{"id":"cd0cd16a-3de3-492b-b4a6-86d31ca07644","arxiv_id":"2412.09396","paper_version":4,"verdict":"REJECT","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper claims eigenvalue lower bounds λ > Ric_h − c|∇h|² and an L_h-stability theorem for h-minimal hypersurfaces, but the proof only supports λ > inf(Ric_h − c|∇h|²).","lead":"This paper claims new lower bounds for the first Dirichlet and Neumann eigenvalues of a weighted Laplacian in terms of Bakry-Emerici Ricci curvature, and uses them to prove a stability condition for h-minimal hypersurfaces. A reader might care because eigenvalue and stability estimates are central tools in geometric analysis, but the main theorem as stated is not well-defined and the proof does not support the claim.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1's proof contradicts the global hypothesis λ≤S everywhere, not the existence of one point where λ≤S; it establishes at most λ>inf S, while the statement and Theorem 2 require λ>S pointwise.","rationale":"The reader's verdict is REJECT, and the load-bearing concern is exactly the quantifier/type error in Theorem 1. The central claim, as stated, compares an eigenvalue to a tensor field, and the proof's contradiction step assumes the inequality fails everywhere rather than at some point. That invalidates the derivation of the pointwise bound. The subsequent application in Theorem 2 depends on replacing λ by the pointwise function S in (2.34), so the stability theorem inherits the error. I checked the surrounding computations in Proposition 2.5 and Theorem 2 and found no independent reason to accept the stability conclusion; even if those computations are correct, the chain breaks at Theorem 1. Thus the rejection is warranted.","tokens_in":11143,"tokens_out":22402,"duration_ms":242361,"concrete_test":"Independently re-derive (2.13)→(2.14) under ¬P: ∃p,v λ≤S(p,v). The replacement Rich(∇f,∇f)≥(c|∇h|²+λ)|∇f|² requires S≥λ on all M, so it fails; a test function near p leaves the sign indeterminate. This settles that the proof yields only λ>inf S. Complementary check: on a weighted disk with h=εr², compute λ_N and compare with max and min of S=Rich−c|∇h|²; the statement requires λ>max S, but the proof only supports λ>min S.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In Theorem 1 the conclusion λ_{1,D} > Ric_h − c|∇h|² (and similarly λ_{1,N}) is not type-correct: the left side is a real number and the right side is a tensor field. The only coherent reading is the pointwise quadratic-form inequality λ > S(p,v) for every p∈M and unit v∈T_pM, where S=Rich−c|∇h|². The proof does not establish this. After (2.13) the authors say 'suppose that this inequality does not hold. That is, suppose that λ_{1,D} ≤ Ric_h − c|∇h|²'. That supposition is the global statement λ≤S everywhere. The actual negation of the desired pointwise claim is ∃p,v with λ≤S(p,v). The global supposition is used to pass from (2.13) to (2.14), replacing Rich(∇f,∇f) by the lower bound (c|∇h|²+λ)|∇f|². A contradiction from this stronger hypothesis only shows ∃p,v with S(p,v)<λ, i.e. λ>inf S; it does not rule out points where S(p,v)>λ. Theorem 2 exposes the gap: in (2.34) the scalar λ is replaced by the function (Rich−c|∇h|²)/2 inside an integrand, which is legitimate only if λ>S pointwise. With only λ>inf S, the integrand can change sign where S exceeds λ, so the Lh-stability conclusion is unsupported. The proof also requires c>0 in the choice m≥n+1/c, although Theorem 1 omits that hypothesis (Theorem 2 states it).","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims a lower bound for the first Dirichlet and Neumann eigenvalues of the drifting Laplacian on a compact weighted manifold with boundary in terms of the Bakry-Émery Ricci tensor: λ > Ric_h − c|∇h|² under the hypothesis Ric_h > c|∇h|² and suitable boundary conditions. It then uses this estimate to prove an Lh-stability criterion for compact h-minimal hypersurfaces with parallel Hessian of the weight. The proofs are based on a weighted Reilly-type formula (Proposition 2.1), a Bochner formula, and an eigenfunction argument; Theorem 2 combines the eigenvalue bound with an identity for L_h(fH) and Young's inequality.","tokens_in":11424,"tokens_out":20813,"duration_ms":203865,"significance":"If correct, the results would supply a new eigenvalue comparison with a curvature tensor and a stability criterion for weighted minimal hypersurfaces. The paper is self-contained and Proposition 2.1 is a standard weighted Reilly inequality that is plausibly correct; the proofs do not assume the desired conclusion, so circularity is not the issue. However, the main theorem is ill-posed as a comparison of a scalar eigenvalue with a tensor field, and the proof establishes at most λ > inf(Ric_h − c|∇h|²), not the pointwise inequality stated. Since Theorem 2 relies on the pointwise form inside an integral, the stability application is not established. The central claim as stated is therefore not credible.","major_comments":[{"comment":"Theorem 1 asserts λ_{1,D} > Ric_h − c|∇h|² with λ a real number and Ric_h a (0,2)-tensor; this inequality is not type-correct. If it is read as the pointwise quadratic-form inequality λ_{1,D} > Ric_h(v,v) − c|∇h|²(p) for every p and unit v, then the proof does not establish it. In the proof, the supposition 'λ_{1,D} ≤ Ric_h − c|∇h|²' is treated as a global pointwise assumption, and the contradiction obtained from (2.13)–(2.15) only rules out the possibility that the pointwise reverse inequality holds everywhere. The argument therefore implies at most λ_{1,D} > inf_{p,v}(Ric_h(v,v) − c|∇h|²(p)); the stated pointwise conclusion does not follow.","section":"§2.1, Theorem 1 and Eqs. (2.13)–(2.15)"},{"comment":"The passage from (2.14) to (2.15) requires the coefficient of ∫|∇f|²|∇h|² to be nonnegative, i.e. c − 1/(m−n) ≥ 0. For c ≤ 0 this fails, and the condition m ≥ n + 1/c is vacuous when c is negative. Since Theorem 1 only says 'there exists a constant c' without any sign condition, the proof does not cover c ≤ 0; if the intended hypothesis is c > 0, it must be stated explicitly.","section":"§2.1, Eqs. (2.14)–(2.15)"},{"comment":"In deriving (2.34), the scalar λ_{1,D} is replaced inside the integrand by the function (Ric_h − c|∇h|²)/2. This replacement is legitimate only if the pointwise inequality λ_{1,D} > Ric_h − c|∇h|² holds on all of M. The proof of Theorem 1 supplies at most λ_{1,D} > inf(Ric_h − c|∇h|²), which does not control the sign of the integrand pointwise. Consequently the passage to (2.34) and the Lh-stability conclusion of Theorem 2 are unsupported.","section":"§2.2, Eq. (2.34)"},{"comment":"Both condition (1.6), 'Rich ≥ 2[|A|² + c|∇h|² + ...]', and the integrand in (2.34), '(Rich − c|∇h|²)/2', treat the Bakry–Émery tensor as a scalar function. If 'Rich ≥ ...' is meant as a quadratic-form inequality, the right-hand side must be tensorial, e.g. the scalar expression times the metric; if 'Rich' is meant as the infimum or the smallest eigenvalue of the tensor, that notion is not defined and is not the object used in Proposition 2.1. As written, the hypotheses and the application of Theorem 1 in Theorem 2 are mathematically ambiguous.","section":"§2.2, Eqs. (1.6) and (2.34)"}],"minor_comments":[{"comment":"There are numerous typographical errors, including 'knwon', 'eingenvalue', 'areises', 'preceeds', and 'obtainded'; these should be corrected in any revision.","section":"Introduction and throughout"},{"comment":"For f vanishing on ∂M, ∇f = f_η η, so the intermediate expression ∇f + f_η η equals 2f_η η and appears inconsistent with the following equality; the final boundary term is correct because of the factor 1/2 in (2.6), but the displayed chain of equalities should be rewritten.","section":"§2.1, Eq. (2.11)"},{"comment":"The notation ∇f = ∇̄f in the Neumann case is not defined; presumably the bar denotes the tangential gradient on the boundary, and this should be stated.","section":"§2.1, Theorem 1 proof, item (2)"},{"comment":"The claim that Theorem 1 generalizes the Ma–Du estimate is not substantiated: with c = 1/(m−n), the conclusion would give λ > a, whereas Ma–Du gives λ ≥ ma/(m−n), which is stronger for a > 0.","section":"Introduction, discussion after Theorem 1"},{"comment":"The text states 0 < λ_1 ≤ λ_2 for the Neumann problem, but the Neumann spectrum contains the zero eigenvalue; presumably λ_{1,N} denotes the first nonzero Neumann eigenvalue, and this should be clarified.","section":"§1, Eq. (1.3)"}],"recommendation":"reject","confidential_remarks":"The central theorem is not merely underproved but ill-posed as a comparison of a scalar with a tensor, and the proof gap between the global and pointwise negations is not a local fix. The stability application would require a different argument even if the eigenvalue statement were repaired as an infimum bound. The paper does contain a plausible weighted Reilly-type formula that could serve as the basis of a revised manuscript, but the current central claims are not supported."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe paper has one new result that looks real—the L_h-stability criterion for h-minimal hypersurfaces under condition (1.6)—but the main eigenvalue theorem it rests on is not proven as stated. The proof only gets λ > inf(Ric_h − c|∇h|²), not the pointwise inequality λ > Ric_h − c|∇h|² that the statement and the stability application require.\n\nThe good part: Proposition 2.5, the computation of L_h(fH), is a serious piece of tensor analysis, and the stability condition (1.6) is new and geometrically meaningful. If the eigenvalue input were valid, Theorem 2 would be a nice application. The authors also correctly identify that their condition is different from Ma-Du's; they do not have the additive constant a, and with c = 1/(m−n) their bound is actually weaker, so the claim that Theorem 1 generalizes Ma-Du is not substantiated.\n\nThe soft spots are load-bearing. Theorem 1 compares a scalar λ to a tensor field; the only coherent reading is the pointwise quadratic-form inequality. The proof's contradiction starts by assuming the global statement λ ≤ S everywhere, which is not the negation of the pointwise claim. It rules out λ ≤ inf S, so at best it proves λ > inf S. Theorem 2's inequality (2.34) needs the integrand ½(Ric_h − c|∇h|²) − ... to be nonnegative pointwise, which only follows from the pointwise eigenvalue inequality. The gap is not a minor detail; the stability theorem is unsupported without it. Also, Theorem 1 omits c > 0, which the proof requires when choosing m ≥ n + 1/c (Theorem 2 states it correctly).\n\nThere are also presentation issues: typos like 'areises', 'preceeds', 'eingenvalue', inconsistent notation for Hessian inner products, and the abstract's 'We will present' reads like a talk announcement rather than a paper.\n\nWho is this for? Geometers working on drifting Laplacians and f-minimal hypersurfaces. A careful reader will see the gap quickly, but the stability calculation itself might be useful for parts. I would not cite the results as stated. If the authors can salvage a correct weaker eigenvalue theorem (λ > inf S) and redo Theorem 2 with that or with a pointwise condition that is actually proved, the paper could become a modest contribution. In current form, it should not be accepted.\n\nRecommendation: send to peer review only if you expect the referee to identify the gap and require major revision; the work is not ready for publication. If you want a cleaner desk decision, reject. My own verdict is reject, but I would let a referee confirm the gap because it is quantitative and fixable.","headline":"The eigenvalue theorem is not proven as stated—only λ > inf S follows—and the stability theorem rests on it; the stability computation itself is real.","tokens_in":12023,"tokens_out":3579,"would_cite":false,"duration_ms":33842,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35P15","53C23","53C42","58K25"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper claims that on compact weighted manifolds, the first Dirichlet and Neumann eigenvalues of the drifting Laplacian are strictly larger than $\\mathrm{Ric}_h - c|\\nabla h|^2$ under the condition $\\mathrm{Ric}_h > c|\\nabla h|^2$ and…","keywords":["weighted manifolds","Bakry-Émery Ricci curvature","drifting Laplacian","first eigenvalue","Dirichlet and Neumann problems","h-minimal hypersurface","L_h-stability","Reilly formula"],"falsifier":"On a Euclidean ball $B_R$ with density $e^{-\\epsilon r^2}$, take $c$ small enough that $\\mathrm{Ric}_h = 2\\epsilon I > 4c\\epsilon^2 r^2 I$ holds and the boundary has nonnegative weighted mean curvature; compute $\\lambda_{1,D}$ numerically and compare it with the infimum and supremum of $2\\epsilon - 4c\\epsilon^2 r^2$ over $B_R$. An eigenvalue below the supremum refutes the pointwise claim, while an eigenvalue below the infimum refutes the weaker bound the proof actually supports.","tokens_in":10898,"feed_emoji":"📐","tokens_out":13576,"duration_ms":120484,"temperature":0.7,"pith_summary":"The paper aims to establish strict lower bounds for the first nonzero eigenvalues of the Dirichlet and Neumann problems for the drifting Laplacian $\\Delta_h$ on a compact weighted manifold $M_h = (M^n, g, e^{-h}dv)$, with the bounds expressed through the Bakry-Émery Ricci curvature $\\mathrm{Ric}_h = \\mathrm{Ric} + \\nabla^2 h$. The central claim is that under the pointwise condition $\\mathrm{Ric}_h > c|\\nabla h|^2$ for some constant $c$, the first eigenvalue is strictly larger than $\\mathrm{Ric}_h - c|\\nabla h|^2$, provided the boundary has nonnegative weighted mean curvature (Dirichlet) or is convex (Neumann). This is meant to generalize earlier estimates of Ma and Du and, for constant weight, to give a strict Reilly-type Lichnerowicz–Obata bound $\\lambda_1 > \\mathrm{Ric}$. The same eigenvalue estimate is then applied to prove a sufficient curvature condition for a compact $h$-minimal hypersurface with boundary to be $L_h$-stable. A reader would care because the bounds connect the spectrum of natural diffusion operators on weighted spaces to the curvature quantity that governs optimal transport and gradient Ricci solitons.","feed_headline":"First eigenvalue beats Bakry-Émery minus weight term","feed_subtitle":"A Reilly-style argument gives strict Dirichlet and Neumann bounds, then a stability test for h-minimal hypersurfaces.","key_machinery":"The engine of the paper is the weighted Reilly formula in Proposition 2.1, obtained by integrating the weighted Bochner formula, with the Hessian term controlled through $|\\nabla^2 f|^2 \\ge \\frac{(\\Delta_h f)^2}{m} - \\frac{\\langle\\nabla f,\\nabla h\\rangle^2}{m-n}$ for $m>n$. Boundary control comes from the boundary integral $\\frac12\\int_{\\partial M_h}\\langle\\nabla|\\nabla f|^2,\\eta\\rangle\\,da_h$, which is shown to be nonpositive for Dirichlet eigenfunctions under $H_h^{\\partial M}\\ge0$ and for Neumann eigenfunctions under convexity of the boundary. For Theorem 2, the key identities are the explicit formula for $L_h(fH)$ in Proposition 2.5, Young's inequality, and the variational characterization of $\\lambda_{1,D}$, which together convert the eigenvalue estimate into the stability inequality.","core_discovery":"On the paper's own terms, the central discovery is Theorem 1: if an $n$-dimensional compact weighted manifold $M_h$ has nonempty smooth boundary, $\\mathrm{Ric}_h > 0$, and $\\mathrm{Ric}_h > c|\\nabla h|^2$ for some constant $c$, then the first Dirichlet eigenvalue satisfies $\\lambda_{1,D} > \\mathrm{Ric}_h - c|\\nabla h|^2$ when the weighted mean curvature of the boundary is nonnegative, and the first Neumann eigenvalue satisfies $\\lambda_{1,N} > \\mathrm{Ric}_h - c|\\nabla h|^2$ when the boundary is convex. The proof derives a weighted Reilly-type inequality and then argues by contradiction, so the inequality is strict. Theorem 2 applies this bound, together with parallelism of $\\nabla^2 h$ and the curvature lower bound (1.6), to conclude that a two-sided compact $h$-minimal hypersurface with $H\\ne0$ and $H_h^{\\partial M}\\ge0$ is $L_h$-stable.","pith_inferences":["The contradiction argument in Theorem 1 actually appears to rule out only eigenvalues at or below $\\inf(\\mathrm{Ric}_h - c|\\nabla h|^2)$; if that reading is right, the stated pointwise inequality is stronger than the proof establishes, and Theorem 2, which uses the pointwise form, inherits the gap.","The weighted Reilly inequality in Proposition 2.1 is not tied to the first eigenvalue; the same boundary-term analysis could be used to estimate higher eigenvalues or the fundamental tone of submanifolds with controlled boundary.","A concrete test of the stability criterion would be to compute the left-hand side of (1.6) for a family of model hypersurfaces (for instance, geodesic spheres in a warped product with a Gaussian weight) to see whether the threshold is sharp or only sufficient.","If the pointwise reading of Theorem 1 is abandoned, the stability conclusion might still survive if $\\mathrm{Ric}_h - c|\\nabla h|^2$ is replaced by its infimum in condition (1.6), suggesting the paper's framework is close to a correct statement."],"forward_implications":["If the pointwise form of Theorem 1 is correct, then whenever $\\mathrm{Ric}_h > c|\\nabla h|^2$ and $H_h^{\\partial M}\\ge0$, the Dirichlet eigenvalue satisfies $\\lambda_{1,D} > \\mathrm{Ric}_h - c|\\nabla h|^2$ at every point, and the analogous Neumann statement holds under convexity of the boundary.","Specializing to constant $h$ recovers strict Reilly-type bounds for the ordinary Laplacian, namely $\\lambda_1 > \\mathrm{Ric}$, for Dirichlet data with nonnegative mean curvature and for Neumann data on a convex boundary.","The authors note that for an appropriate choice of $c$ the estimate recovers the Ma–Du lower bound for drifting Laplacians, placing the new inequalities as strict refinements of previously known non-strict bounds.","Theorem 2 yields a sufficient condition for $L_h$-stability: a compact $h$-minimal hypersurface with nonnegative weighted boundary mean curvature, parallel Hessian of $h$, and $\\mathrm{Ric}_h \\ge 2[|A|^2 + c|\\nabla h|^2 + (|\\nabla^2 h|^2 + |\\nabla H|^2)/H^2]$ is stable under compactly supported variations."],"supporting_citations":[{"why":"Supplies the extended Reilly formula for the drifting Laplacian and the earlier eigenvalue estimates that Theorem 1 generalizes.","marker":"[13]"},{"why":"Introduces the Bakry–Émery Ricci tensor and the weighted Laplacian, which form the paper's curvature measure.","marker":"[4]"},{"why":"Provides the hypersurface Laplacian decomposition used to compute boundary terms in the proof of Theorem 1.","marker":"[3]"},{"why":"Proves sharpness of the Ma–Du lower bounds, which the strict estimates in Theorem 1 refine.","marker":"[11]"},{"why":"Establishes background on $L_h$-stability of weighted minimal surfaces, the notion used in Theorem 2.","marker":"[12]"}],"fun_headline_variants":["Bakry-Émery Ricci yields strict eigenvalue gap","Stability of h-minimal surfaces from eigenvalue bound","Weighted manifold eigenvalue estimate tightens stability test","Dirichlet and Neumann eigenvalues bounded by weighted Ricci","Strict eigenvalue inequality implies h-minimal stability"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that negating the claimed pointwise inequality gives the reverse inequality at every point, so the proof really establishes only $\\lambda_1 > \\inf(\\mathrm{Ric}_h - c|\\nabla h|^2)$ rather than the pointwise comparison stated in the theorem.","fun_headline_variants_meta":{"raw":{"variants":["Bakry-Émery Ricci yields strict eigenvalue gap","Stability of h-minimal surfaces from eigenvalue bound","Weighted manifold eigenvalue estimate tightens stability test","Dirichlet and Neumann eigenvalues bounded by weighted Ricci","Strict eigenvalue inequality implies h-minimal stability"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000198,"raw_usage":{"total_tokens":1290,"prompt_tokens":790,"completion_tokens":500,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":406,"completion_tokens_details":{"reasoning_tokens":425}},"tokens_in":406,"tokens_out":500,"duration_ms":5320,"temperature":1.0,"reasoning_tokens":425,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T17:06:58.457329+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"On a Euclidean ball $B_R$ with density $e^{-\\epsilon r^2}$, take $c$ small enough that $\\mathrm{Ric}_h = 2\\epsilon I > 4c\\epsilon^2 r^2 I$ holds and the boundary has nonnegative weighted mean curvature; compute $\\lambda_{1,D}$ numerically and compare it with the infimum and supremum of $2\\epsilon - 4c\\epsilon^2 r^2$ over $B_R$. An eigenvalue below the supremum refutes the pointwise claim, while an eigenvalue below the infimum refutes the weaker bound the proof actually supports.","supporting_citations":[{"cited_title":"Extension of Reilly formula with applications to eigen- value estimates for drifting Laplacians","cited_arxiv_id":null,"evidence_quote":"Supplies the extended Reilly formula for the drifting Laplacian and the earlier eigenvalue estimates that Theorem 1 generalizes."},{"cited_title":"Diffusions hypercontractives","cited_arxiv_id":null,"evidence_quote":"Introduces the Bakry–Émery Ricci tensor and the weighted Laplacian, which form the paper's curvature measure."},{"cited_title":"P., Manfio, F","cited_arxiv_id":null,"evidence_quote":"Provides the hypersurface Laplacian decomposition used to compute boundary terms in the proof of Theorem 1."},{"cited_title":"f -minimal surface and manifold with positive m- Bakry- ´Emery Ricci curvature","cited_arxiv_id":null,"evidence_quote":"Proves sharpness of the Ma–Du lower bounds, which the strict estimates in Theorem 1 refine."},{"cited_title":"Stable weighted minimal surfaces in manifolds with non-neg ative Bakry–Emery Ricci tensor","cited_arxiv_id":null,"evidence_quote":"Establishes background on $L_h$-stability of weighted minimal surfaces, the notion used in Theorem 2."}],"review_version":1}