dependence: no alternative proof of (15) is given in the paper, and the authors rely on it without comment.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies closed, strictly convex self-similar shrinkers in R^{n+1} for curvature flows whose speed is a power of a quotient of elementary symmetric functions of principal curvature, F^α = (σ_k(κ)/σ_l(κ))^α with 0 ≤ l < k ≤ n. Theorem 1.1 asserts that, for α > 1/(k-l), the only such hypersurfaces are round spheres. Passing to the support function, the equation is rewritten as the PDE σ_k(b)/σ_l(b) = u^{p-1} on S^n, where b_{ij} = u_{ij} + u δ_{ij}. The proof combines a W-function maximum principle adapted from Brendle–Choi–Daskalopoulos with a Z-function introduced by the first author, proves at a maximum point of W that the second fundamental form is scalar, and then uses the strong maximum principle to conclude that u is constant. The final coefficient computation at the end of Section 4 is algebraically correct under the stated normalization.","tokens_in":7070,"tokens_out":14791,"duration_ms":131469,"significance":"The theorem fills a genuine gap in the literature: previous uniqueness results covered σ_k^α flows and the 1-homogeneous case α = 1/(k-l), but not general quotient powers. The W/Z method is a nontrivial adaptation to this setting, and the reduction from the geometric equation to the support-function PDE is conceptually clean. If the missing algebraic estimate is supplied, the result is a solid contribution to the uniqueness theory of self-similar solutions to curvature flows. The main weakness is that a key inequality is quoted from a self-cited preprint without proof; until that is fixed, the proof is not fully self-contained at a load-bearing point.","major_comments":[{"comment":"The proof of inequality (15) is the only support for the nonnegativity of the first two terms in the expression for F^{ij}D_iD_jZ in Proposition 2.3, and it is entirely derived from Lemma 2.1 of the authors' previous preprint [11], which is quoted without proof. This is a load-bearing external dependency: if Lemma 2.1 has hidden hypotheses or is transcribed incorrectly, Lemma 4.1(i) fails and the maximum-principle argument for Theorem 2.1 has no foundation. Please include a self-contained proof of the inequality (or at least of the case needed here) in the paper, or provide a precise reference to a published version with the exact statement.","section":"Section 4, Lemma 4.1(i)"},{"comment":"The proof of (15) divides by l when writing the term with i=l+1 as i σ_i/((i-1)σ_{i-1}); for l=0 this term is σ_1/(0·σ_0), so the argument as written excludes l=0. Since Theorem 2.1 is stated for 0 ≤ l < k, this is a gap in the written proof. The authors should state that l=0 is covered by Remark 2.2 and the result of [9], or they should give a separate limiting argument for l=0.","section":"Section 4, Lemma 4.1(i), l=0 case"}],"minor_comments":[{"comment":"The sentence 'Since For the convenience of discussion, instead of (8), we consider...' is grammatically incomplete; it should be split into two sentences.","section":"Section 2, transition between equations (9) and (10)"},{"comment":"The reduction from Theorem 1.1 to Theorem 2.1 is not fully explicit: for a given equation (7) with indices (k,l), the support-function equation (8) holds for the reversed pair (n-l, n-k) and with p-1 = -1/α, not for the same (k,l). The equivalence is true because both theorems quantify over all pairs, but stating the index reversal and the relation p = 1 - 1/α would prevent confusion.","section":"Section 2, Theorem 2.1 reduction"},{"comment":"Lemma 3.2 does not state the value of β in the definition of W; the proof silently uses β = 1+p0. Please make this explicit in the statement.","section":"Section 3, Lemma 3.2"},{"comment":"Inequality (14) is asserted without proof; it follows immediately from F^{ii} > 0, b_{ii} > 0, and b_{11} ≥ b_{ii}, but a one-line justification would improve readability.","section":"Section 3, equation (14)"},{"comment":"The inequality in part (ii) is attributed to Newton's inequality; a brief explanation of the displayed step would help the reader verify that the constants are correct.","section":"Section 4, Lemma 4.1(ii)"}],"recommendation":"major_revision","confidential_remarks":"To the editor: the main concern is the paper's reliance on Lemma 2.1 of [11] without proof. If the authors supply a self-contained proof or a stable published reference, I would be inclined to accept. The l=0 issue is easy to fix by explicitly invoking [9]. The paper's overlap with [9] and [12] is acceptable: the extension to general quotient powers is new, and I found no circularity, since [11] concerns a different class of flows. The result does not recover the sharpest known threshold in the Gauss-curvature case, but that is not a defect of the stated theorem."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The theorem is a real extension: it covers the full range 0≤l<k≤n for (σ_k/σ_l)^α = ⟨X,ν⟩ with α>1/(k-l), going beyond the α=1/(k-l) cases in Andrews and McCoy, the σ_k^α case in Gao–Li–Ma, and the l=0 case in Chen. The W/Z function strategy is applied carefully, and the maximum-principle argument in Lemma 3.2 is detailed enough to follow. The final coefficient computation at the end of Section 4 checks out under the stated normalization, and Lemma 4.1(ii) is a clean use of Newton's inequality. The paper is well organized and the main ideas are transparent. So credit is due: this is a substantive contribution, not a repackaging.\n\nThe soft spots are real but mostly fixable. The biggest one is Lemma 4.1(i): the proof reduces the needed positivity to inequality (15), which is derived solely from Lemma 2.1 of [11], the authors' own unpublished preprint. That lemma is not proved in the paper, and it is load-bearing. If it has hidden hypotheses (e.g., k≥2 or extra convexity), the maximum-principle step for Z collapses. This is exactly the concern you flagged, and it holds up on reading. The authors should either include a proof of the inequality or cite a published source; as written, the paper is not self-contained on a pivotal point.\n\nThere are also two smaller issues. First, the derivation of (15) divides by l, so it does not cover l=0; the l=0 case is handled only by an earlier remark citing [9], which is fine but should be stated in the proof of Theorem 2.1. Second, Theorem 2.1 states the range as \"1 > p > 1 - k - l,\" which is a typo: it should be 1 > p > 1 - (k-l), consistent with the −1<p0<0 condition used in the proof. Both are minor and easily corrected.\n\nAs for the stress-test, I think it is accurate. The concern is not manufactured; it is a genuine external dependence. But it is not necessarily fatal. The inequality is plausible and likely true, and a referee could verify it. Still, the paper should not be accepted in its current form without that gap being closed.\n\nWho should read this? Geometric analysts working on curvature flows, convex hypersurfaces, and Michael–Simon-type uniqueness problems. It deserves a serious referee, but the referee should demand a proof of the algebraic inequality and a cleanup of the typos before publication.","headline":"A clean, genuine extension of the self-similar uniqueness results to quotient curvature flows, but the proof's key algebraic estimate is imported from a self-cited preprint without proof, so the paper is conditionally correct rather than fully self-contained.","tokens_in":7670,"tokens_out":3931,"would_cite":false,"duration_ms":38256,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35J15","35J60","53C44"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that every closed strictly convex self-similar shrinker under a quotient of elementary-symmetric curvature functions, with exponent above 1/(k−l), is a round sphere.","keywords":["uniqueness","convex hypersurfaces","quotient curvatures","self-similar solutions","support function","maximum principle","elementary symmetric polynomials","curvature flow"],"falsifier":"The theorem predicts no nonconstant positive solutions to $\\sigma_k(b_{ij})/\\sigma_l(b_{ij})=u^{p-1}$ on $S^n$ for $1-(k-l)<p<1$; searching for rotationally symmetric nonconstant solutions reduces this to a one-dimensional ODE, and finding one would refute Theorem 2.1. Independently, the proof's key estimate can be tested directly: for $n=3$, $k=2$, $l=1$ and positive radii, the quoted inequality becomes $2\\sigma_2/\\sigma_1-3\\sigma_3/(2\\sigma_2)\\le \\frac{1}{2}\\sigma_1$, and a single failure among positive triples would invalidate Lemma 4.1(i).","tokens_in":6651,"feed_emoji":"⚪","tokens_out":18143,"duration_ms":153802,"temperature":0.7,"pith_summary":"This paper aims to close a gap in the uniqueness theory of self-similar shrinkers: when a closed strictly convex hypersurface moves by a quotient of elementary-symmetric curvature functions, the only self-similar solutions should be round spheres. The authors prove that a hypersurface satisfying $(\\sigma_k(\\kappa)/\\sigma_l(\\kappa))^\\alpha = \\langle X, \\nu\\rangle$ with $0\\le l<k\\le n$ and $\\alpha>1/(k-l)$ must be a standard sphere. This matters because self-similar solutions describe the singularity profiles of curvature flows, and uniqueness says the profile is always spherical in this family. The proof works in support-function form: for $1\\le l<k$ it runs a maximum-principle argument showing that the only positive solutions of the PDE $\\sigma_k(b_{ij})/\\sigma_l(b_{ij})=u^{p-1}$ are constants, while the $l=0$ case is carried by an earlier result.","feed_headline":"Only round spheres shrink self-similarly under quotient curvature flows","feed_subtitle":"For every ratio σ_k/σ_l with exponent above 1/(k−l), the only closed convex self-similar solution is a sphere.","key_machinery":"The central objects are the two comparison functions $W$ and $Z$. $W$ is the function $u\\lambda_{\\max}(b_{ij})-\\frac{\\beta}{2}(u^2+|Du|^2)$; its maximum points are controlled by Lemma 3.1, a modification of a lemma from [8]. $Z$ is the function $uG-\\frac{n\\beta}{2}(u^2+|Du|^2)$ with $G=\\frac{n}{k}(\\sigma_1-\\frac{(k+1)\\sigma_{k+1}}{\\sigma_k})$, engineered so that $nW\\ge Z$ with equality only when $(b_{ij})$ is a scalar matrix. The heart of the proof is an elliptic differential inequality for $Z$: Proposition 2.3 computes $F^{ij}D_iD_jZ$, and Lemma 4.1 supplies two positivity estimates (one algebraic, based on a quoted inequality from [11], and one from the standard inequality for elementary symmetric means) that make the terms sign-definite. At a maximum point of $W$, where the matrix is scalar, the coefficient of $|Du|^2$ becomes positive, and the strong maximum principle then propagates constancy.","core_discovery":"On the support function $u$ of a strictly convex hypersurface, the self-similar equation becomes $\\sigma_k(b_{ij})/\\sigma_l(b_{ij}) = u^{p-1}$, where $(b_{ij})=(u_{ij}+u\\delta_{ij})$ is the matrix of principal radii, and the desired conclusion is $u\\equiv\\text{const}$. For $1\\le l<k$, the paper proves this by setting $F=(\\sigma_k/\\sigma_l)^{1/(k-l)}$, so that $F(b_{ij})=u^{p_0}$ with $-1<p_0<0$; $F$ is 1-homogeneous, concave, and has positive-definite derivative. Two auxiliary functions are introduced: $W=u\\lambda_{\\max}(b_{ij})-\\frac{\\beta}{2}(u^2+|Du|^2)$ and $Z=uG-\\frac{n\\beta}{2}(u^2+|Du|^2)$, with $G=\\frac{n}{k}(\\sigma_1-\\frac{(k+1)\\sigma_{k+1}}{\\sigma_k})$. A maximum-principle argument shows that at a maximum point of $W$ the matrix $(b_{ij})$ is scalar and $Du=0$; then an elliptic inequality for $Z$, obtained from Proposition 2.3 and Lemma 4.1, forces $Z$ to be constant in a neighborhood, hence $W$ is constant and $u$ is constant. In geometric terms, a constant support function corresponds to a sphere. The case $l=0$ is covered by the earlier result in [9].","pith_inferences":["The same $Z$-function construction may prove uniqueness for other 1-homogeneous concave curvature functions once the two inequalities in Lemma 4.1 are checked, since the maximum-principle part of the proof is written for a general $F$.","If the quoted algebraic inequality admits a strengthening, the method might reach exponents below $1/(k-l)$; the current proof is tied to the range where the sign of a particular coefficient becomes positive.","A natural numerical test is to look for rotationally symmetric nonconstant solutions of the support-function equation near the critical exponent $\\alpha=1/(k-l)$; the theorem predicts none exist above it, and the boundary case is known by separate arguments.","The result supports the broader expectation that concave, homogeneous curvature flows have only spherical convex self-similar shrinkers, though the mechanism here does not obviously extend to nonconvex or noncompact settings."],"forward_implications":["For any $0\\le l<k\\le n$ and any $\\alpha>1/(k-l)$, every closed strictly convex self-similar shrinker of the quotient flow is a sphere.","In support-function form, the only positive solutions of $\\sigma_k(b_{ij})/\\sigma_l(b_{ij})=u^{p-1}$ on $S^n$ with $1-(k-l)<p<1$ are constants.","This extends the previously known uniqueness at $\\alpha=1/(k-l)$ to the entire open range above it.","The proof requires only strict convexity, so it gives a clean classification of convex self-similar solutions for this family of flows."],"supporting_citations":[{"why":"Provides the support-function setup and the relation between principal curvatures and the radii matrix $(b_{ij})$, used to pass from equation (7) to equation (8).","marker":"[3]"},{"why":"Supplies the $W$-function and the maximum-principle strategy at a maximum point of $W$, which the present proof adapts.","marker":"[10]"},{"why":"Supplies the technical lemma (Lemma 3.1 here) describing the behavior of the largest eigenvalue at a maximum point.","marker":"[8]"},{"why":"Introduces the $Z$-function (equation (6)) that plays the central role in forcing constancy.","marker":"[9]"},{"why":"Provides the algebraic inequality quoted as Lemma 2.1, the basis of the positivity estimate Lemma 4.1(i).","marker":"[11]"},{"why":"Proves the predecessor uniqueness result for powers of a single elementary symmetric function, whose method is extended here.","marker":"[12]"},{"why":"Proves uniqueness at the boundary exponent $\\alpha=1/(k-l)$, the case that the present theorem goes beyond.","marker":"[16]"}],"fun_headline_variants":["Spheres are the only self-similar shrinkers here","Quotient curvature flows: only spheres self-shrink","Self-similar shrinking solutions are spheres","Unique self-shrinking spheres in quotient curvature"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"For the $1\\le l<k$ case, the argument depends on an algebraic inequality quoted from [11] (Lemma 2.1 there) bounding $(l+1)\\sigma_{l+1}/(l\\sigma_l)-(k+1)\\sigma_{k+1}/(k\\sigma_k)$ by $(1/l-1/k)\\sigma_1$; this paper gives no proof of it, and Lemma 4.1(i)—hence the whole maximum-principle step for $Z$—rests on that quoted inequality.","fun_headline_variants_meta":{"raw":{"variants":["Spheres are the only self-similar shrinkers here","Quotient curvature flows: only spheres self-shrink","Self-similar shrinking solutions are spheres","Unique self-shrinking spheres in quotient curvature"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001085,"raw_usage":{"total_tokens":4513,"prompt_tokens":903,"completion_tokens":3610,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":519,"completion_tokens_details":{"reasoning_tokens":3548}},"tokens_in":519,"tokens_out":3610,"duration_ms":24605,"temperature":1.0,"reasoning_tokens":3548,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:35:21.893929+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"The theorem predicts no nonconstant positive solutions to $\\sigma_k(b_{ij})/\\sigma_l(b_{ij})=u^{p-1}$ on $S^n$ for $1-(k-l)<p<1$; searching for rotationally symmetric nonconstant solutions reduces this to a one-dimensional ODE, and finding one would refute Theorem 2.1. Independently, the proof's key estimate can be tested directly: for $n=3$, $k=2$, $l=1$ and positive radii, the quoted inequality becomes $2\\sigma_2/\\sigma_1-3\\sigma_3/(2\\sigma_2)\\le \\frac{1}{2}\\sigma_1$, and a single failure among positive triples would invalidate Lemma 4.1(i).","supporting_citations":[{"cited_title":"Andrews, Contraction of convex hypersurfaces in Euclidean space","cited_arxiv_id":null,"evidence_quote":"Provides the support-function setup and the relation between principal curvatures and the radii matrix $(b_{ij})$, used to pass from equation (7) to equation (8)."},{"cited_title":"Brendle, K","cited_arxiv_id":null,"evidence_quote":"Supplies the technical lemma (Lemma 3.1 here) describing the behavior of the largest eigenvalue at a maximum point."},{"cited_title":"Uniqueness of solutions to Lp-Christoffel-Minkowski problem for p<1","cited_arxiv_id":"1905.11043","evidence_quote":"Introduces the $Z$-function (equation (6)) that plays the central role in forcing constancy."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the algebraic inequality quoted as Lemma 2.1, the basis of the positivity estimate Lemma 4.1(i)."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Proves the predecessor uniqueness result for powers of a single elementary symmetric function, whose method is extended here."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Proves uniqueness at the boundary exponent $\\alpha=1/(k-l)$, the case that the present theorem goes beyond."}],"review_version":1}