{"id":"a8c37f19-28f0-430a-8360-26db1d6fcefd","arxiv_id":"2508.20298","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For bounded domains in complete non-compact manifolds, Willmore-type inequalities hold under asymptotic or Lp Ricci curvature bounds, recovering the pointwise Jin-Yin theorem as a limit.","lead":"This paper proves new Willmore-type inequalities for hypersurfaces in spaces where the Ricci curvature is only controlled from below in a weak, non-pointwise sense. The results extend a 2024 theorem of Jin and Yin to asymptotic and integral curvature bounds, with the error term vanishing when the classical pointwise bound holds.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.3's key Jacobian estimate (3.3) is not derived as written: the displayed inequality in Section 3 has mismatched exponents, and the constant C(n,p) later pulled out of the Σ-integral depends on H(x).","rationale":"The reader's weakest-assumption (mean-convexity H≥0 and RV(Ω)<∞) is a legitimate limitation, but it is stated in the theorem and is not where the proof appears to break down. The more immediate load-bearing issue is the derivation of (3.3), which the manuscript leaves with mismatched exponents in the displayed chain; the final inequality is recoverable by a standard correction, and the x-dependence of C(n,p) is patchable by a uniform bound at H=0. Since both gaps are repairable and do not invalidate the comparison strategy, the appropriate verdict remains CONDITIONAL, matching the reader's conclusion, but the specific weakest point differs. The paper deserves acceptance only after the corrected derivation and the uniform bound on C are supplied in a revision.","tokens_in":14380,"tokens_out":25219,"duration_ms":220778,"concrete_test":"Re-derive (3.3) from (3.1) without relying on the displayed implication in Section 3: use d/dt[(J/bJ)^{1/(2p)}] = (1/(2p))(J/bJ)^{1/(2p)-1}(J/bJ)' ≤ (1/(2p))φ(J/bJ)^{1/(2p)} and Hölder with bJ^{-1/(2p-1)}. Verify that the resulting constant is exactly C(n,p) as defined, and check that for every H≥0, C(n,p,H) ≤ C(n,p,0) < ∞. If the corrected derivation instead produces an x-dependent prefactor that cannot be bounded independently of H∈[0,∞) (or requires new integrability of ρ), then Theorem 1.3's volume estimate is not justified as stated.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central bound of Theorem 1.3 is (3.3), obtained from Z=J/bJ via Z'≤φZ. The paper states that (Z^{1/(2p−1)})Z' ≤ φ Z^{1/(2p)} and then integrates to 2p(Z^{1/(2p)}−1) ≤ ∫φ Z^{1/(2p)} dt. These two displayed formulas are inconsistent: multiplying Z'≤φZ by Z^{1/(2p−1)} gives Z^{1/(2p−1)}Z' ≤ φ Z^{2p/(2p−1)}, not φ Z^{1/(2p)}; conversely, the integrated inequality is equivalent to (1/(2p))d/dt Z^{1/(2p)} ≤ (1/(2p))φ Z^{1/(2p)}, whose left factor is Z^{1/(2p)−1}, not Z^{1/(2p−1)}. The intended final estimate is nevertheless obtainable by the correct identity, so this is repairable. A second gap in the same passage: C(n,p) defined after (3.3) contains bJ(x,t) and therefore depends on x through H(x), yet it is subsequently treated as a constant and moved outside the Σ-integral. A uniform bound C(n,p,H(x)) ≤ C(n,p,0) for H≥0 fixes this, but it is not stated. As written, therefore, the proof of the main volume estimate has a genuine algebraic inconsistency plus an unjustified pull-out.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves two Willmore-type inequalities for bounded domains with smooth boundary in complete non-compact (n+1)-manifolds. Theorem 1.1 assumes an asymptotic Ricci lower bound Ric ≥ −n − λ(d(o,·)) with non-increasing λ ∈ L^1, and bounds RV(Ω)ω_n by an integral over {H ≥ −n−2nb} of (1+2b+H/n)^n times e^{2nb}. Theorem 1.3 assumes an L^p integral Ricci bound with p > (n+1)/2, mean-convex boundary, and finite RV(Ω), and bounds RV(Ω)ω_n by (1 + ||ρ||_p^{1/2})∫_{∂Ω}(1+H/n)^n plus an error term that tends to 0 as ||ρ||_p → 0. The proofs use ODE comparison lemmas for ψ_1, ψ_2, a Riccati comparison for the mean curvature of parallel hypersurfaces, Jacobian estimates, and Hölder estimates with a boundedness lemma (Lemma 3.1). The paper also contains corollaries on non-emptiness of {H ≥ −n−2nb} and a compactness criterion for manifolds with boundary.","tokens_in":14705,"tokens_out":11000,"duration_ms":97356,"significance":"If the proof of Theorem 1.3 is corrected, the main results are a genuine extension of the Jin-Yin Willmore inequality to integral and asymptotic Ricci curvature settings, with a quantitative error term in the integral case. The ODE comparison arguments in Section 2 are elementary, clearly presented, and appear correct; they also yield finiteness of the relative volume ratio under asymptotic assumptions. The paper is self-contained modulo standard comparison theorems and does not rely on circular reasoning. The main weakness is that the key Jacobian estimate in Section 3 is not derived correctly as written, and one constant depends on the point x; both issues are locally repairable, so the central claim remains plausible.","major_comments":[{"comment":"The exponent manipulation that yields the integrated Jacobian estimate is algebraically inconsistent. From Z := J/bJ satisfying Z' ≤ φZ, multiplying by Z^{1/(2p−1)} gives Z^{1/(2p−1)}Z' ≤ φ Z^{2p/(2p−1)}, not φ Z^{1/(2p)}. The integrated inequality 2p(Z^{1/(2p)}−1) ≤ ∫ φ Z^{1/(2p)} dt is exactly what follows from (d/dt)Z^{1/(2p)} ≤ (1/(2p))φ Z^{1/(2p)}, whose left factor is Z^{1/(2p)−1}. The displayed line should be corrected; the intended bound is nevertheless obtainable with the correct exponent, so this is repairable.","section":"Section 3, display preceding (3.3)"},{"comment":"The constant C(n,p) depends on x because bJ(x,t) contains H(x), yet it is pulled out of the Σ-integral in the subsequent volume estimate. Since H ≥ 0, one can bound bJ(x,t)^{-1/(2p−1)} ≤ (cosh t)^{-n/(2p−1)}, giving a uniform constant independent of x; this step needs to be stated explicitly. As written, the pull-out is unjustified and must be repaired for the proof of Theorem 1.3 to be complete.","section":"Section 3, definition of C(n,p) after (3.3)"}],"minor_comments":[{"comment":"The quantity \\bar J is used in the integrand without being defined; it should be defined explicitly, presumably as the Jacobian determinant J or its normalized extension.","section":"Theorem 2.6 proof"},{"comment":"The dominance bound (coth t_0 + 2b + H(x)/n)^n can be negative when H(x) is very negative. Since H is bounded on the compact boundary, the proof should choose t_0 after bounding H below, or replace the displayed quantity by its positive part, so that the dominated convergence argument is valid.","section":"Theorem 2.6 proof"},{"comment":"There is a typo: 'Riemaninan' should be 'Riemannian'.","section":"Corollary 2.4"},{"comment":"The proof of Lemma 3.1 could state more explicitly that F_{p,q,ε} attains its maximum on (0,∞) because the limits at 0 and ∞ are finite; the current wording is terse but correct.","section":"Lemma 3.1 proof"}],"recommendation":"major_revision","confidential_remarks":"The paper is a fairly direct extension of known comparison techniques, and the main originality lies in the integral-Ricci error term and the asymptotic statement. The Section 3 errors are local and fixable; I would not reject on their account. The authors should also double-check the uniform bound on C(n,p) and the dominated-convergence estimate before resubmission."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The two main theorems are new and probably correct. Theorem 1.1 handles asymptotic Ricci decay with λ∈L^1, which is weaker than the usual ∫tλ<∞ condition, and Theorem 1.3 gets a Willmore inequality from ∥ρ∥_p with an error term that vanishes as ∥ρ∥_p→0. Both recover Jin–Yin. That is a real step beyond Gallot and Petersen–Sprouse, and the ODE lemmas in Section 2 are clean and well proved.\n\nThe soft spots are in the write-up, mostly in Section 3. The line before (3.3) has a wrong exponent: the author multiplies (J/bJ)' ≤ φ(J/bJ) by (J/bJ)^{1/(2p−1)}, but the integrated inequality that follows needs the multiplier (J/bJ)^{1/(2p)−1}. The result is fine, but as displayed it is algebraically inconsistent. The second issue is that the constant C(n,p) defined after (3.3) depends on x through H(x) in bJ. It is then pulled outside the ∂Ω integral without comment. Under H≥0 it is uniformly bounded by C(n,p,0), so the gap is easy to close, but a referee will want it stated. The reader's minor point about Theorem 2.6 is also real: the dominating function used in dominated convergence can be negative when H<−n−2nb; use the positive part or split the boundary into the two regions. None of this undermines the main claims, in my view.\n\nCitation pattern is honest: the work builds on [12] and [16], and the two self-citations are not load-bearing. The paper is not circular and does not hide anything. Presentation needs a careful proofread—there are typos and the constant-dependence issue obscures the proof.\n\nThis paper is for people working in comparison geometry and Willmore-type inequalities. My recommendation: send it to review. The theorems are worth referee time, and the repairs are local. I'd ask the author to fix the exponent typo, justify the uniform bound on C(n,p,H), and patch the dominated convergence argument. After that, I'd expect it to be acceptable.","headline":"A genuinely new pair of Willmore-type inequalities under asymptotic and L^p integral Ricci bounds, built on sound comparison estimates; the main theorem in Section 3 has two repairable slips in the write-up, not in the underlying argument.","tokens_in":15235,"tokens_out":5619,"would_cite":true,"duration_ms":48375,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53A07","53C21"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves Willmore-type inequalities for bounded domains in complete non-compact manifolds under asymptotic or integral Ricci curvature bounds, continuously recovering the pointwise Ricci lower-bound case.","keywords":["Willmore inequality","integral Ricci curvature","mean curvature","relative volume ratio","comparison geometry","hypersurfaces","Ricci curvature lower bound","tubular neighborhoods"],"falsifier":"Construct an explicit family of complete non-compact metrics $g_\\varepsilon$ on a fixed manifold with $\\|\\rho\\|_p(g_\\varepsilon)\\to0$ and a fixed bounded mean-convex domain $\\Omega$ whose Jacobi fields are computable—for instance a warped product $M=\\mathbb{R}\\times_\\varphi \\Sigma$ with $\\rho$ explicit—and compute both sides of Theorem 1.3. If the left side minus the boundary integral exceeds the sum of the $\\varepsilon^{1/2}$-term and the vanishing $C$-term for some $\\varepsilon$, the theorem fails; if the excess is always bounded by the predicted rate, it supports sharpness of the choice $\\varepsilon=\\|\\rho\\|_p^{1/2}$.","tokens_in":14177,"feed_emoji":"📐","tokens_out":6169,"duration_ms":53905,"temperature":0.7,"pith_summary":"The paper establishes that Willmore-type inequalities—lower bounds on the total bending of a boundary in terms of its mean curvature—hold on complete non-compact manifolds whose Ricci curvature is controlled only in an averaged or asymptotically decaying sense. The main theorems bound the relative volume ratio of a bounded domain by an integral over its boundary of $(1+H/n)^n$, with an explicit error term that vanishes as the integral Ricci curvature defect goes to zero. When the defect is identically zero, both inequalities recover the recent Jin–Yin theorem under the pointwise bound $\\mathrm{Ric}\\ge -n$. The interest is that the proof works without any pointwise lower curvature bound, relying instead on an $L^p$ norm of the negative part of Ricci for $p>(n+1)/2$.","feed_headline":"Willmore inequality survives averaged Ricci curvature bounds","feed_subtitle":"Recovers the pointwise lower-bound case as the curvature defect vanishes, with a controlled error term.","key_machinery":"The engine is a Riccati comparison for the mean curvature $m(t)$ of parallel hypersurfaces along normal geodesics, $m'+m^2/n\\le n+\\rho(\\gamma(t))$. In the asymptotic case, two comparison lemmas for the linear ODE $\\psi''=(1+\\Lambda(t))\\psi$ give sharp control on the ratios $\\psi_2/\\psi_1$ and $\\psi_1/\\sinh t$, yielding a pointwise Jacobian bound of the form $(\\cosh t+(2b+H/n)\\sinh t)^n e^{2nb}$. In the integral case, the deviation $\\phi=\\max\\{m-\\hat m,0\\}$ from the hyperbolic comparison mean curvature is inserted into an $L^p$ estimate: a Hölder argument bounds $\\int_0^r \\phi^{2p}J\\,dt$ by a constant times $\\int_0^r \\rho^p J\\,dt$, and a second Hölder step converts this into a multiplicative correction to the hyperbolic Jacobian $\\hat J$. Lemma 3.1, the elementary inequality $(1+b)^p\\le 1+\\varepsilon+C(p,q,\\varepsilon)\\varepsilon^{-q}b^p$ with $C(p,q,\\varepsilon)\\to0$ as $\\varepsilon\\to0$, controls the error terms; choosing $\\varepsilon=\\|\\rho\\|_p^{1/2}$ produces the final bound.","core_discovery":"The central discovery is that a Willmore-type inequality of the form $$\\mathrm{RV}(\\$\\Omega$)\\cdot\\omega_n \\le \\left(1+\\|\\rho\\|$_p^{{1/2}}$\\right)\\int_{\\partial\\$\\Omega$}\\left(1+\\frac{H}{n}\\right)^n d\\mathrm{vol} + C(n,p,\\|\\rho\\|_p)\\left(1+\\frac{H(\\xi)}{n}\\right)^n$$ holds under the sole assumption that $\\rho=\\max\\{-n-\\mathrm{Ric},0\\}$ lies in $L^p$ with $p>(n+1)/2$, provided the domain is mean-convex and has finite relative volume ratio, and with $C(n,p,\\|\\rho\\|_p)\\to 0$ as $\\|\\rho\\|_p\\to 0$. In the asymptotic setting, a pointwise but decaying bound $\\mathrm{Ric}\\ge -n-n\\lambda(d(o,\\cdot))$ with $\\lambda\\in L^1$ non-increasing yields a similar inequality with the explicit factor $e^{2nb}$ and integration over the region where $H\\ge -n-2nb$. Both results recover the Jin–Yin theorem when the curvature defect is identically zero.","pith_inferences":["The same Riccati-comparison strategy should extend to intermediate Ricci curvature bounds, replacing the ambient dimension $n+1$ by $k+1$ and adjusting the critical exponent accordingly, since only control along normal geodesics is used.","The specific choice $\\varepsilon=\\|\\rho\\|_p^{1/2}$ in the proof suggests the error term is at most of order $\\|\\rho\\|_p^{1/2}$; explicit warped-product examples could test whether this exponent is sharp.","A proof that $\\mathrm{RV}(\\Omega)<\\infty$ follows directly from an integral Ricci bound alone would remove one of the two technical assumptions in Theorem 1.3 and would likely require a volume comparison theorem for tubular neighborhoods under integral curvature.","The theorem quantifies how much the Willmore constant can deteriorate per unit of averaged Ricci curvature below $-n$, which is a natural input for convergence or collapse questions in Riemannian geometry."],"forward_implications":["When $\\rho\\equiv 0$, Theorem 1.3 reduces exactly to the Jin–Yin Willmore-type inequality for $\\mathrm{Ric}\\ge -n$, with no error term.","The inequality is quantitatively stable: as $\\|\\rho\\|_p\\to0$, the correction term disappears and the bound converges continuously to the pointwise one.","The asymptotic Theorem 1.1 implies that $\\partial\\Omega\\cap\\{H\\ge -n-2nb\\}$ is non-empty for every bounded domain, and it yields a compactness criterion for manifolds with boundary whose mean curvature is bounded below.","Under the asymptotic assumptions, the finiteness of $\\mathrm{RV}(\\Omega)$ follows from the curvature bound, so Theorem 1.1 needs no separate finiteness hypothesis.","The integral setting requires mean-convexity of the boundary; without $H\\ge0$ on $\\partial\\Omega$, the key non-negativity step in the proof fails."],"supporting_citations":[{"why":"The pointwise $\\mathrm{Ric}\\ge -n$ Willmore-type inequality that both new theorems recover when the curvature defect vanishes.","marker":"[12]"},{"why":"The relative volume comparison with integral curvature bounds that supplies the $L^p$ machinery and the threshold $p>n/2$ adapted here.","marker":"[16]"},{"why":"An earlier upper bound for tubular neighborhood volumes under integral curvature, restricted to constant mean curvature, which the present proof overcomes.","marker":"[17]"},{"why":"Gallot's isoperimetric bounds from integral Ricci curvature, whose lower bounds can become negative and are therefore insufficient for the desired inequality.","marker":"[9]"},{"why":"The second-order linear ODE comparison theorem used in Lemmas 2.1 and 2.2 to control the Jacobian ratio.","marker":"[18]"},{"why":"Comparison lemmas for asymptotically nonnegative curvature that motivate Lemmas 2.1 and 2.2 but are not directly applicable under the weaker integrability condition.","marker":"[8]"}],"fun_headline_variants":["Willmore inequality holds with integral Ricci curvature","Averaged Ricci curvature extends Willmore-type result","Willmore bound recovers Jin-Yin for weak Ricci","Integral Ricci bounds give Willmore inequality","Weak Ricci curvature suffices for Willmore-type bound"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is mean-convexity of the boundary, $H(x)\\ge0$ on $\\partial\\Omega$, together with the separate assumption that $\\mathrm{RV}(\\Omega)$ is finite, since both are needed for the key estimates and neither is derived under the integral curvature hypothesis.","fun_headline_variants_meta":{"raw":{"variants":["Willmore inequality holds with integral Ricci curvature","Averaged Ricci curvature extends Willmore-type result","Willmore bound recovers Jin-Yin for weak Ricci","Integral Ricci bounds give Willmore inequality","Weak Ricci curvature suffices for Willmore-type bound"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000176,"raw_usage":{"total_tokens":1221,"prompt_tokens":809,"completion_tokens":412,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":425,"completion_tokens_details":{"reasoning_tokens":339}},"tokens_in":425,"tokens_out":412,"duration_ms":4550,"temperature":1.0,"reasoning_tokens":339,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T16:48:14.241204+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Construct an explicit family of complete non-compact metrics $g_\\varepsilon$ on a fixed manifold with $\\|\\rho\\|_p(g_\\varepsilon)\\to0$ and a fixed bounded mean-convex domain $\\Omega$ whose Jacobi fields are computable—for instance a warped product $M=\\mathbb{R}\\times_\\varphi \\Sigma$ with $\\rho$ explicit—and compute both sides of Theorem 1.3. If the left side minus the boundary integral exceeds the sum of the $\\varepsilon^{1/2}$-term and the vanishing $C$-term for some $\\varepsilon$, the theorem fails; if the excess is always bounded by the predicted rate, it supports sharpness of the choice $\\varepsilon=\\|\\rho\\|_p^{1/2}$.","supporting_citations":[{"cited_title":"Petersen and G","cited_arxiv_id":null,"evidence_quote":"The relative volume comparison with integral curvature bounds that supplies the $L^p$ machinery and the threshold $p>n/2$ adapted here."},{"cited_title":"Integral curvature bounds, distance esti- mates and applications","cited_arxiv_id":null,"evidence_quote":"An earlier upper bound for tubular neighborhood volumes under integral curvature, restricted to constant mean curvature, which the present proof overcomes."},{"cited_title":"Isoperimetric inequalities based on integral norms of Ricci cur- vature","cited_arxiv_id":null,"evidence_quote":"Gallot's isoperimetric bounds from integral Ricci curvature, whose lower bounds can become negative and are therefore insufficient for the desired inequality."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The second-order linear ODE comparison theorem used in Lemmas 2.1 and 2.2 to control the Jacobian ratio."},{"cited_title":"Sobolev inequalities in manifolds with asymptotically nonnegative curvature","cited_arxiv_id":null,"evidence_quote":"Comparison lemmas for asymptotically nonnegative curvature that motivate Lemmas 2.1 and 2.2 but are not directly applicable under the weaker integrability condition."}],"review_version":1}