{"id":"03836749-23de-4a3f-bba8-e4a7b3ed1a46","arxiv_id":"2411.14667","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For a flat torus boundary, the average boundary mean curvature of any fill-in with scalar curvature at least -n(n-1) is bounded by an explicit function of the shortest noncontractible circle length.","lead":"This paper proves a sharp upper bound on how much a torus boundary can bend in a fill-in when the interior scalar curvature is bounded from below. It resolves a special case of Gromov's conjecture and connects fill-in geometry to systolic inequalities.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.7's concluding rescaling changes the boundary metric from γ to c^2γ; as written the proof fails when K>n(n−1), so the arbitrary-boundary result Theorem 1.8 is not established.","rationale":"The reader's weakest assumption is exactly the point I find most load-bearing. In Theorem 1.6, the global rescaling after gluing does not endanger the theorem because the original boundary condition is no longer used; the systolic inequality is applied to a distant level set. In Theorem 1.7, however, the boundary condition is the object of the theorem. The collar metric from Lemma 3.1 has scalar curvature exactly −K, with K chosen as sup |R_{γ_t}| so that the monotonicity of the total mean curvature (property (v)) holds. There is no freedom to lower K without risking failure of (v). Since K can exceed n(n−1) for generic metrics, the glued metric can fall below the allowed scalar curvature threshold, and the only repair offered, a global rescale, changes the boundary metric. Thus the proof of Theorem 1.7 is incomplete. I do not see an analogous internal gap in Theorem 1.6; the apparent inversion in the final inequality is a typographical error, and the corrected ratio (1−ε/2)/(1−ε/4) < 1 yields the intended contradiction. The reliance on the Brendle–Hung preprint is a verification concern rather than an internal inconsistency; if that preprint is correct, Theorem 1.6 is plausible. Therefore the verdict remains CONDITIONAL: the flat-torus result appears sound modulo external verification and a repairable typo, but the arbitrary-metric generalization is not proved as written.","tokens_in":12767,"tokens_out":24165,"duration_ms":215328,"concrete_test":"Take n = 3, Σ = T², choose γ̂ a flat metric and γ a metric on T² with a localized high-curvature bump so that K := sup_{t∈[0,1]} |R_{(1−t)γ̂+tγ}| > 6. Run the construction in Lemma 3.1: the collar metric g′ has R_{g′} = −K. Glue it to any fill-in of γ̂ with R ≥ −6, obtaining a metric g with R_g ≥ −K−1 < −6. Now track the proposed global rescaling: the scalar curvature repair requires c² ≥ (K+1)/6, while the boundary condition g|∂Ω = γ forces c = 1. Verify that these constraints are incompatible; this directly demonstrates that the final step of Theorem 1.7's proof is invalid and no global rescaling can salvage it.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing gap is in the proof of Theorem 1.7 (Section 3). After gluing (Ω, ĝ) to the collar constructed in Lemma 3.1, the scalar curvature is at least min{−K, −n(n−1)} − 1, where K = sup_{Σ×[0,1]} |R_{γ_t}|. The last paragraph asserts: 'Finally, we can rescale g so that the scalar curvature is bounded from below by −n(n−1).' A global rescaling g ↦ c²g changes the boundary metric from γ to c²γ. Since Theorem 1.7 requires the boundary metric to be exactly γ, one must have c = 1. But when K > n(n−1), repairing the scalar curvature lower bound requires c² ≥ (K+1)/(n(n−1)) > 1. Such pairs (γ̂, γ) exist in abundance on tori, for instance when the linear interpolation has a localized curvature bump. Hence the proof does not produce the claimed fill-in, and Theorem 1.8, which depends on Theorem 1.7, is unsupported. This is distinct from Theorem 1.6, where the rescaling occurs after the original boundary is no longer used.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies fill-ins of the torus Σ = S^1 × T^{n-2} (3 ≤ n ≤ 7) with scalar curvature bounded below by -n(n-1). Theorem 1.6 establishes, for flat boundary metrics γ, a sharp upper bound on the total mean curvature of the boundary, with the constant expressed in terms of the systolic length σ; sharpness is claimed via Horowitz–Myers metrics. Theorem 1.7 is an interpolation result: from a fill-in of a smaller metric γ̂ one constructs a fill-in of a larger metric γ with controlled total mean curvature. Theorem 1.8 then derives a general total-mean-curvature bound for arbitrary boundary metrics by combining Theorems 1.6 and 1.7. The proofs use the prescribed scalar curvature equation, a priori estimates, a gluing construction, and the Brendle–Hung systolic inequality.","tokens_in":12967,"tokens_out":17580,"duration_ms":147842,"significance":"If correct, Theorem 1.6 resolves a special case of Gromov's conjecture on total mean curvature for fill-ins with scalar curvature bounded below, in the setting of flat tori, and provides a sharp constant. The use of the Brendle–Hung systolic inequality in place of a direct positive mass argument is a genuine novelty. The paper also gives a route toward general boundary metrics via Theorem 1.8. The arguments are largely coherent, and the dependence on machine-checkable or previously established estimates (parabolic a priori estimates, gluing lemmas) is explicit. However, two issues affect the proofs as written: a typo in the final inequality of Theorem 1.6, and a more substantial gap in the rescaling step of Theorem 1.7.","major_comments":[{"comment":"The assertion 'Finally, we can rescale g so that the scalar curvature is bounded from below by -n(n-1)' is not justified. A global rescaling g ↦ c²g changes the boundary metric from γ to c²γ, so the required boundary condition g|_{∂Ω} = γ forces c = 1. When the constant K = sup_{Σ×[0,1]} |R_{γ_t}| > n(n-1), the glued metric satisfies only R_g ≥ -K - 1 < -n(n-1), and no constant rescaling can raise the scalar curvature lower bound to -n(n-1) while preserving the boundary metric exactly. Since Theorem 1.8 relies on Theorem 1.7, this is a load-bearing gap. The statement of Theorem 1.7 may still be true, but the proof as written does not establish it.","section":"Section 3, Proof of Theorem 1.7, last paragraph"},{"comment":"The displayed inequality 'hat σ λ^{-1} ≤ (1 - ε/4)/(1 - ε/2) σ' has the ratio inverted. Combining the lower bound on \\tilde H - (n-1) with the systolic upper bound yields hat σ λ^{-1} ≤ (1 - ε/2)/(1 - ε/4) σ, not the displayed inequality. With the corrected ratio, choosing λ large gives hat σ/λ ≤ (1 - ε/2)/(1 - ε/4) σ < σ, which contradicts the convergence of hat σ/λ to σ as λ → ∞. As written, the displayed inequality does not yield a contradiction, so the proof is incomplete at this point; however, the error appears to be a localized algebra mistake that is repairable from the preceding inequalities.","section":"Proof of Theorem 1.6, final displayed inequality"}],"minor_comments":[{"comment":"The notation 'C = C(n,γ,γ′)' should be 'C = C(Σ,γ,\\hat γ,n)' or similar; also the proof refers to 'γ′' which is not defined at that point.","section":"Proof of Theorem 1.8"},{"comment":"In the display of the comparison ODEs, there is an extra comma after 'w_-(0) = min Σ v0, ,' and the expression for w_+ should be 'w_+(0) = max_Σ v0'. These typos should be corrected for clarity.","section":"Lemma 3.1, proof"},{"comment":"Please clarify the precise form of the Brendle–Hung systolic inequality used. The proof applies it as a pointwise bound on \\tilde H - (n-1). If the original statement is an integral inequality, the pointwise consequence should be justified using the near-constancy of the mean curvature on \\hat Σ_λ established in Proposition 2.8.","section":"Proof of Theorem 1.6"},{"comment":"There are numerous typographical issues, e.g., 'g|∂Ω =gRn|∂Ω' in Theorem 1.1, 'the g outward unit normal vector' in Theorem 1.6, and 'Letfε to be the unique solution' in the proof of Theorem 1.6. A careful proofreading pass is recommended.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The main result, Theorem 1.6, appears essentially correct modulo the final-inequality typo, which is easily repairable. The more serious issue is the rescaling step in Theorem 1.7, which is load-bearing for Theorem 1.8. The author should be asked to either provide a correct argument that yields the scalar curvature lower bound without changing the boundary metric, or restructure the proof (e.g., by a conformal deformation that fixes the boundary) to establish the interpolation result. Also, the paper depends on the advisor's preprint [5] for the systolic inequality; the editor may wish to confirm that this preprint is publicly available and has been verified. The manuscript is promising and the methods are of interest to the scalar-curvature and quasi-local mass communities."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper has two distinct layers. The core result, Theorem 1.6, is a genuinely new sharp total-mean-curvature estimate for flat torus boundaries in dimensions 3 through 7. The proof is a coherent combination of the Shi–Wang–Wei prescribed scalar curvature equation, a priori estimates, gluing, and the Brendle–Hung systolic inequality. The perturbation argument via the function f is clean, and the Horowitz–Myers example for sharpness is a good check. This layer is solid, and I would trust it after a routine fix of one typo: the final displayed inequality in the proof of Theorem 1.6 has the ratio inverted. Swapping (1−ε/4)/(1−ε/2) for (1−ε/2)/(1−ε/4) makes the contradiction go through, so this is repairable and not a real mathematical flaw.\n\nThe soft spot is exactly where the stress-test lands. In the proof of Theorem 1.7, after gluing the collar, the scalar curvature is only bounded below by min{−K, −n(n−1)} − 1. The last paragraph says: “Finally, we can rescale g so that the scalar curvature is bounded from below by −n(n−1).” A global rescaling multiplies the boundary metric by c². Since Theorem 1.7 requires the boundary to be exactly γ, this step is unjustified whenever the constant K is large. Such pairs (γ̂, γ) exist on tori without any special pathology. Consequently, Theorem 1.7 is not established, and Theorem 1.8, which depends on it, is unsupported. This is not a minor gap: the abstract advertises a general total-mean-curvature bound, and that claim rests on Theorem 1.8.\n\nThe reliance on the Brendle–Hung preprint [5] is real but not circular. It is an external dependency on an unpublished proof, which makes the result contingent, but the paper does not assume its own conclusion. The author should cite the published version when it appears.\n\nWho is this for? People working on scalar curvature fill-ins and quasi-local mass. The flat-torus case is the clean result; the interpolation claim needs more work. I would send it to a serious referee because the core theorem is important and likely correct, and the gap is addressable. The author should fix the typo, repair or restrict the rescaling argument in Theorem 1.7, and resubmit.","headline":"The flat-torus sharp estimate in Theorem 1.6 is likely correct and worth refereeing, but the advertised general torus-boundary result (Theorem 1.8) is not proven as written because the final rescaling step in the proof of Theorem 1.7 changes the prescribed boundary metric.","tokens_in":13543,"tokens_out":8653,"would_cite":true,"duration_ms":71697,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53C20","53C21","53C24"],"pacs":[],"model":"deepseek-v4-flash","headline":"For flat tori $S^1 \\times T^{n-2}$ with $3 \\le n \\le 7$, fill-ins with scalar curvature $\\ge -n(n-1)$ and positive mean curvature obey a sharp total-mean-curvature bound whose constant is attained in the limit.","keywords":["scalar curvature","total mean curvature","fill-in","flat torus","systolic inequality","prescribed scalar curvature","quasi-local mass","total mean curvature conjecture"],"falsifier":"Produce a fill-in of a flat metric $\\gamma$ on $S^1 \\times T^{n-2}$ with $R_g \\ge -n(n-1)$ and $H_{\\Sigma} > 0$ for which the normalized total mean curvature exceeds $\\frac{1}{2}\\left(\\frac{4\\pi}{n\\sigma}\\right)^n$; a numerical construction in the allowed dimensions would disprove the sharp estimate. Separately, check the rescaling step in the interpolation theorem: for any glued metric $g'$ with boundary $\\gamma$, a global rescaling $c^2g'$ has boundary $c^2\\gamma$, so preserving the boundary metric forces $c=1$; verifying whether a different deformation can raise the scalar-curvature lower bound to $-n(n-1)$ would decide whether the arbitrary-boundary theorem stands.","tokens_in":12513,"feed_emoji":"📐","tokens_out":10382,"duration_ms":87017,"temperature":0.7,"pith_summary":"This paper establishes a sharp upper bound on the total mean curvature of the boundary of a Riemannian fill-in of a flat torus $\\Sigma = S^1 \\times T^{n-2}$ with $3 \\le n \\le 7$, assuming the interior scalar curvature is at least $-n(n-1)$ and the boundary mean curvature is positive. The bound depends only on the dimension and the flat boundary metric, through the length of the shortest non-contractible circle in the $S^1$ factor. The result settles a special case of a conjecture about total mean curvature of fill-ins with scalar curvature bounded from below. The proof combines the prescribed scalar curvature equation with a recently proved systolic inequality for tori.","feed_headline":"Total mean curvature of torus fill-ins has sharp upper bound","feed_subtitle":"A sharp constant controls boundary mean curvature under a lower scalar-curvature bound, settling a conjecture for flat tori.","key_machinery":"The central object is the prescribed scalar curvature equation on the hyperbolic cylinder $\\Sigma \\times [\\rho_0, \\infty)$ with metric $g = \\rho^2\\gamma + u^2\\rho^{-2}d\\rho^2$; the scalar-curvature condition $R_g = -n(n-1)$ reduces to $(n-1)\\rho\\,\\partial_\\rho u = u^2\\Delta_{\\Sigma_\\rho}u - \\frac{1}{2}n(n-1)(u^3-u)$. Solving this equation with boundary data coming from the fill-in yields the asymptotic expansion $u = 1 + \\mu(x)\\rho^{-n} + O(\\rho^{-n-2})$. A perturbation of the level sets of $\\rho$ by a function that solves a Laplacian equation adjusts the mean curvature of a large torus slice to the desired constant. A gluing lemma attaches this cylinder to the fill-in while nearly preserving the lower scalar-curvature bound, and a systolic inequality on the torus converts any mean-curvature excess into a contradiction unless the claimed estimate holds.","core_discovery":"The central claim is Theorem 1.6: if $\\gamma$ is flat on $\\Sigma$, $g$ fills $\\Omega = B^2 \\times T^{n-2}$, $R_g \\ge -n(n-1)$, $g|_{\\partial\\Omega} = \\gamma$, and $H_{\\Sigma} > 0$, then $\\frac{1}{\\mathrm{vol}(\\Sigma)} \\int_{\\Sigma} (H_{\\Sigma} - (n-1))\\, d\\mathrm{vol}_{\\gamma} \\le \\frac{1}{2} \\left(\\frac{4\\pi}{n\\sigma}\\right)^n$, where $\\sigma$ is the length of the shortest closed curve in $\\Sigma$ whose integral of $\\Xi$ is nonzero, and $\\Xi$ is the pullback of the volume form on the circle factor. The constant is sharp: taking larger and larger pieces of the model metrics on $\\mathbb{R}^2 \\times T^{n-2}$ makes the left-hand side approach the bound. The paper also proves an interpolation theorem comparing different boundary metrics and uses it to obtain a uniform total-mean-curvature bound for arbitrary boundary metrics on the same torus.","pith_inferences":["A direct proof via a positive mass theorem may be possible; the paper notes it used the systolic inequality instead, so recovering the same constant from a mass inequality would likely extend the result beyond the dimension range allowed by the gluing construction.","The interpolation mechanism suggests a route to sharp constants for non-flat boundary metrics: compare any boundary metric with a flat one, so the flat constant may control the whole conformal class up to a geometric factor.","The construction is testable numerically for small $n$: a sequence of fill-ins approaching the bound should develop a long cylindrical neck whose length is controlled by the systolic length $\\sigma$ of the boundary torus.","The dimension restriction $3 \\le n \\le 7$ appears to come from the gluing lemma, so a different gluing argument is a natural route to higher-dimensional analogues."],"forward_implications":["The sharp estimate settles the flat-torus case of the total-mean-curvature conjecture for fill-ins with scalar curvature bounded below.","For any boundary metric on $S^1 \\times T^{n-2}$, the interpolation theorem yields a constant $C = C(n, \\gamma)$ such that every admissible fill-in has $\\int_{\\partial\\Omega} H\\, d\\mathrm{vol} \\le C$.","The constant in the flat case is optimal, approached by the model metrics on $\\mathbb{R}^2 \\times T^{n-2}$ filling larger tori.","The proof demonstrates that the systolic inequality can replace the positive mass theorem in deriving total-mean-curvature estimates in this setting."],"supporting_citations":[{"why":"provides the systolic inequality for toral boundaries that produces the contradiction if the mean-curvature bound is exceeded.","marker":"[5]"},{"why":"supplies the gluing construction used to attach the hyperbolic cylinder to the fill-in.","marker":"[6]"},{"why":"formulates the total-mean-curvature conjecture that the paper addresses.","marker":"[10]"},{"why":"introduces the model metrics used to show sharpness of the constant.","marker":"[12]"},{"why":"contains the monotonicity formula and boundary estimate being generalized to the torus setting.","marker":"[20]"},{"why":"supplies the prescribed scalar curvature equation and the interpolation method used in Theorems 1.7 and 1.8.","marker":"[21]"},{"why":"provides the earlier non-sharp fill-in estimate that motivates the constant-bounds approach.","marker":"[22]"}],"fun_headline_variants":["Sharp bound for torus fill-in curvature","Flat torus fill-ins: sharp curvature bound","Optimal fill-in mean curvature bound for tori","Gromov conjecture case proven for flat tori","Sharp total mean curvature bound for tori"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof of the interpolation theorem ends by asserting that the glued metric can be rescaled so its scalar curvature is at least $-n(n-1)$, but a constant rescaling changes the boundary metric from $\\gamma$ to $c^2\\gamma$, so that step is not justified as written; if it cannot be repaired, the uniform bound for arbitrary boundary metrics does not follow.","fun_headline_variants_meta":{"raw":{"variants":["Sharp bound for torus fill-in curvature","Flat torus fill-ins: sharp curvature bound","Optimal fill-in mean curvature bound for tori","Gromov conjecture case proven for flat tori","Sharp total mean curvature bound for tori"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000839,"raw_usage":{"total_tokens":3679,"prompt_tokens":991,"completion_tokens":2688,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":607,"completion_tokens_details":{"reasoning_tokens":2617}},"tokens_in":607,"tokens_out":2688,"duration_ms":21760,"temperature":1.0,"reasoning_tokens":2617,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T15:04:28.418926+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Produce a fill-in of a flat metric $\\gamma$ on $S^1 \\times T^{n-2}$ with $R_g \\ge -n(n-1)$ and $H_{\\Sigma} > 0$ for which the normalized total mean curvature exceeds $\\frac{1}{2}\\left(\\frac{4\\pi}{n\\sigma}\\right)^n$; a numerical construction in the allowed dimensions would disprove the sharp estimate. Separately, check the rescaling step in the interpolation theorem: for any glued metric $g'$ with boundary $\\gamma$, a global rescaling $c^2g'$ has boundary $c^2\\gamma$, so preserving the boundary metric forces $c=1$; verifying whether a different deformation can raise the scalar-curvature lower bound to $-n(n-1)$ would decide whether the arbitrary-boundary theorem stands.","supporting_citations":[{"cited_title":"C., and Neves, A","cited_arxiv_id":null,"evidence_quote":"supplies the gluing construction used to attach the hyperbolic cylinder to the fill-in."},{"cited_title":"Scalar curvature of manifolds with boundaries: Natural que stions and artificial constructions, 2019","cited_arxiv_id":null,"evidence_quote":"formulates the total-mean-curvature conjecture that the paper addresses."},{"cited_title":"T., and Myers, R","cited_arxiv_id":null,"evidence_quote":"introduces the model metrics used to show sharpness of the constant."},{"cited_title":"Positive mass theorem and the boundary behaviors of compact manifolds with nonnegative scalar curvature","cited_arxiv_id":null,"evidence_quote":"contains the monotonicity formula and boundary estimate being generalized to the torus setting."},{"cited_title":"Total mean curvature of the boundary and nonnegative scalar curvature fill-ins","cited_arxiv_id":null,"evidence_quote":"supplies the prescribed scalar curvature equation and the interpolation method used in Theorems 1.7 and 1.8."},{"cited_title":"On the fill-in of nonnegative scalar curvature metrics","cited_arxiv_id":null,"evidence_quote":"provides the earlier non-sharp fill-in estimate that motivates the constant-bounds approach."}],"review_version":1}