{"id":"aaf16c2e-c37b-4ab8-a2ea-9b12a1c325e7","arxiv_id":"2607.16025","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Sharp p-capacity comparison inequalities in terms of boundary mean curvature on manifolds with Ric ≥ -ng and Ric ≥ 0, with equality forcing warped-product rigidity, plus optimal normalization thresholds for scale-invariant relative-capacity lower bounds.","lead":"This mathematical paper proves sharp bounds on p-capacity — a nonlinear measure of how much \"charge\" a bounded region can hold — for curved, infinite spaces whose Ricci curvature is bounded below, with the bound expressed through the boundary's mean curvature and sharpened to an exact description of the spaces where equality occurs. It also pins down when a scale-invariant version of relative capacity must stay positive, with a clean threshold answer: the normalization expone","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.6's positive lower bounds rest entirely on p-eigenvalue estimates from unpublished preprints [7],[8]; if those fail, the claimed optimal μ≥1 thresholds lose their proof.","rationale":"The reader's weakest_assumption identified exactly the same load-bearing point: Theorem 1.6's lower bounds chain through (4.1) to λ_{1,p} estimates in the unpublished preprints [7] and [8]. I agree this is the single most load-bearing concern because it underpins an entire advertised theorem (the optimal normalization thresholds), whereas the equality-case gap—though real—affects only the rigidity half of Theorem 1.3 and can likely be filled with a standard matrix-Riccati argument. My verification of the upper-bound chain (2.2)–(2.8) and the model capacities found no algebraic error, so the main capacity inequalities (1.1)–(1.4) appear sound. Thus the paper is correctly assessed as CONDITIONAL: the central upper bounds are solid, but Theorem 1.6 is conditional on external claims that must be made available and checked. I do not recommend changing the verdict; the existing CONDITIONAL already reflects this concern.","tokens_in":15413,"tokens_out":18300,"duration_ms":172392,"concrete_test":"Obtain [7] and [8] (Jin–Lü, 2026) and verify the exact statements used: (i) λ_{1,p}(Ω) ≥ (p-1)(π_p/(2 diam Ω))^p for all domains O in complete Ric≥0 manifolds; (ii) Corollary 1.5 in [8] giving λ_{1,p}(O) ≥ C(p,n,κ,D_0)/D^p for diam(O)≤D_0 under Ric≥-nκ, and the exponential lower bound used in (1.8). Independently re-derive (i) for p=2 as a sanity check; if the claimed bound fails for a specific domain (e.g., a thin dumbbell with arbitrary diameter), Theorem 1.6(1) is refuted. For the equality-case gap, derive from the Riccati equation for the second fundamental form that equality in the volume comparison forces |A|² = H²/n and Ric(ν,ν) = -n; if this derivation cannot be completed, the rigidity claim in Theorem 1.3 is unproven.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 1.6 is a central advertised result, but its 'if' direction is not proven in the paper. For the Ric≥0 case with μ=1, the proof invokes 'the estimate in [7]' that λ_{1,p}(O) > (p-1)(π_p/(2 diam(O)))^p (Section 4, after (4.1)). For κ>0 with diam(O)≤D_0, it invokes 'Corollary 1.5 in [8]' for λ_{1,p}(O) ≥ C(p,n,κ,D_0)/D^p, and for (1.8) a lower bound λ_{1,p}(O) ≥ C(n,p)e^{-n diam(O)}. Neither the statements nor proofs are reproduced; [7] and [8] are unpublished 2026 preprints by two of the present authors. If either estimate is false or has stricter hypotheses (e.g., requires a two-sided Ricci bound or convexity), then (1.5), (1.7), and (1.8) lose their support, and the sharp thresholds 'iff μ≥1' are void. This is the most load-bearing missing support. A secondary gap: in the equality case of Theorem 1.3, the step from J(x,r)=e^{nr} to 'D²r = (J'/(nJ))g = g' (Section 2.2) is unjustified without the matrix Riccati equation; the scalar Jacobian equality gives only the mean curvature, not the full second fundamental form. This affects the rigidity statement but not the capacity bound itself.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves sharp upper bounds for the absolute p-capacity of compact smooth domains in complete non-compact manifolds with Ric ≥ -ng, expressed through the boundary mean curvature. The main results are three comparison estimates (Theorem 1.3), a Ric ≥ 0 variant (Theorem 1.5), and a family of scale-invariant lower bounds for relative p-capacity with supposedly optimal normalization exponents (Theorem 1.6). The upper-bound proofs use a Jacobian comparison estimate (2.1) from the authors' related work [12], construct explicit radial test functions, and characterize equality cases as warped-product models. The lower-bound part reduces capacity to the first p-Laplace Dirichlet eigenvalue and invokes eigenvalue estimates from two unpublished preprints [7], [8] by two of the authors.","tokens_in":15690,"tokens_out":3043,"duration_ms":33083,"significance":"If correct, the comparison inequalities are substantial: they give sharp, curvature-dependent bounds for p-capacity in terms of boundary mean curvature alone, with rigidity forcing warped-product ends, and they identify optimal ranges of normalization parameters for a scale-invariant relative capacity. The upper-bound chain, especially equations (2.2), (2.4), (2.6) and (2.8), is carefully computed and the model computations in Example 1.2 and Remark 1.4 are consistent. The paper is less self-contained in its second half: Theorem 1.6, a central advertised result, rests on eigenvalue estimates from two unpublished preprints by the same authors, and the rigidity claims for equality contain a gap in passing from scalar Jacobian equality to the full second fundamental form. These issues are fixable but must be addressed before the claims can be considered fully supported.","major_comments":[{"comment":"The positive lower bounds in (1.5), (1.7) and (1.8) are proved by combining (4.1) with eigenvalue estimates from [7] and [8], both unpublished 2026 preprints by two of the present authors. Neither the statements (beyond a brief inline quotation) nor the proofs are reproduced. If those estimates are false or carry additional hypotheses, the 'if μ≥1' directions of the sharp thresholds lose their proof. Please include full statements and sufficient proofs of the eigenvalue bounds, or rework the argument to rely on published results.","section":"Section 4, Theorem 1.6, eqs. (4.1)-(4.2) and (1.5)-(1.8)"},{"comment":"In each equality case, the step from the scalar Jacobian equality J(x,r)=e^{nr} (or its analogues) to 'D²r = J'(x,r)/(nJ(x,r)) g = g' is not justified. The Riccati comparison for J gives information about the mean curvature (the trace of D²r), not the full second fundamental form. Equality in the scalar comparison does not by itself imply the matrix Riccati equality. A separate matrix comparison argument is needed to conclude that the level sets have isotropic second fundamental form and hence that the metric splits as the advertised warped product. The same gap appears in the rigidity statements for (1.2), (1.3), (1.4), and Theorem 3.1.","section":"Section 2.2, equality cases of Theorem 1.3"}],"minor_comments":[{"comment":"The connectedness of ∂Ω is argued by invoking Theorem 3 of [3]. The hypotheses of that reference (dealing with boundaries in an AdS/CFT setting) should be checked or stated explicitly; otherwise the equality characterization is incomplete.","section":"Section 2.2, first equality case"},{"comment":"The phrase 'the same analysis in Section 3' should refer to Section 2.2; the referred analysis is in the preceding section of the current numbering.","section":"Section 3.1, proof of Theorem 3.1"},{"comment":"The paper cites two 2026 preprints by two of the authors for the key eigenvalue estimates. These should either be replaced by published sources, or their precise statements and necessary hypotheses should be stated in the text, as they are not yet publicly verifiable.","section":"References [7], [8]"}],"recommendation":"major_revision","confidential_remarks":"The upper-bound part of the paper is sound and the main comparison estimates are valuable. However, Theorem 1.6's proof is not self-contained: it depends on two unpublished preprints by two of the authors, and the eigenvalue bounds are not reproduced. In addition, the rigidity claims contain a technical gap in the Jacobian-to-second-fundamental-form step. Both are repairable, but they are load-bearing for the advertised theorems. I recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The main thing to know: Theorems 1.3 and 1.5 are the real content. The sharp comparisons for p-capacity in terms of boundary mean curvature under Ric ≥ -ng and Ric ≥ 0 are new, and the test-function computations are carefully done. I checked the chains (2.2), (2.4), (2.6), and (2.8); they are correct, including the sign-sensitive step in (2.8). The model capacities in Example 1.2 align, and the normalization thresholds in Theorem 1.6 are plausible.\n\nThe serious soft spot is Theorem 1.6. The positive lower bounds (1.5), (1.7), and (1.8) go through eigenvalue estimates quoted from [7] and [8], two unpublished preprints by two of the authors. Neither statements nor proofs are reproduced. That is load-bearing: if those bounds are false or carry extra hypotheses, the μ≥1 thresholds lose their support. The attached stress-test note is right about this. A referee should be able to see [7] and [8], or the paper should prove the needed lemmas.\n\nSecond, the equality analysis. In Section 2.2, the step from J(x,r)=e^{nr} to D²r=g is not justified. The scalar Jacobian equality gives only the mean curvature; the full second fundamental form requires the matrix Riccati equation and a comparison for the shape operator. The same issue appears in the H>n and Ric≥0 cases. This affects the rigidity statements, not the capacity bounds themselves. It is fixable, likely by invoking standard rigidity in the equality case of the Hessian comparison.\n\nThird, in case (1.2) the connectedness argument cites Theorem 3 of [3] for level sets with mean curvature n coth(r+θ₀)>n, but that theorem may be stated for H=n. I cannot inspect [3] from here, but the application looks suspicious. The Jacobian bound (2.1) is also cited to the authors' own [12] rather than proved or credited classically.\n\nNone of this makes me doubt the main inequalities (1.1)-(1.4). The paper deserves a serious referee. I would send it to review, with instructions to demand either reproduction of [7]-[8] or explicit statements of their results, and to fix the rigidity arguments.","headline":"Sharp p-capacity bounds check out, but the lower-bound half leans on unpublished preprints and the rigidity proofs skip a needed step.","tokens_in":16344,"tokens_out":7538,"would_cite":true,"duration_ms":70788,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53C21","31C15"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves sharp upper bounds for the p-capacity of a compact domain in a complete non-compact (n+1)-manifold with Ricci curvature at least −ng, expressed purely through the boundary's mean curvature, and characterizes equality by is","keywords":["p-capacity","p-Laplacian","Ricci curvature","mean curvature","warped product","relative capacity","condenser","eigenvalue estimate"],"falsifier":"On the warped product $(R^{n+1}, dr^2 + f(r)^2 g_{S^n})$ with $f(r) = \\sinh r$ for $r \\leq 1/2$ and $f(r) = C e^{-r}$ for $r \\geq 1$ (the example built in Section 4), compute the first Dirichlet $p$-eigenvalue $\\lambda_{1,p}$ of the annulus $\\{R \\leq r \\leq R+D\\}$ for large $R$. If $\\lambda_{1,p}$ decays faster than $e^{-n(R+D)}$ as $R \\to \\infty$, the key eigenvalue input from the companion paper fails and the uniform lower bound (1.8) collapses.","tokens_in":15171,"feed_emoji":"⚡","tokens_out":5904,"duration_ms":66519,"temperature":0.7,"texified_at":"2026-08-05T21:26:41.754694+00:00","pith_summary":"This paper proves sharp upper bounds for the $p$-capacity of a compact domain in a complete non-compact $(n+1)$-manifold with Ricci curvature at least $-ng$. The bound is expressed solely through the mean curvature of the boundary and matches three warped-product models: an exponential end, a hyperbolic end, and a cosh-type end, depending on whether the mean curvature equals, exceeds, or falls below $n$. Equality is rigid: it forces the exterior of the domain to be isometric to the corresponding model end, with the boundary mean curvature constant. The paper also studies scale-invariant lower bounds for relative capacity of condensers and shows a uniform positive lower bound exists exactly when the normalization exponent $\\mu$ satisfies $\\mu \\geq 1$, for both $\\mathrm{Ric} \\geq 0$ and $\\mathrm{Ric} \\geq -n\\kappa$ settings (with a diameter constraint or an exponential factor in the negative-curvature case). A sharp Euclidean lower bound for the normalized capacity is derived as a corollary.","texify_model":"deepseek-v4-flash","texify_usage":{"total_tokens":5767,"prompt_tokens":916,"completion_tokens":4851,"prompt_tokens_details":{"cached_tokens":0},"prompt_cache_hit_tokens":0,"prompt_cache_miss_tokens":916,"completion_tokens_details":{"reasoning_tokens":3887}},"feed_headline":"Boundary mean curvature sets sharp p-capacity bound","feed_subtitle":"Equality forces the exterior to be a warped model end, and optimal condenser normalizations are found.","key_machinery":"The proof rests on the normal Jacobian comparison $J(x, r) \\leq (\\cosh r + \\frac{H(x)}{n} \\sinh r)^n$ along geodesics emanating from $\\partial \\Omega$, which follows from the Riccati equation under $\\mathrm{Ric} \\geq -ng$. This inequality converts the $p$-Dirichlet energy of a radial test function into an explicit integral over $\\partial \\Omega$, yielding the three sharp bounds. The test functions are radial $p$-harmonic functions from the model spaces: $e^{nr/(1-p)}$ for the exponential end, $v_p(r)$ for the hyperbolic model, and $w_p(r)$ for the cosh model. Equality in the estimates reduces to equality in the Jacobian comparison and constant mean curvature, and the rigidity conclusions then follow from splitting theorems. For the lower-bound half, the inequ","core_discovery":"On the paper's own terms: for any $p > 1$ and any compact smooth domain $\\Omega$ in a complete non-compact $(n+1)$-manifold with $\\mathrm{Ric} \\geq -ng$, the $p$-capacity satisfies $\\mathrm{Cap}_p(\\Omega) \\leq \\left(\\frac{n}{p-1}\\right)^{p-1} \\int_{\\partial \\Omega} \\max\\{1, H/n\\}^n d\\sigma$, with equality exactly when $\\partial \\Omega$ is connected, $H \\equiv n$, and $M \\setminus \\Omega$ is isometric to $([0,\\infty) \\times \\partial \\Omega, dr^2 + e^{2r} g_{\\partial \\Omega})$. Analogous sharp estimates hold when $H > n$ (with a hyperbolic warped-product model) and when $0 \\leq H < n$ (with a cosh-type model), each with its own rigidity statement. For $\\mathrm{Ric} \\geq 0$ and $p \\in (1, n+1)$, a separate sharp bound holds with an asymptotically flat model end. On the lower-bound side, the paper introduces scale-invariant normalizations $C_{p,\\mu}$ and $D_{p,\\mu}$ for relative capacity and","pith_inferences":["The sharp upper bounds likely extend to boundary terms involving higher-order curvature invariants, since the Jacobian comparison only needs the Riccati equation and scalar mean curvature—higher-order terms might give refined versions of the same comparison.","The phase transition at µ = 1 suggests a sharp functional-analytic threshold: below it, capacity can be made arbitrarily small relative to volume and diameter, which may reflect the parabolic/hyperbolic nature of the manifold class M_κ.","The Ric ≥ 0 rigidity gives a global model for an asymptotically flat end without assuming scalar curvature bounds, hinting at possible connections to mass-capacity inequalities on asymptotically flat manifolds.","A testable numerical extension: on a warped product with nonconstant boundary mean curvature, the ratio Cap_p(Ω) divided by the right-hand side of (1.1) should lie in [0,1] and approach 1 as the mean curvature approaches a constant; computing this ratio for smooth perturbations could verify the stability of the inequality."],"forward_implications":["If the bounds hold, the p-capacity of a domain is controlled by its boundary mean curvature alone, with constants matching the model ends—useful for nonlinear potential theory on general manifolds.","The rigidity statements mean that any domain attaining equality is visibly a warped-product end, providing a geometric characterization that could be used to identify asymptotic models from capacity data.","The threshold µ ≥ 1 is optimal for the normalized relative capacity, separating manifolds of nonnegative Ricci curvature (where volume growth is polynomial) from those with negative lower bounds (where exponential volume growth dominates).","In the Ric ≥ 0 case, the sharp Euclidean lower bound for C_{p,1} gives an explicit, computable constant for condensers, with equality for concentric balls at a specific radius ratio depending on p and n."],"fun_headline_variants":["Mean curvature yields sharp p-capacity, equality rigid","Equality in p-capacity forces warped exterior","Scale-invariant bounds for condenser p-capacity","Ricci lower bound pins sharp p-capacity"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The lower-bound half of the paper borrows its decisive eigenvalue estimates from two unpublished companion preprints by the same authors; if those estimates fail, the optimality of $\\mu \\geq 1$ in (1.5), (1.7), and (1.8) loses its support.","fun_headline_variants_meta":{"raw":{"variants":["Mean curvature yields sharp p-capacity, equality rigid","Equality in p-capacity forces warped exterior","Scale-invariant bounds for condenser p-capacity","Ricci lower bound pins sharp p-capacity"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001459,"raw_usage":{"total_tokens":5718,"prompt_tokens":761,"completion_tokens":4957,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":505,"completion_tokens_details":{"reasoning_tokens":4896}},"tokens_in":505,"tokens_out":4957,"duration_ms":43562,"temperature":1.0,"reasoning_tokens":4896,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T21:40:23.911263+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"On the warped product $(R^{n+1}, dr^2 + f(r)^2 g_{S^n})$ with $f(r) = \\sinh r$ for $r \\leq 1/2$ and $f(r) = C e^{-r}$ for $r \\geq 1$ (the example built in Section 4), compute the first Dirichlet $p$-eigenvalue $\\lambda_{1,p}$ of the annulus $\\{R \\leq r \\leq R+D\\}$ for large $R$. If $\\lambda_{1,p}$ decays faster than $e^{-n(R+D)}$ as $R \\to \\infty$, the key eigenvalue input from the companion paper fails and the uniform lower bound (1.8) collapses.","supporting_citations":[],"review_version":1}