{"id":"f0d60611-3b6a-48a8-bdf9-c75bd48fbaca","arxiv_id":"2509.00743","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"If the CR Yamabe invariant of a Sasaki manifold attains the minimum determined by its Reeb cone, the manifold is K-semistable, linking CR analysis to algebraic stability.","lead":"This mathematics paper connects the CR Yamabe invariant, a number measuring how much scalar curvature a space can carry, to the existence of special Sasaki metrics and to K-stability, an algebraic stability notion. It offers a new criterion for when polarized complex manifolds admit constant scalar curvature metrics, a central problem in differential geometry.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.4 depends on convexity/slope lemmas that are asserted, not proven, for non-regular Reeb fields","rationale":"The reader identified the weakest assumption as the dependence of Corollary 5.8 on convexity of the action functional and the slope inequality, both sketched and deferred to prior work. This is exactly the most load-bearing concern. The paper's Theorem 1.4 is the advertised link between the CR Yamabe invariant and K-semistability, and its proof is a direct application of Corollary 5.8 via Lemma 5.4. If Theorem 5.6 or Proposition 5.7 is not established for arbitrary (possibly irregular) Reeb vector fields, the inequality Y_sup^T ≤ EH^χ_s does not follow, and the K-semistability conclusion is unsupported. The text gives a one-line derivation of Theorem 5.6 from Proposition 5.11, but the displayed formula does not manifestly imply convexity; the sign of the curvature term dd^c Ψ is not positive in general. The approximation argument from A^Ψ to A is not detailed, and the paper explicitly refers to [LLS23, Ino21] for the hard analytic content. This is not an internal contradiction, but it means the central result is conditional on technical results that are not proven in this manuscript. The reader's CONDITIONAL verdict is appropriate; our concern does not move the verdict. We also note the secondary gap about the reduction to smooth, ample, dominant test configurations, but that is a standard algebraic step and less concerning than the missing analytic input.","tokens_in":37499,"tokens_out":10930,"duration_ms":129742,"concrete_test":"Take the toric Sasaki manifold over P^1×P^1 with Kähler class p c1(O(2)) + q c1(O(2)) studied in §2.4, choose the irregular Reeb vector field χ = ξ+ (Lemma 2.10), and consider a product test configuration induced by a C^* action on one factor. Using the toric formulas of §5.2, compute the action functional A^χ(s,t) explicitly (e.g., by numerical integration of (5.9)) and verify that for s>0 small, ∂t^2 A^χ(s,t) ≥ 0 for the corresponding weak geodesic ribbon. If a negative second derivative is found, Theorem 5.6 is false and Corollary 5.8 collapses; if convexity holds, the authors should still provide the missing proof for all non-regular χ.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim (Theorem 1.4) follows from Corollary 5.8: Y_sup^T(X,L) ≤ EH^χ_s(X,L) for every smooth, ample, dominant test configuration. The proof of Corollary 5.8 (Section 5.1) explicitly invokes Theorem 5.6 (convexity of the action functional A^χ(s,t) along weak geodesic ribbons) and Proposition 5.7 (slope inequality). The text gives only a sketch: Theorem 5.6 is said to follow from the second-derivative formula in Proposition 5.11, but the expression for dd^c A^Ψ_χ contains the term ns∫ dd^c Ψ∧(ω+ddcΦ)^n / f^{n+1}, whose sign is not evidently nonnegative for the metric Ψ defined by Ω^{n+1} in (5.24). No positivity argument or integration-by-parts is supplied. Moreover, the passage from the modified functional A^Ψ to A via approximating sequences of metrics is mentioned but not carried out. The paper also states that Theorem 5.6 generalizes [LLS23, Theorem 1.4] and invokes [Ino21] without proof. Consequently, the inequality Y_sup^T ≤ EH^χ_s, and hence Theorem 1.4, rests on unverified analytic results. An additional, secondary gap is the unstated reduction from all test configurations to smooth, ample, dominant ones with reduced central fibre when concluding K-semistability.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the CR Yamabe energy on the Boothby-Wang circle bundle N associated to a polarized manifold (X,L), and connects it with constant transversal scalar curvature Sasaki (cscS) structures and Sasaki K-stability. The main invariant is Y^T_sup(X,L), the supremum of the equivariant CR Yamabe energy over Sasaki forms with a fixed regular Reeb field, while EH_min is the minimum of the Einstein-Hilbert functional over the Sasaki-Reeb cone. The central results are: (i) if Y^T_sup(X,L)=EH_min, then the Sasaki manifold with the minimizing Reeb field is K-semistable (Theorem 1.4); (ii) under non-positive average scalar curvature, existence of L^p-approximate cscS structures implies the equality Y^T_sup=EH_min (Corollary 1.8); (iii) a partial converse is discussed but explicitly not completed. The proof of Theorem 1.4 proceeds through an inequality Y^T_sup(X,L) ≤ EH^χ_s(X,L) for test configurations, proved using an action functional along weak geodesic ribbons. The paper also contains regularity results for the CR Yamabe energy and an isolation theorem for cscS structures.","tokens_in":37921,"tokens_out":5139,"duration_ms":66397,"significance":"If fully established, the paper gives a new and potentially important bridge between the CR Yamabe invariant and Sasaki K-stability, with a concrete numerical criterion for K-semistability of polarized manifolds. The approach is original: it uses the global conformal invariant rather than prescribing a Reeb vector field. Several computations are explicit and non-tautological, e.g. the test-configuration slopes in Propositions 5.12 and 5.13, and the example on P^1×P^1 is a nice illustration. However, the central implication rests on analytic convexity and slope results (Theorem 5.6 and Proposition 5.7) that are only sketched and deferred to the authors' related work and to Inoue. Until those arguments are supplied, the main theorem is conditional.","major_comments":[{"comment":"These two results are the load-bearing analytic inputs for Corollary 5.8 and hence Theorem 1.4. The text only sketches Theorem 5.6 as a consequence of Proposition 5.11. But the formula for dd^c A^Ψ_χ contains the term ns∫ dd^c Ψ∧(ω+ddcΦ)^n / f^{n+1}; no argument is given that this term is nonnegative, nor is an integration-by-parts or sign estimate provided. Moreover, the passage from the regularized functional A^Ψ to A by approximating metrics is asserted to be 'exactly the same as in [Ino21, LLS23]' but is not carried out. Since [LLS23] corresponds to the regular Reeb case, the new non-regular statement is not actually proved here.","section":"§5.1–5.2, Theorem 5.6 and Proposition 5.7"},{"comment":"The proof of K-semistability only treats smooth, ample, dominant T-equivariant test configurations with reduced central fibre. The definition of Sasaki K-semistability in [CS18] concerns all test configurations; the manuscript does not justify that the restricted class is sufficient. If this reduction is standard, a precise reference or a short argument is needed; otherwise the conclusion of Theorem 1.4 exceeds what has been shown.","section":"§5.1, proof of Theorem 1.4"},{"comment":"The abstract advertises 'a partial converse' to Corollary 1.8, but Section 4.1 explicitly states 'We are not yet able to obtain such bounds' after Lemma 4.5. Lemmas 4.4 and 4.5 give only weak estimates and do not imply the existence of L^p-approximate cscS structures. The reader should be told precisely which statement is conjectural and which is proved; as written, the advertised partial converse is not a theorem.","section":"§4.1, approximate cscS converse"}],"minor_comments":[{"comment":"The inequality Y^T_sup(X,L) ≤ EH^χ_s(X,L) is stated as Theorem 1.5 in the Introduction but appears as Corollary 5.8 in Section 5.1. Please renumber consistently.","section":"Introduction vs. §5.1"},{"comment":"The case n=1 requires separate wording: q=4 and the Sobolev embedding W^{1,2}⊂L^q is compact on the 2-dimensional quotient, but this is not made explicit. The current sentence comparing q with the critical Sobolev exponent assumes n>1.","section":"§2.1, proof of Proposition 2.3"},{"comment":"The phrase 'we freely use results proved for A^Ψ_χ, such as Proposition 5.11, for Aχ without further comment' is too quick; the approximation argument should at least state what convergence properties are preserved (lower semicontinuity, convexity, slopes).","section":"§5.2, after Proposition 5.11"},{"comment":"The reference [RT1105] appears to have a formatting error in the year/volume fields. Please correct.","section":"References"},{"comment":"There are several typos and grammatical slips, e.g. 'of of' near the beginning of Section 2, and 'The converse holds, when EH_min ≤0' with an awkward comma. These do not affect the mathematics.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The paper's main theorem depends heavily on results the authors have stated elsewhere ([LLS23]) and on Inoue's preprint. In its current form, the referee cannot fully verify the central implication. The editor may wish to request either a complete proof of the convexity and slope statements for arbitrary Reeb fields, or a clearly stated theorem limiting the scope to the regular case. The overlap with the authors' own prior work should also be clarified, since the novelty assessment is difficult without knowing precisely which parts of §5.2 are new."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague, here's my read of arXiv:2509.00743. The paper has one genuinely new idea worth your time: the CR Yamabe invariant, as a functional on the space of contact forms over a polarized manifold, provides a numerical criterion for Sasaki K-semistability. The main theorem (1.4) — equality Y_sup = EHmin forces K-semistability — is new, and it's a real condition, not a tautology. The authors also prove an isolation result for cscS structures (Theorem 1.3) and a negative-average-curvature characterization (Theorem 1.6) that look solid.\n\nThe good part: the paper is transparent. Section 4.1 says outright that they can't yet prove the bounds for the approximate converse. The debt to [LLS23] and [Ino21] is acknowledged. The variational setup is clean, and the regularity results in Section 3 are more than sketches.\n\nThe soft spot is the load-bearing Section 5. Corollary 5.8, which yields Theorem 1.4, depends on Theorem 5.6 (convexity of the action functional along weak geodesic ribbons) and Proposition 5.7 (slope inequality). Both are deferred to previous work and earlier papers. The sketch of convexity goes through Proposition 5.11, but the sign of the term ns∫ dd^cΨ∧(ω+ddcΦ)^n / f^{n+1} is not evidently nonnegative; no positivity argument is supplied. The approximation step from the modified A^Psi to A is mentioned but not carried out. For the general non-regular Reeb case, these proofs need to be written out before Theorem 1.4 is fully established. There's also an unstated reduction from all test configurations to smooth, ample, dominant ones with reduced central fibre; it's probably standard, but it's not pointed to.\n\nNone of this is fatal. The gaps are addressable, and the central claim is plausible. But as written, the K-stability theorem rests on asserted lemmas.\n\nWho should read this? Specialists in Sasaki geometry and K-stability. It deserves a serious referee: an editor should send it out. I'd mark the key section as needing real verification. I'd cite the isolation result and the negative-curvature characterization, but hold off citing the K-semistability theorem until the convexity proof lands.","headline":"A plausible and genuinely new link between CR Yamabe invariant and Sasaki K-stability, but the key analytic lemma is only sketched.","tokens_in":38309,"tokens_out":2931,"would_cite":true,"duration_ms":34583,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53C25","53D10"],"pacs":[],"model":"deepseek-v4-flash","headline":"A Sasaki manifold is K-semistable whenever its CR Yamabe invariant equals the minimum of the Einstein-Hilbert functional on the Reeb cone.","keywords":["CR Yamabe invariant","Sasaki geometry","constant transversal scalar curvature","K-semistability","Sasaki-Futaki invariant","Einstein-Hilbert functional","test configurations","cscK metrics"],"falsifier":"On the toric circle bundles over P^1×P^1 with polarization pO(2)+qO(2), q>5p, discussed in Section 2.4, compute the CR Yamabe invariant from the L^{n+1} formula of Theorem 1.6 over the explicit toric Sasaki metrics. If Y_sup equals EH(ξ0) while some explicit test configuration has negative Sasaki-Futaki invariant, Theorem 1.4 is contradicted; if Y_sup is strictly below EH(ξ0), the equality criterion has real content and can be compared with the known three cscS structures.","tokens_in":37483,"feed_emoji":"⚖️","tokens_out":10202,"duration_ms":112079,"temperature":0.7,"pith_summary":"This paper tries to establish a bridge between the CR Yamabe invariant of a compact Sasaki manifold and the existence and stability of constant scalar curvature Sasaki (cscS) metrics. Its central claim is that if the CR Yamabe invariant attains the minimum value of the Einstein-Hilbert functional on the Sasaki-Reeb cone, then the Sasaki manifold is K-semistable, meaning every test configuration has nonnegative Sasaki-Futaki invariant. The authors show that equality is also detected by approximate cscS metrics when the average scalar curvature is nonpositive, and they give a numerical criterion for K-semistability of polarized complex manifolds. The approach matters because it detects cscS structures without first prescribing their Reeb vector fields, and it provides a Yamabe-style route to the algebraic stability side of the cscS/cscK existence problem.","feed_headline":"Yamabe equality forces Sasaki K-semistability","feed_subtitle":"When the CR Yamabe invariant hits its Einstein-Hilbert minimum, all test configurations must have nonnegative Futaki invariant.","key_machinery":"The load-bearing object is the CR Yamabe energy Y_CR(η)=inf_f EH(f^{-1}η), where EH(α) is the CR Einstein-Hilbert functional (total Tanaka-Webster scalar curvature divided by a power of volume) on the conformal class of a Sasaki contact form η. Varying η over Kähler forms in c1(L) gives the invariant Y_sup^T(X,L). The key inequality chain is Y_sup^T≤EH_s^χ(X,L) for test configurations, obtained by extending EH along weak geodesic ribbons via an action functional; differentiating EH_s^χ at s=0 recovers the Sasaki-Futaki invariant of the test configuration. Equality of the two ends of the chain converts a curvature-bound inequality into an algebro-geometric stability statement.","core_discovery":"On the circle bundle over a polarized manifold, the paper defines the CR Yamabe energy Y_CR(η)=inf_f EH(f^{-1}η) and shows its critical points are exactly cscS structures whose Reeb field minimizes the Einstein-Hilbert functional EH on the Sasaki-Reeb cone. The invariant Y_sup^T(X,L)=sup_η Y_CR^T(η) is then compared with EH_min. The main theorem: if Y_sup^T=EH_min, the Sasaki manifold (N,I,χ_min) is K-semistable—every test configuration has nonnegative Sasaki-Futaki invariant. The proof refines Y_sup^T≤EH_min to Y_sup^T≤EH_s^χ(X,L) for smooth ample dominant test configurations, and the s-derivative of EH_s^χ at s=0 is the Sasaki-Futaki invariant, so equality forces nonnegativity. When EH_min","pith_inferences":["The equality Y_sup^T=EH_min is probably closer to an existence criterion than to a semistability criterion: in the nonpositive case the paper proves equality from approximate solutions, so a full converse would make CR Yamabe equality equivalent to the existence of cscS metrics, i.e. a weak existence-stability correspondence for this curvature condition.","Because only minimizers of EH are detected, cscS structures whose Reeb fields are non-minimizing (like the three toric examples on P^1×P^1) will not be visible to the CR Yamabe invariant; a moduli or isolation theory would need a different functional.","The L^{n+1}-norm characterization invites explicit computations of Y_sup on toric or join Sasaki manifolds with negative average curvature; such computations could turn the K-semistability criterion into a practical numerical test.","The stated regularity assumption (existence of a regular Reeb vector field in the cone) is likely unnecessary for the K-stability part: the same action-functional argument should extend to quasi-regular orbifold quotients, as the paper itself hints."],"forward_implications":["If Y_sup^T(X,L)=EH_min, the Sasaki manifold (N,I,χ_min) is K-semistable: all Sasaki-Futaki invariants of T-equivariant test configurations are nonnegative.","A single contact form η with Y_CR^T(η)=EH_min produces an actual cscS metric, η(χ_min)^{-1}η, with Reeb field a minimizer of EH; when EH_min≤0, the converse holds.","When EH_min≤0, |Y_sup^T| equals the infimum over conformal classes of the L^{n+1} norm of Tanaka-Webster scalar curvature, giving a numerical handle on the invariant.","Under EH(ξ0)=EH_min and c1(X)·c1(L)^{n-1}≤0, the existence of L^p-approximate cscK metrics implies K-semistability of the polarized manifold (X,L).","A cscS structure satisfying the eigenvalue bound λ_1^T>c_η/2(n+1) is isolated in the T-invariant space of cscS structures."],"supporting_citations":[{"why":"Supplies the action-functional machinery and the ξ0 case of the slope and convexity results that Theorem 5.3 generalises.","marker":"[LLS23]"},{"why":"Provides the Perelman-entropy analogue and the approximation arguments used for weak geodesic ribbons.","marker":"[Ino21]"},{"why":"Defines the global Sasaki-Futaki invariant appearing in Lemma 5.4 whose nonnegativity is K-semistability.","marker":"[ACL21]"},{"why":"Defines Sasaki K-semistability, the target notion of Theorem 1.4.","marker":"[CS18]"},{"why":"Solves the CR Yamabe problem, producing the minimizers that define the CR Yamabe energy.","marker":"[JL87]"},{"why":"Shows transverse scalar curvature and volume only depend on the Reeb field, giving EH on the Sasaki-Reeb cone.","marker":"[FOW09]"},{"why":"Establishes that EH attains a minimum on the Sasaki-Reeb cone, so EH_min exists.","marker":"[BHL18]"}],"fun_headline_variants":["CR Yamabe minimum forces Sasaki K-semistability","Yamabe equality stabilizes Sasaki manifolds","CR invariant at EH bound implies Sasaki K-semistability","Sasaki K-semistability from CR Yamabe equality"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The argument's load-bearing premise is that the action functional along the contact-form families induced by test configurations is convex and satisfies a slope inequality; the paper only sketches these proofs and defers them to earlier work, so the equality-to-K-semistability implication would fail if that deferred analysis is wrong.","fun_headline_variants_meta":{"raw":{"variants":["CR Yamabe minimum forces Sasaki K-semistability","Yamabe equality stabilizes Sasaki manifolds","CR invariant at EH bound implies Sasaki K-semistability","Sasaki K-semistability from CR Yamabe equality"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000279,"raw_usage":{"total_tokens":1494,"prompt_tokens":741,"completion_tokens":753,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":485,"completion_tokens_details":{"reasoning_tokens":695}},"tokens_in":485,"tokens_out":753,"duration_ms":7946,"temperature":1.0,"reasoning_tokens":695,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T13:14:53.002671+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"On the toric circle bundles over P^1×P^1 with polarization pO(2)+qO(2), q>5p, discussed in Section 2.4, compute the CR Yamabe invariant from the L^{n+1} formula of Theorem 1.6 over the explicit toric Sasaki metrics. If Y_sup equals EH(ξ0) while some explicit test configuration has negative Sasaki-Futaki invariant, Theorem 1.4 is contradicted; if Y_sup is strictly below EH(ξ0), the equality criterion has real content and can be compared with the known three cscS structures.","supporting_citations":[],"review_version":1}