{"id":"a481f058-49a0-47ff-8995-ca950a0bfb23","arxiv_id":"2509.01023","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Every complete mean convex asymptotically conical self-expander in R^{n+1}, n≥3, with a mean convex and weakly convex asymptotic cone is strictly convex.","lead":"This paper proves that mean convex self-expanders of the mean curvature flow that asymptote to convex cones must be strictly convex in every dimension n≥3. It extends two-dimensional and 2-convex results to the full mean convex case using a new pinching argument.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No internal flaw found; the most load-bearing point is the imported decay estimate Lemma 2.6, on which the infimum-at-infinity case of §4 depends and which is not re-derived here.","rationale":"I agree with the reader that the weakest assumption is Lemma 2.6. It is genuinely load-bearing: without it, the infimum-at-infinity case in Theorem 1.1's proof is not contradicted, and the theorem would reduce to a statement about interior minima. My own check of the internal lemmas found no algebraic or logical error: Lemma 3.2 is a correct pairing/cancellation identity, Lemma 3.3 follows with the stated signs, and Lemma 3.4's contradiction is valid once the elliptic inequality and smoothness of λk/H are granted. The strict-convexity step is briefer, but it uses standard tools (Hamilton's strong maximum principle, parallel nullity, de Rham splitting) and is not where I would stake an objection. Since Lemma 2.6 is published in the author's prior work [35] and I have no evidence that its hypotheses exclude the present mean-convex setting, I do not change the ACCEPT verdict. The proposed check would settle the one residual doubt and would raise confidence beyond MODERATE.","tokens_in":16246,"tokens_out":26053,"duration_ms":332682,"concrete_test":"Locate the proof of Lemma 4.2 in Xie-Yu [35] and verify its hypotheses: does it require 2-convexity of the expander, or only asymptotic conicality? Independently, using the graph parametrization of Lemma 2.5, expand the second fundamental form of Σ near infinity: for each cone direction with a zero principal curvature, compute the corresponding eigenvalue of A_Σ and confirm it is o(r^{-1}) (ideally O(r^{-3})) while H≈r^{-1}. If any such eigenvalue is instead comparable to r^{-1}, the limit in (4.2) could be negative and Case 2 would not be ruled out.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of weak convexity in §4 splits into two cases. Case 1 (interior infimum) is handled internally via Lemmas 3.2–3.4 and appears correct. Case 2 (infinity infimum) rules out a negative infimum at infinity by invoking Lemma 2.6, imported from [35, Lemma 4.2] and not re-derived. Lemma 2.6 asserts that a principal curvature of Σ asymptotic to a zero principal curvature of the cone decays as O(r^{-3}), while H≈r^{-1}, forcing κ1/H→0 at infinity. This is the sole step preventing a mean-convex self-expander from having a negative infimum at infinity, and the only link from cone weak convexity to the global infimum. If Lemma 2.6 in fact required the 2-convexity hypothesis of [35], or if the correspondence between ordered principal curvatures of Σ and those of C is not maintained along the graph parametrization near infinity, the contradiction in Case 2 fails and Theorem 1.1 is not established. The rest of the argument—Lemma 2.3, Lemma 3.2, Lemma 3.4, and the strict-convexity reduction—appears sound; the strict-convexity step relies on standard maximum-principle/splitting facts. This is a verification concern about a published lemma, not a demonstrated error.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves Theorem 1.1: for n ≥ 3, any complete immersed two-sided mean convex self-expander in R^{n+1} that is asymptotic to a regular, mean convex, weakly convex cone must be strictly convex. The proof first establishes weak convexity by contradiction, assuming a negative infimum of κ1/H. Case 1 (interior infimum) is ruled out via a new elliptic differential inequality for λk/H = (κ1+···+κk)/H, derived from a purely algebraic identity (Lemma 3.2) and a strong maximum principle argument (Lemma 3.4). Case 2 (infimum at infinity) is ruled out using a decay estimate (Lemma 2.6) imported from the author's prior work [35]. Strict convexity is then obtained from weak convexity by Hamilton's strong maximum principle and a splitting argument.","tokens_in":16472,"tokens_out":7623,"duration_ms":94225,"significance":"If the proof is correct, the result is a notable advance: it extends Smoczyk's two-dimensional convexity theorem to all dimensions and removes the 2-convexity assumption used in prior work [35], replacing it with mean convexity plus convexity of the asymptotic cone. The paper brings a genuinely new tool to the problem: Lemma 3.2, an algebraic identity for sums of the smallest principal curvatures that yields an elliptic inequality even when principal curvatures have multiplicities. The use of Derdziński/Singley and Kato's spectral projection to handle multiplicities is elegant and likely useful beyond this paper. The main theorem is falsifiable and would, if established, provide a clear picture of how curvature at infinity controls interior curvature for self-expanders.","major_comments":[{"comment":"The entire Case 2 relies on Lemma 2.6, imported from [35, Lemma 4.2], which asserts that principal curvatures of the expander decay as O(r^{-3}) when the corresponding cone curvature is zero, and ≈ r^{-1} otherwise. This lemma is not re-derived here, and its statement as given does not specify (i) the exact hypotheses needed (in particular, whether it holds for all mean convex expanders or only under the 2-convexity assumption of [35]) or (ii) how the 'corresponding' principal curvatures are ordered along the graph parametrization of Lemma 2.5. Since this is the only step that rules out a negative infimum at infinity, a failure of either point would invalidate Theorem 1.1. Please provide a proof or a precise statement of Lemma 2.6 in the context of this paper, including a verification that the correspondence between ordered principal curvatures is maintained. The alternative argument via","section":"Section 4, Case 2 (Infinity Infimum), eqs. (4.1)–(4.2)"},{"comment":"After showing rank(A) is locally constant and ker(A) is parallel, the paper asserts that 'by completeness, the local splitting extends globally.' Completeness alone does not imply a global product splitting unless the universal cover is trivial or the parallel distribution is globally integrable with a simply connected leaf. The argument should be expanded: for example, pass to the universal cover, obtain a global Euclidean factor, and then argue that the asymptotic cone would split off a line, contradicting the isolated singularity, as the author sketches. Without this justification, the conclusion that the self-expander splits off a line is not fully established.","section":"Section 4, strict-convexity step"}],"minor_comments":[{"comment":"The text says 'LH + H(|A|^2 + 1) = 0, hence H ≡ 0.' More precisely, once H is constant, LH=0, so the equation gives H(|A|^2+1)=0, and since H>0 this forces H≡0. Please rephrase for clarity.","section":"Lemma 3.4, final lines"},{"comment":"The alternative argument using the cone's ratio lacks a proof of the claimed limit (4.2). If the main argument via Lemma 2.6 is retained, this alternative can be removed or expanded into a self-contained argument.","section":"Section 4, eq. (4.2)"},{"comment":"The condition 'mean convex and weakly convex cones' is defined only in a remark. Since it is a central hypothesis, consider defining it directly in the theorem statement or immediately before it, so that the reader does not have to wait for the remark.","section":"Theorem 1.1 and Remark 1.1"}],"recommendation":"major_revision","confidential_remarks":"The main unresolved issue is the applicability of Lemma 2.6. The author should be asked to confirm whether [35, Lemma 4.2] holds without 2-convexity and to specify the correspondence between the ordered principal curvatures of the expander and the cone. If this is confirmed, the theorem is very likely correct and the paper would be a strong contribution. The splitting step in the strict-convexity argument also needs a more careful global justification, though this is likely fixable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Xie proves that a complete immersed two-sided mean convex self-expander in R^{n+1}, n≥3, asymptotic to a mean convex and weakly convex cone, must be strictly convex. That is a real advance: it removes the 2-convex hypothesis from the author's earlier theorem with Yu and extends Smoczyk's surface result to all dimensions. I read the proof in some detail and I think it holds up.\n\nThe main new ingredients are good. The function λ_k/H is shown to be smooth on the region where the k-th and (k+1)-st principal curvatures separate, using Kato's spectral projection; that is a clean way around the multiplicity problem. The algebraic identity in Lemma 3.2 is the heart of the matter: it turns a sum that could have either sign into a manifestly nonpositive expression. I checked the cancellation and it works. Lemma 3.4 then gets a contradiction from a negative interior minimum in the expected way: after the strong maximum principle forces λ_k/H to be constant, the Codazzi equations plus (3.5) force ∇λ_k and ∇H to vanish, and the self-expander equation gives H=0. Sound.\n\nThe soft spot is the infinity case. To rule out a negative infimum at infinity, the proof uses Lemma 2.6, taken from the author's earlier paper [35], which says the principal curvature corresponding to a zero principal curvature of the cone decays like O(r^{-3}) while H≈r^{-1}. This estimate is not re-derived here, and it is load-bearing. The alternative argument in Section 4 is a compact version of the same claim: it asserts that along the graph parametrization the limit of κ_1/H is the cone's radial curvature over H, i.e. zero. I suspect this is true, but a referee should verify that Lemma 2.6 does not secretly require the 2-convexity assumption of [35] and that the ordering of principal curvatures is preserved near infinity. This is a verification concern, not a demonstrated error.\n\nEverything else is standard. The strict-convexity step uses real analyticity and the parallel nullity argument; it is brief but acceptable. The self-citation is appropriate: the borrowed lemma is published and the rest of the paper does not depend on unpublished work.\n\nThis paper is for specialists in geometric flows. It deserves a serious referee. I would recommend sending it out, with an explicit request to scrutinize Lemma 2.6 and the infinity-infimum argument. My own verdict is close to the reader's: accept with moderate confidence.","headline":"Xie removes the 2-convex hypothesis from the convexity theorem for self-expanders; the proof is sound, with one imported decay estimate that deserves referee scrutiny.","tokens_in":17043,"tokens_out":6302,"would_cite":true,"duration_ms":72216,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53C44","53C42"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that a complete mean convex self-expander asymptotic to a mean convex weakly convex cone is strictly convex.","keywords":["self-expanders","mean curvature flow","convexity","principal curvatures","asymptotically conical","maximum principle","Codazzi tensors","strict convexity"],"falsifier":"The direct falsifier is an example: an n≥3 complete two-sided mean convex self-expander asymptotic to a mean convex weakly convex cone whose smallest principal curvature is negative somewhere. A concrete route is to compute κ1 along a rotationally symmetric self-expander asymptotic to a strictly convex double cone; finding a negative value at any interior radius would contradict Theorem 1.1.","tokens_in":16049,"feed_emoji":"📐","tokens_out":8861,"duration_ms":100691,"temperature":0.7,"pith_summary":"The paper proves that, in dimensions n≥3, any complete mean convex self-expander of the mean curvature flow that is asymptotic to a mean convex and weakly convex cone must be strictly convex: every principal curvature is positive everywhere. This upgrades an earlier weak-convexity theorem that required 2-convexity, and extends the known two-dimensional convexity statement to all higher dimensions. The argument rules out a negative infimum of the ratio of the smallest principal curvature to the mean curvature, both at interior points and at infinity. Interior minima are excluded by a maximum-principle argument applied to a smooth partial sum of the smallest principal curvatures; the infinity case is excluded by the convexity of the asymptotic cone, which forces the ratio to tend to zero at infinity.","feed_headline":"Mean-convex expanders over weakly convex cones are strictly convex","feed_subtitle":"A curvature-maximum argument rules out negative curvature and upgrades mean convexity to strict convexity for all n≥3.","key_machinery":"The central object is the partial sum λ_k = κ1 + ... + κ_k of the ordered principal curvatures. Because the ratio κ1/H can fail to be smooth where multiplicities of principal curvatures vary, the proof shifts to λ_k/H, which is smooth on the open set {κ_k < κ_{k+1}} via a spectral-projection argument. A key algebraic identity for Codazzi tensors shows that a curvature-gradient term in the drift Laplacian of the cutoff function φ(λ_k/H) is nonpositive, yielding an elliptic differential inequality. Combined with the strong maximum principle, this rules out negative interior infima. At infinity, the imported decay estimate—principal curvatures in the cone's flat directions decay like r^{-3} whi","core_discovery":"The paper's central result is Theorem 1.1: for n≥3, every complete immersed two-sided mean convex self-expander Σ^n in R^{n+1} that is asymptotic to a cone which is mean convex and weakly convex is strictly convex, meaning all principal curvatures are positive at every point. The proof first establishes weak convexity by contradiction, ruling out a negative infimum of the ratio κ1/H. Ruling out an interior infimum uses a maximum-principle argument for the smooth partial-sum ratio λ_k/H on the region where the k-th and (k+1)-th principal curvatures are separated. Ruling out an infimum at infinity uses the convexity of the asymptotic cone together with the decay rates of the principal curvatur","pith_inferences":["The paper's elliptic inequality and algebraic identity are remarked to hold for self-shrinkers and translators as well; if the analogous asymptotics at infinity are available, similar strict-convexity statements should follow for those solitons.","The proof's dependence on the O(r^{-3}) decay suggests that dropping weak convexity of the cone should allow genuinely mean convex but non-convex self-expanders; the paper itself anticipates such examples but does not construct one.","The spectral-projection smoothing of partial sums of ordered principal curvatures is likely a standalone tool: in any setting with a Codazzi tensor whose ordered eigenvalues have gaps, partial sums are smooth across those gaps, which may help in other maximum-principle rigidity proofs."],"forward_implications":["Under the theorem, every such self-expander is a strictly convex hypersurface, so all n principal curvatures are positive at every point.","Because the expander is self-similar, the entire mean curvature flow {√tΣ} starting from it consists of strictly convex hypersurfaces, so positive curvature persists from the cone through the whole flow.","The ratio κ1/H can never have a negative infimum, neither at an interior point nor at infinity; the convexity of the asymptotic cone forces the ratio to zero at infinity.","The theorem extends the known two-dimensional convexity statement to all dimensions n≥3 and removes the stronger 2-convexity assumption used in the earlier weak-convexity proof."],"supporting_citations":[{"why":"Supplies the earlier weak-convexity result for 2-convex asymptotically conical expanders and the decay estimate (Lemma 2.6) that rules out the infimum-at-infinity case.","marker":"[35]"},{"why":"Supplies the cutoff-function and contradiction strategy used to exclude negative interior infima of the smallest-curvature ratio.","marker":"[31]"},{"why":"Supplies the theorem that principal curvatures and eigendistributions of a Codazzi tensor are differentiable on an open dense subset, enabling computations when multiplicities vary.","marker":"[13]"},{"why":"Also supplies the regularity of principal curvatures and eigendistributions on an open dense subset, used in the local curvature-frame computations.","marker":"[28]"},{"why":"Supplies the spectral-projection construction proving that the partial sum λ_k of ordered principal curvatures is smooth on the set where the k-th and (k+1)-th curvatures separate.","marker":"[23]"},{"why":"Supplies the strong maximum principle used to exclude negative interior minima and, via invariance of the kernel under parallel translation, to upgrade weak convexity to strict convexity.","marker":"[18]"},{"why":"Supplies the graph parametrisation of asymptotically conical self-expanders over the asymptotic cone and the rigidity statement used to rule out a Euclidean-factor split.","marker":"[15]"},{"why":"Supplies the argument that the nullspace of the second fundamental form is invariant under parallel translation for weakly convex flow, the key step in proving strict convexity.","marker":"[8]"},{"why":"Supplies the evolution equation for the second fundamental form under mean curvature flow, used together with the maximum principle in the strict-convexity step.","marker":"[19]"}],"fun_headline_variants":["Mean convex self-expanders over convex cones are strictly convex","Self-expanders asymptotic to convex cones gain strict convexity","Strict convexity for mean convex self-expanders with convex cones","Convex cones ensure strict convexity in mean convex self-expanders"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The proof imports a decay estimate from an earlier paper: along an asymptotically conical self-expander, the principal curvatures in the cone's flat directions decay like r^{-3} while the mean curvature decays like r^{-1}; if this failed, the ratio κ1/H could remain negative at infinity and the argument would not close.","fun_headline_variants_meta":{"raw":{"variants":["Mean convex self-expanders over convex cones are strictly convex","Self-expanders asymptotic to convex cones gain strict convexity","Strict convexity for mean convex self-expanders with convex cones","Convex cones ensure strict convexity in mean convex self-expanders"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000753,"raw_usage":{"total_tokens":3108,"prompt_tokens":588,"completion_tokens":2520,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":332,"completion_tokens_details":{"reasoning_tokens":2449}},"tokens_in":332,"tokens_out":2520,"duration_ms":23523,"temperature":1.0,"reasoning_tokens":2449,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T12:58:21.045976+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"The direct falsifier is an example: an n≥3 complete two-sided mean convex self-expander asymptotic to a mean convex weakly convex cone whose smallest principal curvature is negative somewhere. A concrete route is to compute κ1 along a rotationally symmetric self-expander asymptotic to a strictly convex double cone; finding a negative value at any interior radius would contradict Theorem 1.1.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the earlier weak-convexity result for 2-convex asymptotically conical expanders and the decay estimate (Lemma 2.6) that rules out the infimum-at-infinity case."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the cutoff-function and contradiction strategy used to exclude negative interior infima of the smallest-curvature ratio."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the theorem that principal curvatures and eigendistributions of a Codazzi tensor are differentiable on an open dense subset, enabling computations when multiplicities vary."},{"cited_title":"1, 135–144","cited_arxiv_id":null,"evidence_quote":"Also supplies the regularity of principal curvatures and eigendistributions on an open dense subset, used in the local curvature-frame computations."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the spectral-projection construction proving that the partial sum λ_k of ordered principal curvatures is smooth on the set where the k-th and (k+1)-th curvatures separate."},{"cited_title":"Differential Geom","cited_arxiv_id":null,"evidence_quote":"Supplies the strong maximum principle used to exclude negative interior minima and, via invariance of the kernel under parallel translation, to upgrade weak convexity to strict convexity."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the graph parametrisation of asymptotically conical self-expanders over the asymptotic cone and the rigidity statement used to rule out a Euclidean-factor split."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the argument that the nullspace of the second fundamental form is invariant under parallel translation for weakly convex flow, the key step in proving strict convexity."},{"cited_title":"Differential Geom","cited_arxiv_id":null,"evidence_quote":"Supplies the evolution equation for the second fundamental form under mean curvature flow, used together with the maximum principle in the strict-convexity step."}],"review_version":1}