{"id":"6a7df181-01de-457d-8d69-a72a3cd65cf4","arxiv_id":"2502.03357","paper_version":1,"verdict":"CONDITIONAL","confidence":"LOW","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"The authors construct the first rationally convex surfaces in C2 with only hyperbolic complex tangencies, in fillable and non-fillable versions.","lead":"The paper constructs the first examples of rationally convex surfaces in C2 whose only complex tangencies are hyperbolic, not elliptic. One family admits holomorphic handlebody fillings and lives in the round sphere; the other is exact and admits no attached compact Riemann surfaces, and can be knotted in many ways.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Rational convexity rests on an unverified flatness hypothesis: Theorem 3.1 invokes Slapar's theorem for the §4.1 standard tangency, but the paper only asserts that the model is flat. Both surface families inherit this dependence.","rationale":"The single most load-bearing step is Theorem 3.1, since both families of rationally convex surfaces ultimately rely on it. The theorem's proof depends on Slapar's theorem producing a psh function ψ with ψ^{-1}(0)=Σ=dψ^{-1}(0). The authors do not state Slapar's flatness condition, nor do they verify that their standard hyperbolic tangency from Section 4.1 satisfies it. This is not a matter of internal consistency: the construction of Σ0 and ρ in Proposition 4.1 is explicit, and the Lagrangian condition is checked; the missing piece is the external flatness hypothesis. If the model is flat, the proof likely works; if not, the central claim is unsupported. The reader identified the same weakest assumption. I found no other issue that is more fundamental: the surgery construction is plausible, the Bedford-Klingenberg step is standard, and the exactness argument for non-fillability is sound. The paper should remain conditional pending verification of this external input and an expansion of the proof around the extension of ψ.","tokens_in":23262,"tokens_out":24984,"duration_ms":234427,"concrete_test":"Consult [Sla04, Theorem 2] and its definition of a flat hyperbolic complex tangency, then verify by direct coordinate computation whether the local surface Σ0 = {v=(Re u)^2-(Im u)^2} from Section 4.1 satisfies that definition. If it does not, determine whether the model can be C^2-approximated by a flat tangency while preserving the exact zero-set property ψ^{-1}(0)=Σ=dψ^{-1}(0) required by Theorem 3.1, and re-check Lemma 3.2's extension of ψ to a global psh function on C^2.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central rational-convexity engine is Theorem 3.1, which requires the hyperbolic tangencies to be 'flat' in the sense of Slapar [Sla04, Theorem 2], so that a psh function ψ exists with ψ^{-1}(0)=Σ=dψ^{-1}(0), strictly psh away from H. Section 4.1 states that the standard model v=(Re u)^2-(Im u)^2 is flat in that sense, but this is asserted, not proved, and the actual definition of flatness from [Sla04] is not reproduced. If the model is not flat, Theorem 3.1 cannot be applied: ψ is used to make φ=C·tilde_ρ+ψ strictly psh outside H and to control the equality |h|=e^φ along Σ. The non-fillable family (Theorem 5.1, Corollary 5.2) uses exactly this model in each Darboux ball; the fillable family (Section 2.4) also relies on Theorem 3.1 through the deformation argument. A related gap is that the extension of the local Slapar function to a compactly supported psh function on C^2 is described only as 'we extend ψ', which is not automatic for psh functions. This is load-bearing: without a valid ψ, the rational-convexity proof does not go through, even if the surfaces themselves are correctly constructed.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper constructs what it claims are the first examples of rationally convex surfaces in C^2 with hyperbolic complex tangencies. Two families are given: (i) genus g≥1 surfaces inside the round S^3 that are unknotted and admit holomorphic handlebody fillings (Theorem 1.1), and (ii) genus g≥2 exact surfaces with no non-constant compact Riemann surfaces attached (Theorems 5.1, 5.5, 6.2), which can be produced in arbitrarily many smooth isotopy classes for sufficiently large genus. The first family is built by an ambient 0-surgery on Lagrangian tori that adds pairs of hyperbolic tangencies, followed by an analysis of holomorphic disc fillings following Bedford–Klingenberg. The second family is built from Lin's exact Lagrangian caps, a local model of a flat hyperbolic tangency, and a generalized Duval–Sibony criterion (Theorem 3.1) for rational convexity of surfaces that are Lagrangian for a Kähler form with prescribed degeneracies.","tokens_in":23480,"tokens_out":5964,"duration_ms":54870,"significance":"If the gaps identified below are properly closed, this would be a substantial advance: it would provide the first rationally convex surfaces with hyperbolic complex tangencies, resolve a natural existence question, and exhibit a stark contrast between fillable and non-fillable rationally convex surfaces. The non-fillable examples are exact for a degenerate Kähler form, yielding a strong rigidity statement (no attached compact Riemann surfaces) that is new in this setting. The paper also introduces a useful general surgery for adding hyperbolic tangencies and gives an explicit local symplectic model. The constructions are explicit and geometric, and the exposition is mostly clear. The main theorems are stated with precise hypotheses, and the core constructions are visible, not black-boxed.","major_comments":[{"comment":"The proof of rational convexity relies on Slapar's theorem [Sla04, Theorem 2] to obtain a plurisubharmonic function ψ with ψ^{-1}(0)=Σ=dψ^{-1}(0), strictly psh away from the hyperbolic set H. Section 4.1 merely asserts that the standard model v=(Re u)^2-(Im u)^2 is 'flat' in Slapar's sense; the definition of flatness is not reproduced and no verification is given. If this assertion is false or unverified, Theorem 3.1 cannot be applied, and both the fillable and non-fillable families inherit this failure. Additionally, in the proof of Theorem 3.1 the sentence 'We extend ψ to a smooth and compactly supported function defined on all of C^2' is not justified: a psh function defined on a neighbourhood of Σ does not automatically extend to a global psh function with compact support, and the subsequent use of ϕ=C·tilde_ρ+ψ requires that ψ be psh (not merely smooth) on all of C^2. This step needs either a precise extension theorem or an explicit construction.","section":"Sec. 3 (Theorem 3.1) and Sec. 4.1"},{"comment":"The proof that each genus-0 surface Σ_j^i admits a holomorphic filling that coincides with the disc family T_i near ∂Σ_j^i is not complete. Lemma 2.3 invokes an auxiliary sphere with four elliptic complex tangencies and then a 'standard argument involving positivity of intersection' to conclude that the Bedford–Klingenberg filling of that sphere restricts to the prescribed disc family near the boundary. Neither the construction of the auxiliary sphere nor the positivity argument is provided. The alternative suggestion of 'running the argument of Bedford–Klingenberg' directly is not a proof. Since Corollary 2.4 and hence Theorem 1.1 depend on this lemma, this is a load-bearing gap that must be filled.","section":"Sec. 2.1 (Lemma 2.3, Corollary 2.4)"},{"comment":"Proposition 6.1 asserts that the rationally convex surfaces Σ_g constructed in Theorem 5.1 all bound embedded handle-bodies. The proof relies on the claim that Lin's exact Lagrangian genus-two cap has slices whose knot projections satisfy a certain nesting property, justified only by 'inspection' of [Lin16, Figures 23, 24 and 25]. The same property is asserted for the explicit Legendrian surgeries used earlier in §5. These claims are geometric statements about the actual slice presentations and are used as the key input for unknottedness, which in turn is used in Theorem 6.2. The inspection argument is not sufficient for a rigorous proof; a precise lemma with either a proof or a complete reference should be added.","section":"Sec. 6 (Proposition 6.1)"}],"minor_comments":[{"comment":"The title as printed in the arXiv source is garbled ('RA TIONALL Y CONVEX'); the final version should correct this typographical error.","section":"Title"},{"comment":"The notation 'h±' is used both for the singular points and for the two orientation signs; the figure and surrounding text would benefit from a clearer distinction (e.g., 'the points h_+ and h_-').","section":"Sec. 2 (Lemma 2.1)"},{"comment":"The phrase 'for any choice of primitive η of ω = dη' is stronger than needed, since on C^2 any closed 1-form is exact and all primitives differ by exact forms; this can be simplified or clarified.","section":"Theorem 5.1"},{"comment":"The proof of Lemma 2.5 is a short sketch; the relationship between the orientation of the characteristic distribution and the orientation of the disc boundary (dt∧dθ) should be expanded for readability, since the orientation convention is crucial for Lemma 2.6 and Proposition 2.7.","section":"Sec. 2.2 (Lemma 2.5)"}],"recommendation":"major_revision","confidential_remarks":"The paper is a strong and novel contribution, and the gaps identified are fixable within the manuscript's scope: the flatness of the standard model can be verified by a direct local calculation or by a precise citation of Slapar's definition, and the Bedford–Klingenberg filling arguments can be made rigorous. I recommend major revision rather than rejection. The self-citations [DR16] and [DR24] are used for legitimate prior constructions, not as restatements of the paper's own results."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe core of this paper is a pair of genuinely new constructions: rationally convex surfaces in C2 whose only complex tangencies are hyperbolic. That has been open enough to matter. The fillable family (Section 2) is built by surgery on the Clifford torus and gives unknotted genus-g surfaces in S3 with 2(g−1) hyperbolic tangencies. The non-fillable family (Section 5) uses Lin's exact Lagrangian caps, gives surfaces with no nonconstant compact Riemann surfaces attached, and via [DR24] can be produced in many isotopy classes. If these hold, they settle the question the abstract advertises.\n\nThe paper also does good work on the general mechanism. Theorem 3.1 is a Duval–Sibony-type criterion: a surface that is Lagrangian for a Kähler form degenerating at flat hyperbolic tangencies is rationally convex. The converse in Theorem 3.3 is a natural counterpart. The local model in Section 4.1, with the explicit psh function making the standard model Lagrangian, is concrete and checkable. I want to credit that: the authors give real formulas, not just a sketch.\n\nNow the soft spots. The biggest one is also flagged by the stress-test and by the reader: Theorem 3.1 leans on Slapar's theorem [Sla04] that the standard model is “flat” in his sense, but the paper only asserts flatness, it does not verify the hypothesis from Slapar's paper. This matters because without a valid ψ the psh function φ = C·ρ̃ + ψ may not control |h| ≤ eφ along Σ, and rational convexity via [Gay00] doesn't follow. The same missing verification is inherited by both families. I would guess the assertion is true, but a referee should ask for the actual verification or a precise reference to the exact statement in [Sla04]. Related minor gaps: the “we extend ψ” step after Slapar's theorem is not automatic for psh functions, and Lemma 2.3's Bedford–Klingenberg argument is compressed under a “standard positivity of intersection” phrase. Proposition 6.1 relies on inspecting figures in [Lin16]; that is acceptable for a construction paper but leaves the unknottedness claim less airtight than the rest.\n\nThe self-citations [DR16] and [DR24] are not a red flag: they are separate construction papers used as black boxes, and the dependence is legitimate.\n\nBottom line: this is a serious paper, likely correct in its main claims, with one load-bearing assumption that should be made fully explicit. It deserves peer review and a careful referee who knows CR geometry. I would bring it to a reading group of symplectic/CR people, and I would cite it if I worked in this area.\n\nRecommendation: engage with it, but make the referee check the Slapar flatness hypothesis before signing off.","headline":"A serious construction paper that likely delivers the first examples of rationally convex surfaces with hyperbolic tangencies, with one load-bearing flatness assumption that should be checked carefully before acceptance.","tokens_in":24096,"tokens_out":1035,"would_cite":true,"duration_ms":12454,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["32E20","53D12"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper constructs the first rationally convex surfaces in C2 whose complex tangencies are all hyperbolic, in two classes: unknotted fillable surfaces of every genus g >= 1, and exact non-fillable surfaces of genus g >= 2 that admit no…","keywords":["rationally convex","hyperbolic complex tangency","exact Lagrangian","holomorphic filling","Legendrian surgery","plurisubharmonic potential","polynomial hull","symplectic topology"],"falsifier":"Compute directly in the four-ball model the plurisubharmonic potential required by the paper for the surface v=(Re u)^2-(Im u)^2: verify that its zero set is exactly the surface, that its critical set is the single tangency, and that strict plurisubharmonicity holds away from that point. Alternatively, search the moduli graph of holomorphic discs produced by the filling argument for a disc whose boundary meets both stable manifolds of a hyperbolic point; such a disc realises configuration (b) and would invalidate the rational-convexity proof.","tokens_in":22973,"feed_emoji":"📐","tokens_out":6527,"duration_ms":56667,"temperature":0.7,"pith_summary":"The paper establishes the first rationally convex surfaces in C2 whose complex tangencies are only hyperbolic. Two independent families are constructed: for every genus g >= 1, an unknotted surface in the round sphere S3 with 2(g-1) hyperbolic tangencies and a holomorphic handlebody filling; and for every g >= 2, an exact surface with the same tangency count but no nonconstant compact Riemann surface attached, realisable in many smooth isotopy classes for large genus. The key move is to view such a surface as Lagrangian for a Kähler form that degenerates precisely at the hyperbolic tangencies, with a local plurisubharmonic potential controlling rational convexity. A sympathetic reader should take away that hyperbolic tangencies are compatible with polynomial-hull triviality, while elliptic tangencies are not.","feed_headline":"First rationally convex surfaces with hyperbolic tangencies built","feed_subtitle":"Fillable and non-fillable examples are constructed; the non-fillable ones admit no attached Riemann surfaces.","key_machinery":"Two mechanisms carry the argument. The first is the ambient 0-surgery along a Legendrian arc: replacing a neighbourhood of the arc by a concave cylinder adds one handle and creates one positive and one negative hyperbolic tangency, so each handle contributes a pair. The second is the standard flat hyperbolic tangency v=(Re u)^2-(Im u)^2, which is flat in the sense that it admits a plurisubharmonic potential whose zero set is exactly the surface and whose differential vanishes exactly at the tangency; the Kähler form i∂∂ρ is then degenerate precisely there, and the surface is Lagrangian on the complement. Theorem 3.1 converts this singular-Lagrangian condition into rational convexity. The non-fillable examples are assembled by Weinstein 1-handle attachments, Legendrian ambient surgery, and exact Lagrangian caps, giving an exact surface outside the hyperbolic points.","core_discovery":"The paper's central discovery is an existence theorem with two faces. Theorem 1.1: for each genus g >= 1 there is a rationally convex surface in S3 ⊂ C2 with exactly 2(g-1) hyperbolic complex tangencies, filled by holomorphic discs into a genus-g handlebody, including the standard Heegaard-splitting surface. Theorem 1.3 (combining Theorem 5.1, Corollary 5.5, and Theorem 6.2): for each g >= 2 there are exact rationally convex surfaces with 2(g-1) hyperbolic tangencies that admit no nonconstant compact Riemann surface attached, and for sufficiently large genus these surfaces occupy arbitrarily many smooth isotopy classes with distinct fundamental groups. The unifying claim is that a surface which is Lagrangian away from its hyperbolic tangencies for a Kähler form degenerating exactly there is rationally convex, and that such singular Lagrangian surfaces can be built in abundance.","pith_inferences":["One can test whether the same ambient 0-surgery produces rationally convex surfaces in arbitrary contact three-manifolds, since the local construction is contact-invariant; the paper only needs the surgery in S3.","The converse direction (Theorem 3.3) suggests a classification: rationally convex surfaces with only flat hyperbolic tangencies are exactly the singular Lagrangians of this degenerate type, so the flatness hypothesis may be removable.","The exact non-fillable surfaces could provide new constraints on symplectic embedding capacities or displaceability in C2, since their exactness is relative to a degenerate form rather than to the standard symplectic form."],"forward_implications":["Closed orientable surfaces of every genus g >= 1 can be rationally convex while carrying the maximal allowed number 2g-2 of hyperbolic tangencies.","The fillable examples are unknotted and live in the round sphere, giving high-genus analogues of Lagrangian tori that bound holomorphic handlebodies.","The non-fillable examples show that rational convexity does not force a holomorphic filling; exactness relative to a degenerate Kähler form blocks every attached compact Riemann surface.","For large genus, rational convexity is compatible with knotted embeddings whose complements have prescribed knot-group fundamental groups.","A global Kähler form that degenerates exactly at the hyperbolic points is the right substitute for the Duval-Sibony Lagrangian criterion when complex tangencies are present."],"supporting_citations":[{"why":"Supplies the equivalence between rational convexity and being Lagrangian for a global Kähler form in the totally real case, which the paper generalises.","marker":"[DS95]"},{"why":"Provides the local Stein neighborhood basis for flat hyperbolic tangencies used to produce the plurisubharmonic potential in Theorem 3.1.","marker":"[Sla04]"},{"why":"Supplies the rational-convexity criterion (Lemme 1) that the paper verifies to conclude convexity.","marker":"[Gay00]"},{"why":"Adapts Gayet's condition and the Lagrangian-inclusion argument used in Theorem 3.1.","marker":"[SS16]"},{"why":"Gives the filling theorem for spheres with hyperbolic tangencies used to construct the holomorphic handlebody fillings.","marker":"[BK91]"},{"why":"Defines Legendrian ambient surgery, the operation used to build exact Lagrangian cobordisms in Section 5.","marker":"[DR16]"},{"why":"Produces the exact Lagrangian caps and singular exact Lagrangians at the core of the non-fillable construction.","marker":"[Lin16]"},{"why":"Implies no closed exact Lagrangian exists for a nondegenerate Kähler form, motivating the degenerate form and the no-Riemann-surface conclusion.","marker":"[Gro85]"}],"fun_headline_variants":["First hyperbolic tangency surfaces: both fillable and not","Rationally convex surfaces: fillable vs no Riemann surfaces","New surfaces with hyperbolic tangencies, fillable or not","First examples: hyperbolic tangencies, with and without fillings","Hyperbolic tangency surfaces: two types, one unknotted"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument rests on the cited local model where a flat hyperbolic tangency has a plurisubharmonic potential whose zero set is exactly the surface and whose critical set is exactly the tangency; if that local potential fails for the standard tangency used here, the rational-convexity proof does not go through.","fun_headline_variants_meta":{"raw":{"variants":["First hyperbolic tangency surfaces: both fillable and not","Rationally convex surfaces: fillable vs no Riemann surfaces","New surfaces with hyperbolic tangencies, fillable or not","First examples: hyperbolic tangencies, with and without fillings","Hyperbolic tangency surfaces: two types, one unknotted"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000743,"raw_usage":{"total_tokens":3241,"prompt_tokens":795,"completion_tokens":2446,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":411,"completion_tokens_details":{"reasoning_tokens":2362}},"tokens_in":411,"tokens_out":2446,"duration_ms":18013,"temperature":1.0,"reasoning_tokens":2362,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-09T04:58:51.140794+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute directly in the four-ball model the plurisubharmonic potential required by the paper for the surface v=(Re u)^2-(Im u)^2: verify that its zero set is exactly the surface, that its critical set is the single tangency, and that strict plurisubharmonicity holds away from that point. Alternatively, search the moduli graph of holomorphic discs produced by the filling argument for a disc whose boundary meets both stable manifolds of a hyperbolic point; such a disc realises configuration (b) and would invalidate the rational-convexity proof.","supporting_citations":[],"review_version":1}