{"id":"2837bee8-95e3-49c9-8807-715a67f8c09e","arxiv_id":"2608.08748","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For Witten-Laplacians with radial log-concave measures on space forms, the geodesic ball uniquely minimizes the harmonic mean of the first n nonzero Neumann eigenvalues.","lead":"This preprint proves a sharp isoperimetric inequality for the harmonic mean of the first n nonzero Neumann eigenvalues of a Witten-Laplacian on origin-symmetric domains in constant-curvature spaces with radial log-concave weights. It shows the centered ball is the unique minimizer whenever the radial weight is convex and satisfies a natural monotonicity condition.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 3.1 omits the comparison of angular modes k>=2, so the proof that μ1(B_R)=λ1 and that the first eigenfunctions have the form T1(r)ψ_i is incomplete; the gap is repairable but load-bearing.","rationale":"I re-derived the key monotonicity Lemma 3.3 and found the maximum-principle argument and the sign in equation (3.15) correct. The externally cited matrix trace inequality, Lemma 2.1, can be justified by convexity/Jensen, so dependence on [12] is not a major correctness risk. The most substantive flaw is in Lemma 3.1: the proof that the first nonzero eigenvalue on the ball comes from the k=1 angular mode is incomplete, because the k≥2 radial eigenvalues are never compared with λ1. This is load-bearing because the T1 used in Lemmas 3.2, 3.3, and throughout Section 4 is defined as the k=1 radial eigenfunction; if the true μ1(B_R) came from a higher angular mode, the trial construction would collapse. The gap is easily closed by the stated Rayleigh-quotient monotonicity, so the verdict remains CONDITIONAL rather than ACCEPT. The n≥2 issue is real but secondary, since the n=1 case is degenerate and the inequality there is trivial.","tokens_in":12993,"tokens_out":38250,"duration_ms":419286,"concrete_test":"Verify the missing comparison: for k≥2, set α_k = k(k+n−2) and let λ_{k,1} be the first eigenvalue of −(pT')' + α_k p T/Sκ^2 = μ p T with T(0)=0, T'(R)=0. Since α_k ≥ α_1 = n−1 and the Rayleigh quotient ∫p(T')^2 + α_k∫p T^2/Sκ^2 over ∫pT^2 is monotone increasing in α_k, conclude λ_{k,1} ≥ λ_{1,1} = λ1. Then re-examine Lemma 3.2 for n=1 to confirm that the boundary term at r=0 requires n≥2 and add that hypothesis to the theorem statement if needed.","verdict_should_be":"UNCHANGED","load_bearing_attack":"After separation of variables on B_R, each angular eigenvalue k(k+n−2) gives a radial problem of the form (3.5) with coefficient k(k+n−2)/Sκ(r)^2. To conclude that the first nonzero Neumann eigenvalue of (3.1) is the k=1 eigenvalue λ1, the paper proves only λ1<τ2, where τ2 is the second k=0 Neumann radial eigenvalue, and then asserts the conclusion. It never compares λ1 with the first radial eigenvalues λ_{k,1} for k≥2. If some λ_{2,1} were smaller than λ1, the function T1 used in Lemmas 3.2–3.3 and in the trial construction would not be the first eigenfunction, and the monotonicity T1/Sκ decreasing would not be the correct one. The missing comparison is true by the variational formula λ_{k,1} = min_T [∫ p(T')^2 + k(k+n−2)∫ p T^2/Sκ^2] / ∫ p T^2, which is increasing in k, so Lemma 3.1 can be repaired, but as written the proof is incomplete. Separately, the proof drops boundary terms at r=0 using Sκ(0)=0, which requires n≥2; the theorem does not state n≥2, although the n=1 case is trivial.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript proves a sharp isoperimetric inequality for the harmonic mean of the first n nonzero Neumann eigenvalues of the Witten-Laplacian on origin-symmetric Lipschitz domains in space forms, under a radial log-concave measure dγ_κ = e^{-φ(r)} dVol_κ. The main theorem, Theorem 1.1, states that if φ is convex and ((C_κ/S_κ)φ')' ≥ 0 on (0,R), then for Ω ∈ E_κ with the same weighted volume as the centered ball B_R, one has ∑_{i=1}^n 1/μ_i(Ω) ≥ n/μ_1(B_R), with equality only for the ball. The proof combines separation of variables on balls, monotonicity properties of the first radial eigenfunction T_1, the bathtub principle for radial rearrangements, a Hersch-type trace formulation, and a matrix trace inequality from He-Li-Tang. The result is presented as an extension of Gaussian and Euclidean results by Chiacchio, Gao-Wang, and Chen-Mao.","tokens_in":13242,"tokens_out":12824,"duration_ms":121444,"significance":"If correct, Theorem 1.1 gives a substantial unified sharp inequality for a broad class of weighted manifolds, removing the non-increasing assumption on φ that was present in prior work. It recovers known Gaussian-space inequalities and yields a new family of Szegő-Weinberger-type bounds. The proof strategy is conceptually clear and the paper is careful in connecting the monotonicity of T_1/S_κ to matrix inequalities. The main strengths are the parameter-free sharp inequality, the explicit equality statement, and the breadth of admissible weights (for example, φ(r)=r^{2k} in R^n). However, the current version has a load-bearing gap in the identification of the first eigenfunction on balls and relies on an unpublished lemma for the decisive trace step, so the verification is not yet complete.","major_comments":[{"comment":"The proof establishes λ1 < τ2, where τ2 is the second radial eigenvalue for k=0, and then immediately concludes that μ1(B_R)=λ1. This conclusion is not justified, because the separated spectrum also contains, for each angular quantum number k≥2, the first radial eigenvalues λ_{k,1} of the problems T'' + ((n-1)C_κ/S_κ - φ')T' + (μ - k(k+n-2)/S_κ^2)T = 0. The paper never compares λ1 with λ_{k,1} for k≥2. This comparison is load-bearing: the subsequent Lemmas 3.2 and 3.3 and the trial functions in Section 4 all use that T_1 is a genuine first eigenfunction. The gap is repairable: the Rayleigh quotient for the k-th angular mode, namely [∫ p(T')^2 + k(k+n-2)∫ p T^2/S_κ^2]/∫ p T^2, is increasing in k, so λ1 ≤ λ_{k,1} for all k≥1; this argument should be included explicitly.","section":"Section 3, Lemma 3.1"},{"comment":"The decisive matrix inequality (2.2), used to pass from the bounds J ⪰ aI + cZ and K ⪯ λaI - dZ to the final trace lower bound, is quoted from the unpublished preprint [12] by He-Li-Tang. Because this lemma is the central engine of the proof and its equality statement is also used, the manuscript should provide a proof in an appendix or cite a peer-reviewed version. As it stands, a reader cannot verify the key inequality except by trusting an unpublished source.","section":"Section 2, Lemma 2.1"},{"comment":"The paper does not state n≥2, but several arguments rely on n≥2. In particular, Lemma 3.2 drops the boundary term at r=0 using S_κ(0)=0, which fails for n=1 because S_κ(0)^{n-1}=1; Lemma 3.1's multiplicity-n statement is also false in dimension one. The n=1 case is presumably trivial or otherwise known, but it should be explicitly excluded or treated separately.","section":"Throughout, Theorem 1.1 and Lemmas 3.1–3.3"}],"minor_comments":[{"comment":"The derivation of (3.15) from (3.14) is omitted. The identity is correct after using (C_κ/S_κ)' = -1/S_κ^2 and collecting terms, but this algebra is central to the maximum-principle argument and should be displayed or at least sketched.","section":"Section 3, Lemma 3.3, identity (3.15)"},{"comment":"In (4.9) the factor S_κ(r)^2 should presumably be S_κ(r)^{n-1} to match the polar-coordinate integration dγ_κ = S_κ^{n-1}e^{-φ} dr dθ and the subsequent expression in (4.12); as printed the inequality is dimensionally inconsistent.","section":"Section 4, inequality (4.9)"},{"comment":"The text says that H'(y) is strictly decreasing, but Lemma 3.3 only establishes that T_1/S_κ is monotonically decreasing, i.e. non-increasing. If strictness is needed for the equality case or for the strict version of the bathtub principle, it should be proved or the statements should be weakened to non-strict monotonicity.","section":"Section 4, after (4.5)–(4.6)"},{"comment":"The sentence 'Hence, in all cases under consideration, for every r∈(0,R), 0>∫_0^r (μ-(n-1)/S_κ^2)T_1 p dr' is not self-evident; it relies on the fact that the integrand's indefinite integral has total mass zero at r=R and that μ-(n-1)/S_κ^2 is strictly increasing, forcing the cumulative integral to be strictly negative for r∈(0,R). This one-sentence justification would improve readability.","section":"Section 3, Lemma 3.2"},{"comment":"There are several typos, including 'muliplicity' and 'eigfenfunction' in Lemma 3.1 and 'domian' in Section 2; these should be corrected.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The paper depends on two recent arXiv preprints for core ingredients: [9] (Chiacchio) and especially [12] (He-Li-Tang). If the journal requires load-bearing external results to be published or proved in the paper, this needs resolution before acceptance. The main mathematical gap in Lemma 3.1 is easily repairable, and the rest of the proof appears coherent; once the angular-mode comparison, the n≥2 handling, and the proof or published reference for Lemma 2.1 are supplied, the result would be suitable for publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This paper proves that, for origin-symmetric Lipschitz domains in space forms with radial log-concave measure, the ball minimizes the harmonic mean of the first n nonzero Witten-Laplacian Neumann eigenvalues (equivalently maximizes μ1), under convex φ and ((Cκ/Sκ)φ')' ≥ 0. That genuinely extends Chen–Mao, which needed φ non-increasing, and recovers the Gaussian results of Chiacchio–Di Blasio, Gao–Wang, and Chiacchio. The condition covers φ = r^{2k} on R^n, so this is not a cosmetic generalization. The proof is the right kind: separation on balls, a comparison λ1 < τ2, a monotonicity lemma for T1/Sκ, then a Hersch-type trace inequality with the He–Li–Tang matrix inequality. The structure is coherent and the main theorem looks correct.\n\nCredit where due: the authors identify the correct convexity condition on the weight, and the monotonicity lemma (Lemma 3.3) is the technical heart; the maximum principle argument is plausible and the examples are useful.\n\nNow the soft spots. The most serious is in Lemma 3.1. To conclude μ1(B_R) = λ1, they only show λ1 < τ2, where τ2 is the second k = 0 radial eigenvalue. They never compare λ1 with the first radial eigenvalues for k ≥ 2. The comparison is true — the radial problem's eigenvalue increases with k(k+n−2) — but it is not written. As is, the proof of the multiplicity and form of the first eigenfunctions is incomplete. This is repairable, but it is load-bearing because the rest of the paper uses T1 and its monotonicity.\n\nSecond, the boundary argument at r = 0 uses Sκ(0) = 0 and drops terms; that requires n ≥ 2. The theorem never states n ≥ 2, though the n = 1 case is trivial. Minor but should be stated.\n\nThird, identity (3.15) is asserted after \"further simplification\" — the reader cannot verify it without doing the algebra. Given the importance of that identity, showing the steps would help. There are also two notational slips: in the displayed definition of d they write G(R)^2/R^2 instead of G(R)^2/Sκ(R)^2 (only the same for κ = 0), and (4.9) writes Sκ(r)^2 where the context needs Sκ(r)^{n-1}. These are typos, not mathematical errors.\n\nThe reliance on Lemma 2.1 from an unpublished preprint [12] is a citation concern, but the inequality itself is plausible and, if needed, can be proved separately; it is not a defect of this paper's argument.\n\nOverall: the main theorem is very likely correct, and the proof strategy is sound modulo the Lemma 3.1 gap. An editor should send this to a serious referee. I would bring it to the reading group and would cite it once the gap is closed.","headline":"New sharp harmonic-mean inequality for Witten-Laplacian Neumann eigenvalues under general convex radial weights; proof mostly sound but Lemma 3.1 has a repairable gap.","tokens_in":13839,"tokens_out":2735,"would_cite":true,"duration_ms":26441,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35P15","35J05","33A40"],"pacs":[],"model":"deepseek-v4-flash","headline":"For radial log-concave weights on space forms, the ball uniquely minimizes the sum of reciprocals of the first $n$ nonzero Witten-Laplacian Neumann eigenvalues among origin-symmetric Lipschitz domains.","keywords":["Neumann eigenvalues","Witten-Laplacian","harmonic mean","isoperimetric inequality","log-concave measure","space forms","Szegő-Weinberger inequality","reciprocal eigenvalue sums"],"falsifier":"Numerically compute the first eigenfunction on a geodesic ball for a smooth convex radial weight violating $((C_\\kappa/S_\\kappa)\\phi')'\\ge 0$, such as $\\phi(r)=e^r$ on a small Euclidean ball, and check whether $T_1(r)/S_\\kappa(r)$ is decreasing on $(0,R)$. If it increases somewhere, Lemma 3.3 is false and the proof collapses; if it stays monotone, the hypothesis is stronger than needed.","tokens_in":12753,"feed_emoji":"🔵","tokens_out":15727,"duration_ms":146127,"temperature":0.7,"pith_summary":"The paper proves a sharp isoperimetric inequality for the first $n$ nonzero Neumann eigenvalues of the Witten-Laplacian (a Laplacian with a drift induced by the weight) on origin-symmetric Lipschitz domains in the sphere, Euclidean space, and hyperbolic space, equipped with a radial log-concave measure. The central statement is that, under a convexity-plus-derivative condition on the radial weight, the ball of the same weighted volume uniquely minimizes the sum of the reciprocals of the first $n$ nonzero eigenvalues. This simultaneously yields a Szegő-Weinberger type bound: the ball maximizes the first nonzero eigenvalue. The main advance is that the weight no longer has to be non-increasing, so the Gaussian measure and many other radial log-concave weights are covered in a single framework.","feed_headline":"Ball minimizes reciprocal sum of weighted Neumann eigenvalues","feed_subtitle":"For radial log-concave measures, the ball uniquely minimizes the sum of the first n reciprocal Witten-Laplacian eigenvalues.","key_machinery":"The carrying object is the first nonzero eigenfunction $T_1(r)$ of the Witten-Laplacian on a geodesic ball, together with the sine-type radius function $S_\\kappa(r)$ ($\\sin r$, $r$, or $\\sinh r$) and its derivative $C_\\kappa=S_\\kappa'$. Lemma 3.3 is the load-bearing step: under the condition $((C_\\kappa/S_\\kappa)\\phi')'\\ge 0$, the ratio $T_1(r)/S_\\kappa(r)$ is decreasing on $(0,R)$. This monotonicity makes the accumulated function $A$ convex and $H$ strictly concave, so a bathtub-type rearrangement principle turns radial-slice integrals into affine bounds in the volume variable; those bounds become the matrix inequalities $J\\succeq aI+cZ$ and $K\\preceq \\mu_1(B_R)aI-dZ$ with $\\mathrm{tr}\\,Z=0$. A matrix trace inequality then yields $\\mathrm{tr}(K^{-1}J)\\ge n/\\mu_1(B_R)$, which, combined with the variational principle for reciprocal eigenvalue sums, gives the theorem.","core_discovery":"The paper's central claim is Theorem 1.1: let $M_\\kappa$ be a space form (the sphere for $\\kappa=1$, Euclidean space for $\\kappa=0$, hyperbolic space for $\\kappa=-1$), let $d\\gamma_\\kappa=e^{-\\phi(r)}\\,d\\mathrm{Vol}_\\kappa$ be a radial log-concave measure, and let $\\Omega$ be a connected origin-symmetric Lipschitz domain with the same weighted volume as the origin-centered geodesic ball $B_R$. If $\\phi$ is convex and $((C_\\kappa/S_\\kappa)\\phi')'\\ge 0$ on $(0,R)$, where $S_\\kappa(r)$ is $\\sin r$, $r$, or $\\sinh r$ and $C_\\kappa=S_\\kappa'$, then $\\sum_{i=1}^n 1/\\mu_i(\\Omega)\\ge n/\\mu_1(B_R)$, with equality if and only if $\\Omega=B_R$. In particular $\\mu_1(\\Omega)\\le \\mu_1(B_R)$: among such domains the ball is the unique maximizer of the first nonzero Witten-Laplacian Neumann eigenvalue and the unique minimizer of the sum of reciprocals of the first $n$ nonzero eigenvalues. The proof achieves this without any non-increasing assumption on $\\phi$, extending prior results that required monotone weights or the special structure of the Gaussian density.","pith_inferences":["The monotonicity condition $((C_\\kappa/S_\\kappa)\\phi')'\\ge 0$ may be more than a proof artifact: it is exactly what makes $T_1/S_\\kappa$ monotone. Testing smooth convex weights that violate it, such as $\\phi(r)=e^r$ on small Euclidean balls, could reveal whether the ball remains extremal under weaker hypotheses.","The same trial-function construction should transfer to any rotationally symmetric model space whose radial function satisfies the identities used here, so the constant-curvature setting is likely not essential.","Using higher spherical harmonics as trial functions might extend the reciprocal-sum bound to more than $n$ eigenvalues, provided the matrix trace inequality admits a generalized form."],"forward_implications":["Among origin-symmetric Lipschitz domains of fixed weighted volume satisfying the hypotheses, the ball is the unique maximizer of the first nonzero Witten-Laplacian Neumann eigenvalue.","The Gaussian-space reciprocal-sum inequalities for the first $n$ nonzero eigenvalues follow as a special case, since the Gaussian weight satisfies the derivative condition with equality.","The condition is satisfied by many convex weights, including $\\phi(r)=r^{2k}$ with $k\\ge 1$ in Euclidean space, so the result is not confined to monotone or Gaussian weights.","The inequality $n/\\mu_1(B_R)\\le \\sum_{i=1}^n 1/\\mu_i(\\Omega)\\le n/\\mu_1(\\Omega)$ gives two-sided control: the ball eigenvalue bounds the first eigenvalue of every admissible domain."],"supporting_citations":[{"why":"supplies Lemma 2.1, the matrix trace inequality that converts the two matrix bounds into $\\mathrm{tr}(K^{-1}J)\\ge n/\\lambda$.","marker":"[12]"},{"why":"supplies Lemma 2.2, the variational principle connecting reciprocal sums of Neumann eigenvalues to $\\mathrm{tr}(K^{-1}J)$.","marker":"[13]"},{"why":"supplies Lemma 2.3, the one-dimensional bathtub principle used to compare radial slices of $\\Omega$ with initial intervals of the ball.","marker":"[16]"},{"why":"the Gaussian-space first-eigenvalue result that Theorem 1.1 recovers as a special case.","marker":"[8]"},{"why":"the Gaussian first-$n$ eigenvalue reciprocal-sum result that the theorem extends to general radial log-concave measures.","marker":"[9]"},{"why":"the Gaussian reciprocal-sum bound for the first $n-1$ eigenvalues that the theorem generalizes.","marker":"[11]"},{"why":"the prior space-form Witten-Laplacian result with non-increasing convex weights that the present theorem improves by removing the monotonicity assumption.","marker":"[7]"}],"fun_headline_variants":["Ball minimizes sum of reciprocals of Witten-Laplacian eigenvalues","Weighted Neumann isoperimetric: ball uniquely optimal","No monotone weight needed: ball extrema for Witten-Laplacian","Radial log-concave measures: ball minimizes eigenvalue reciprocals"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is the monotonicity lemma: under $((C_\\kappa/S_\\kappa)\\phi')'\\ge 0$, the ratio $T_1(r)/S_\\kappa(r)$ of the first ball eigenfunction to the sine-type radius is decreasing on $(0,R)$, and every later comparison—convexity of $A$, concavity of $H$, and the two matrix bounds—depends on this fact.","fun_headline_variants_meta":{"raw":{"variants":["Ball minimizes sum of reciprocals of Witten-Laplacian eigenvalues","Weighted Neumann isoperimetric: ball uniquely optimal","No monotone weight needed: ball extrema for Witten-Laplacian","Radial log-concave measures: ball minimizes eigenvalue reciprocals"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000273,"raw_usage":{"total_tokens":1633,"prompt_tokens":940,"completion_tokens":693,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":556,"completion_tokens_details":{"reasoning_tokens":619}},"tokens_in":556,"tokens_out":693,"duration_ms":6518,"temperature":1.0,"reasoning_tokens":619,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T04:28:05.028205+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Numerically compute the first eigenfunction on a geodesic ball for a smooth convex radial weight violating $((C_\\kappa/S_\\kappa)\\phi')'\\ge 0$, such as $\\phi(r)=e^r$ on a small Euclidean ball, and check whether $T_1(r)/S_\\kappa(r)$ is decreasing on $(0,R)$. If it increases somewhere, Lemma 3.3 is false and the proof collapses; if it stays monotone, the hypothesis is stronger than needed.","supporting_citations":[{"cited_title":"Caract´ erisation variationnelle d’une somme de valeurs propres cons´ ecutives; g´ en´ eralisation d’in´ egalit´ es de P´ olya-Schiffer et de Weyl.C","cited_arxiv_id":null,"evidence_quote":"supplies Lemma 2.2, the variational principle connecting reciprocal sums of Neumann eigenvalues to $\\mathrm{tr}(K^{-1}J)$."},{"cited_title":"Lieb and Michael Loss.Analysis, volume 14 ofGraduate Studies in Mathematics","cited_arxiv_id":null,"evidence_quote":"supplies Lemma 2.3, the one-dimensional bathtub principle used to compare radial slices of $\\Omega$ with initial intervals of the ball."},{"cited_title":"Chiacchio and G","cited_arxiv_id":null,"evidence_quote":"the Gaussian-space first-eigenvalue result that Theorem 1.1 recovers as a special case."},{"cited_title":"A sharp Gaussian harmonic-mean inequality for Neumann eigenvalues of the Ornstein-Uhlenbeck operator","cited_arxiv_id":"2607.28328","evidence_quote":"the Gaussian first-$n$ eigenvalue reciprocal-sum result that the theorem extends to general radial log-concave measures."},{"cited_title":"An isoperimetric inequality for lower order Neumann eigenvalues in Gauss space","cited_arxiv_id":null,"evidence_quote":"the Gaussian reciprocal-sum bound for the first $n-1$ eigenvalues that the theorem generalizes."}],"review_version":1}