{"id":"ee00680e-f270-4590-bddf-9ae0c25c476f","arxiv_id":"2505.18320","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Connected sums preserve the spectral Ricci bound lambda1(-gamma Delta + Ric) > lambda for n >= 3 and gamma > (n-1)/(n-2), and this range is sharp.","lead":"The paper proves that if two complete Riemannian manifolds satisfy the same spectral Ricci lower bound, then their connected sum also admits a metric with that bound, provided the parameter gamma exceeds (n-1)/(n-2). It extends scalar curvature surgery ideas to a weaker, spectral notion of Ricci curvature and shows the threshold is sharp.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The gluing computation is internally coherent, but the construction rests on an imported two-pole Green's function (2) whose hypotheses are not verified for possibly non-connected M; this is the least secure step.","rationale":"I checked the gluing computation in Section 2.2. In Region I, the cancellation is valid: the coefficient γ(n−2)−(n−1) is positive by the hypothesis, Lemma 6 controls the error terms near x=1, and in Region II the term f''/(r0^2 f) dominates. Lemma 8 supplies the required asymptotics from Serrin's theorems provided the Green's function exists. Thus no internal error appears in the tunnel argument itself. Separately, the abstract's claim that γ>(n−1)/(n−2) is sharp for the connected-sum theorem is not proved in the body; the body only establishes sharpness for the Bour–Carron Betti number bound and refers to [2] for warped-product counterexamples. That overclaim is real but secondary. The load-bearing point is the imported existence result, which the paper does not verify in the possibly non-connected, two-pole setting. The reader's weakest_assumption identified the same Green's function step, and the reader's rationale also flagged the sharpness overclaim; my read therefore agrees with the conditional verdict until the Green's function hypotheses and the sharpness statement are either verified or revised.","tokens_in":11905,"tokens_out":23978,"duration_ms":191160,"concrete_test":"Read [5, Theorem 2.3] and verify three points: (a) it applies to each connected component of a possibly disconnected complete Riemannian manifold; (b) for the lower-order term Ric−(λ−ε/2), with only λ1(−γ∆+Ric)≥λ on the original manifold, it yields a positive Green's function with two point sources of equal weight; (c) its regularity gives u∈C^{2,α} away from the poles. As a complementary check, run the proof with M = M1⊔M2 and x_i in separate components, constructing u as the sum of one-pole Green's functions on each component; if the sum is positive and satisfies (2), the non-connected case is resolved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central argument requires a positive C^{2,α} solution u of −γ∆u+Ric·u = (λ−ε/2)u + δ_{x1}+δ_{x2} on a possibly non-connected complete M, with the sharp r^{2−n} asymptotics of Lemma 8. The paper imports existence from [5, Theorem 2.3] and does not reproduce the theorem or check its hypotheses in this exact setting. Criticality theory is typically formulated for connected manifolds, while Theorem 1 allows M to have several components and x1,x2 may lie in different components. The shift λ−ε/2 makes the operator strictly positive even when the initial data only give λ1≥λ, so that part is fine, but the cited theorem's coverage of compact components, multiple poles, and disconnected domains needs to be checked. Lemma 8 proves the asymptotic expansion from Serrin's theorems, but it still assumes that such a Green's function exists. If [5, Theorem 2.3] does not cover the stated generality, Proposition 1 and hence Theorem 1 are not established as written. This is a verifiable citation gap rather than an internal contradiction.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves a connected-sum (tunnel-gluing) theorem for complete n-manifolds satisfying a spectral Ricci lower bound of the form λ1(−γΔ + Ric) ≥ λ. In the supercritical range γ > (n−1)/(n−2), given any two points x1, x2 on such a manifold M, the authors construct a complete metric on the manifold obtained by removing two small balls and inserting a tunnel S^{n−1}×[0,1], preserving the bound with an arbitrarily small loss ε (Theorem 1). The proof is based on solving a two-pole Poisson equation for the operator −γΔ + Ric − (λ−ε/2) with Green's function asymptotics, then interpolating the Green's function with a model warped-product eigenfunction f^{2−n} on the tunnel. Corollaries assert invariance under connected sums and under taking connected sums with S^{n−1}×S^1, and the abstract claims that the threshold γ=(n−1)/(n−2) is sharp.","tokens_in":12098,"tokens_out":11493,"duration_ms":116375,"significance":"If correct, the result is a clean and non-obvious stability property for spectral Ricci lower bounds under codimension-one surgery, complementing the Betti-number and rigidity results of Bour–Carron and the criticality framework of Catino–Mari–Mastrolia–Roncoroni. The paper's main strength is its explicit local analysis: Lemmas 2–6 give careful estimates for the Ricci curvature, Laplacian, and error terms on the tunnel, and the construction of the interpolated metric and eigenfunction is concrete. The argument is not fully self-contained, but the external inputs are standard and the local computations are detailed enough to be checkable. The paper also usefully identifies the same threshold γ=(n−1)/(n−2) that appears in previous rigidity results.","major_comments":[{"comment":"The proof imports the existence of a two-pole positive Green's function u solving −γΔu + Ric·u = (λ−ε/2)u + δ_{x1}+δ_{x2} from [5, Theorem 2.3 (i) ⇔ (iv)], but the theorem is not stated and its hypotheses are not verified in the present setting. In particular, M is allowed to be non-connected, x1 and x2 may lie in different components or in compact components, and the source is a sum of two Dirac masses. Since Proposition 1 and Theorem 1 depend critically on this u and on its sharp r^{2−n} asymptotics, this is a load-bearing citation gap. Please either quote the relevant theorem and check its hypotheses explicitly, or give a short direct construction (for instance, as a sum of one-pole Green's functions on the relevant components).","section":"Section 2.1, Eq. (2)"},{"comment":"The abstract's claim that the range γ>(n−1)/(n−2) is 'sharp' for the connected-sum preservation is not proved. The examples from [2, Remark 4] cited in Section 1 are counterexamples to the volume-comparison theorem, not to the preservation of the spectral Ricci bound under connected sums. No example is given with γ≤(n−1)/(n−2) for which the conclusion of Theorem 1 fails. Please either prove sharpness of the connected-sum statement itself, or rephrase the abstract to say that the construction requires γ>(n−1)/(n−2) and that this threshold is the one appearing in the associated rigidity phenomena.","section":"Abstract and Section 1"}],"minor_comments":[{"comment":"In the reversed coordinate r∈(−r0,0) on U1, the expression r^{2−n} should be interpreted as |r|^{2−n} (or the notation should be adjusted), since u is positive while r^{2−n} would be signed for odd n.","section":"Eq. (3)"},{"comment":"The connected sum M1#M2 is not well-defined when M1 or M2 is non-connected without specifying which components are being summed; please clarify the statement (for instance, by taking the connected sum of chosen components, or by first noting the hypothesis passes to components).","section":"Corollary 1"},{"comment":"The smoothness of g′ and u′ across the gluing boundaries r=±r0 is asserted but not explicitly checked. It follows because f is linear in a neighborhood of ±1 and the new objects agree with the old ones in a neighborhood of the boundaries, but a one-sentence justification would make the proof complete.","section":"Proof of Theorem 1"},{"comment":"The applications of [15, Theorem 10] and [16, Theorem 1] are terse; please state the exact hypotheses used (e.g., the removability theorem and the two-sided isolated-singularity bound) so the reader can verify that the regularity and the r^{2−n} bounds indeed hold in this setting.","section":"Lemma 8"},{"comment":"To conclude sharpness of the constant in Bour–Carron's Betti number bound with λ=n−1, one should mention the scaling argument: if a metric satisfies λ1(−γΔ+Ric)>0, scaling it by a constant makes the spectral lower bound equal to n−1. As written, the text jumps from positivity to the Betti number statement.","section":"Section 1, discussion after Corollary 2"}],"recommendation":"major_revision","confidential_remarks":"The main theorem is plausible and the local computations are careful, but the unverified application of [5, Theorem 2.3] to a possibly disconnected manifold with two point sources is a genuine gap in the written proof. I believe it is fixable by stating the theorem and checking its hypotheses, or by a short reduction to one-pole Green's functions. The sharpness claim in the abstract should also be softened unless a counterexample for γ≤(n−1)/(n−2) is supplied. The paper fits the scope of math.DG and, after these revisions, could be a nice contribution."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my take. The paper gives a genuine new result: connected sums preserve the spectral Ricci lower bound λ1(-γ∆+Ric) ≥ λ for γ > (n-1)/(n-2), via a tunnel construction that merges Green's functions rather than pointwise curvature functions. That's a real departure from Gromov-Lawson/Schoen-Yau and from the previous spectral Ricci literature, and the proof is careful: the estimates on the warped tunnel, the asymptotics of the Green's function, and the algebraic Lemma 6 all work. The corollary improving Bour-Carron's Betti number sharpness is a nice payoff.\n\nThe soft spots are real but minor. First, the abstract says the range γ > (n-1)/(n-2) is 'sharp for this to hold' — meaning the connected sum theorem itself. The body only proves sharpness of that threshold in Bour-Carron's Betti number bound, as a consequence of the new theorem; it does not show the connected sum property fails below the threshold. The abstract should be reworded. Second, the proof imports a positive Green's function with two poles from [5, Theorem 2.3], but that theorem is presumably stated for connected manifolds and a single pole. Since M is allowed to be disconnected and x1,x2 may lie in different components, you need to apply the theorem component-wise and add the Green's functions, or otherwise justify the superposition. It's an easy fix, but as written it's a citation gap. Third, the smooth matching at the tunnel ends is asserted without proof; it follows because f is affine for |x|≥1 and the asymptotics of w_i give matching derivatives, but a remark would help.\n\nI don't think any of this threatens the main theorem. The cited result [5] does cover the supercritical shift that makes the operator strictly positive, and the two-pole issue is a linearity footnote. The proof of Proposition 1 and Lemma 8 are internally coherent.\n\nWho should read it: people working on spectral Ricci bounds, stable minimal hypersurfaces, and geometric surgery. It deserves a serious referee; the result is new and the argument is largely self-contained. I'd recommend sending it to review, with a request to fix the abstract and add the Green's-function justification.","headline":"New connected sum theorem for spectral Ricci bounds, sound up to minor presentational gaps; the abstract overclaims sharpness.","tokens_in":12646,"tokens_out":4288,"would_cite":true,"duration_ms":34666,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53C21","53C20","58J50"],"pacs":[],"model":"deepseek-v4-flash","headline":"Connected sums preserve spectral Ricci lower bounds above the critical ratio; the range is sharp.","keywords":["spectral Ricci lower bound","connected sum","Gromov-Lawson tunnel","Green's function","warped product metric","sharp threshold","Betti number bounds"],"falsifier":"Run the tunnel construction at $\\gamma=(n-1)/(n-2)$: the coefficient $\\gamma(n-2)-(n-1)$ multiplying the main term vanishes, so the test function gives no positive contribution and the proof's key inequality reduces to $0\\ge o(r_0^{-2})$. Exhibiting a complete connected-sum metric with $\\lambda_1(-\\gamma\\Delta+\\mathrm{Ric})>\\lambda$ at this critical value would show the claimed sharpness is false.","tokens_in":11660,"feed_emoji":"🔗","tokens_out":13693,"duration_ms":101911,"temperature":0.7,"pith_summary":"Let $n\\ge 3$ and take $\\gamma>(n-1)/(n-2)$. The paper proves that if two complete $n$-manifolds admit metrics with $\\lambda_1(-\\gamma\\Delta+\\mathrm{Ric})>\\lambda$, then their connected sum admits a complete metric with the same spectral lower bound; it also proves a one-manifold version in which a tunnel is glued between any two points with only an $\\varepsilon$ loss. This matters because the spectral condition is a global, functional-analytic notion of Ricci curvature, and topological operations that preserve it are rare. The proof works by solving an elliptic equation with two point sources, whose Green's function has the same $r^{2-n}$ singularity that the model tunnel function has, and then gluing the two via a warped product. The number $(n-1)/(n-2)$ is shown to be sharp: for larger $\\gamma$ the connected sum operation always preserves the bound, while for the critical value the argument's main coefficient vanishes.","feed_headline":"Spectral Ricci bounds survive connected sums above a sharp cutoff","feed_subtitle":"Above the critical coefficient, two manifolds with this spectral bound can be joined by a tunnel without losing it.","key_machinery":"The load-bearing objects are the Green's function for the operator $-\\gamma\\Delta+\\mathrm{Ric}$ with two point sources and a warped-product tunnel metric. The Green's function $u$ solves $-\\gamma\\Delta u+\\mathrm{Ric}\\,u=(\\lambda-\\varepsilon/2)u+\\delta_{x_1}+\\delta_{x_2}$ and has the sharp asymptotic form $u\\sim r^{2-n}w_i(r,\\theta)$ near each pole; the tunnel metric $\\tilde g=dr^2+r_0^2 f(r/r_0)^2\\tilde h$ interpolates the two local metrics through a round slice, and the candidate tunnel eigenfunction $\\tilde u=r_0^{2-n}f(r/r_0)^{2-n}\\tilde w/(\\gamma(n-2)|S^{n-1}|)$ interpolates the two Green's functions. The calculation reduces the inequality $-\\gamma\\Delta_{\\tilde g}\\tilde u+\\mathrm{Ric}_{\\tilde g}\\tilde u\\ge(\\lambda-\\varepsilon)\\tilde u$ to the model identity for $f^{2-n}$ plus two estimates: the asymptotics of the polar metric coefficients and of the Green's function, and a convexity inequality for the profile $f$ that converts the supercritical condition $\\gamma>(n-1)/(n-2)$ into a sign condition on the only surviving principal term.","core_discovery":"The central discovery is that the spectral Ricci bound $\\lambda_1(-\\gamma\\Delta+\\mathrm{Ric})\\ge\\lambda$ is a connected-sum invariant in the supercritical range $\\gamma>(n-1)/(n-2)$: given a complete manifold satisfying the bound, one may remove two small balls around arbitrary points and glue back a tunnel $S^{n-1}\\times[-r_0,r_0]$ carrying a warped metric $dr^2+r_0^2 f(r/r_0)^2\\tilde h$, on which a positive function $\\tilde u = r_0^{2-n}f^{2-n}\\tilde w/(\\gamma(n-2)|S^{n-1}|)$ continues to satisfy $-\\gamma\\Delta \\tilde u+\\mathrm{Ric}\\,\\tilde u\\ge(\\lambda-\\varepsilon)\\tilde u$. The proof's quantitative heart is an explicit identity: for the model metric $dr^2+f(r)^2g_{S^{n-1}}$, the function $f^{2-n}$ satisfies $-((n-1)/(n-2))\\Delta f^{2-n}+\\mathrm{Ric}(\\partial_r,\\partial_r)f^{2-n}=0$ for arbitrary $f$, so the error terms in the tunnel are controlled purely by the asymptotic agreement of the metric and Green's function with their Euclidean prototypes. Consequently, the threshold $\\gamma=(n-1)/(n-2)$ is the exact place where the stabilization coefficient $\\gamma(n-2)-(n-1)$ changes sign.","pith_inferences":["The same tunnel-and-Green's-function strategy has an evident higher-codimension analogue: for a surgery of codimension $k$ with $2<k\\le n-1$, the stabilizing coefficient would be $\\gamma-(k-1)/(k-2)$, so the natural conjecture is that spectral Ricci bounds are stable under such surgeries exactly when $\\gamma>(k-1)/(k-2)$.","Because the proof only needs $f$ convex near the transition rather than a solution of a special differential equation, the tunnel profile is highly flexible; perturbing $f$ inside the allowed class should preserve the bound, suggesting the construction is stable under small geometric perturbations of the neck.","The threshold prediction is testable in low dimensions: at $\\gamma=(n-1)/(n-2)$ the principal term vanishes, so any connected-sum metric satisfying the bound at the critical value would have to come from a mechanism entirely different from the Green's-function tunnel described here."],"forward_implications":["Any two complete manifolds satisfying the bound can be joined by a tunnel, and the resulting connected sum still satisfies the same spectral Ricci lower bound.","For every $n\\ge 3$, $\\gamma>(n-1)/(n-2)$, and $k\\ge 1$, the iterated connected sum $\\#_k(S^{n-1}\\times S^1)$ admits a complete metric with $\\lambda_1(-\\gamma\\Delta+\\mathrm{Ric})>0$, making the known Betti-number bound sharp.","A tunnel can be inserted between any two points of a single admissible manifold with only an arbitrarily small loss: the new metric satisfies the bound with $\\lambda-\\varepsilon$ in place of $\\lambda$.","The proof's principal term carries the factor $\\gamma(n-2)-(n-1)$, so the construction works exactly when $\\gamma>(n-1)/(n-2)$ and its main term vanishes at the critical value."],"supporting_citations":[{"why":"defines the spectral condition via positive supersolutions, allowing the proof to verify only a pointwise inequality for the tunnel function.","marker":"[8]"},{"why":"supplies the existence and $C^{2,\\alpha}$ regularity of the two-point Green's function in equation (2).","marker":"[5]"},{"why":"provides the isolated-singularity removability result needed before comparing the Green's function to $r^{2-n}$.","marker":"[15]"},{"why":"gives the sharp two-sided bounds and asymptotics for isolated singularities used in Lemma 8.","marker":"[16]"},{"why":"contains the toy identity for $u=f^{2-n}$ on warped products, which is the local model for the tunnel.","marker":"[3]"},{"why":"provides the warped-product examples that make the threshold $(n-1)/(n-2)$ visible in the broader volume and Betti-number picture.","marker":"[2]"}],"fun_headline_variants":["Connected sums preserve spectral Ricci bounds above sharp cutoff","Sharp cutoff: spectral Ricci bound survives connected sum","Supercritical γ: connected sum keeps spectral Ricci bound","Above the critical γ, connected sum preserves spectral Ricci bound"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof depends on the existence of a positive Green's function for $-\\gamma\\Delta+\\mathrm{Ric}$ with two point sources whose singularity is exactly of order $r^{2-n}$; if the leading singularity differed, the tunnel eigenfunction would not match the two copies and the gluing inequality would break.","fun_headline_variants_meta":{"raw":{"variants":["Connected sums preserve spectral Ricci bounds above sharp cutoff","Sharp cutoff: spectral Ricci bound survives connected sum","Supercritical γ: connected sum keeps spectral Ricci bound","Above the critical γ, connected sum preserves spectral Ricci bound"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000627,"raw_usage":{"total_tokens":2899,"prompt_tokens":943,"completion_tokens":1956,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":559,"completion_tokens_details":{"reasoning_tokens":1894}},"tokens_in":559,"tokens_out":1956,"duration_ms":11808,"temperature":1.0,"reasoning_tokens":1894,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T14:34:56.461419+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the tunnel construction at $\\gamma=(n-1)/(n-2)$: the coefficient $\\gamma(n-2)-(n-1)$ multiplying the main term vanishes, so the test function gives no positive contribution and the proof's key inequality reduces to $0\\ge o(r_0^{-2})$. Exhibiting a complete connected-sum metric with $\\lambda_1(-\\gamma\\Delta+\\mathrm{Ric})>\\lambda$ at this critical value would show the claimed sharpness is false.","supporting_citations":[{"cited_title":"Local behavior of solutions of quasi-linear equations","cited_arxiv_id":null,"evidence_quote":"provides the isolated-singularity removability result needed before comparing the Green's function to $r^{2-n}$."},{"cited_title":"Isolated singularities of solutions of quasi-linear equations","cited_arxiv_id":null,"evidence_quote":"gives the sharp two-sided bounds and asymptotics for isolated singularities used in Lemma 8."}],"review_version":1}