Rationally Convex Surfaces with hyperbolic complex tangencies

Georgios Dimitroglou Rizell; Mark G. Lawrence

REVIEW 3 major objections 4 minor 23 references

Rationally Convex Surfaces with hyperbolic complex tangencies

T0 review · 3 major / 4 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read 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…

desk verdict 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. read the letter →

arxiv 2502.03357 v1 pith:XW2VIJU7 submitted 2025-02-05 math.SG math.CV

Georgios Dimitroglou Rizell , Mark G. Lawrence This is my paper · ORCID

classification math.SGmath.CV MSC 32E2053D12

keywords rationallyconvexhyperboliccomplextangencyexactLagrangianholomorphicfillingLegendriansurgeryplurisubharmonicpotentialpolynomialhullsymplectictopology

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

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.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

The 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.

The 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.

Now 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.

The self-citations [DR16] and [DR24] are not a red flag: they are separate construction papers used as black boxes, and the dependence is legitimate.

Bottom 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.

Recommendation: engage with it, but make the referee check the Slapar flatness hypothesis before signing off.

Referee Report

3 major / 4 minor

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.

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 (3)

[Sec. 3 (Theorem 3.1) and Sec. 4.1] 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.
[Sec. 2.1 (Lemma 2.3, Corollary 2.4)] 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.
[Sec. 6 (Proposition 6.1)] 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.

minor comments (4)

[Title] The title as printed in the arXiv source is garbled ('RA TIONALL Y CONVEX'); the final version should correct this typographical error.
[Sec. 2 (Lemma 2.1)] 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_-').
[Theorem 5.1] 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.
[Sec. 2.2 (Lemma 2.5)] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central rational-convexity proofs rely on external theorems and independent constructions, not on the paper's own conclusions.

full rationale

The paper's central claim—existence of rationally convex surfaces with hyperbolic complex tangencies—is not assumed in any hypothesis. Theorem 3.1 invokes Slapar's external theorem [Sla04, Theorem 2] to obtain a local plurisubharmonic potential for flat hyperbolic tangencies, and then uses Gayet's criterion [Gay00] / Shafikov–Sukhov [SS16] to conclude rational convexity from that potential plus a holomorphic-extension argument. The local model in Section 4.1 is asserted to be 'flat' in Slapar's sense; this is an unverified assumption and hence a possible correctness gap, but it is not circular, since flatness is a geometric property of the model rather than a restatement of the target theorem. The fillable family is shown rationally convex via the filling analysis (Lemmas 2.9 and 2.10) and the graph-structure argument (Proposition 2.7), which reduce to Bedford–Klingenberg [BK91] and standard positivity-of-intersection arguments, not to the desired conclusion. The non-fillable family (Theorems 5.1, 5.5, and 6.2) uses Lin's exact Lagrangian caps [Lin16] and the first author's prior constructions [DR16] and [DR24] as building blocks; these are independent theorems that do not assume the present result. No fitted parameter is renamed as a prediction, and no displayed equation is equivalent to its input by construction. The self-citations [DR16] and [DR24] are load-bearing for the existence of certain Lagrangian cobordisms and knotted concordances, but they are external constructions with their own proofs, not restatements of the rational-convexity theorem. Thus the derivation chain is not circular; any weakness lies in the unproved flatness assertion and in the reliance on external results, not in circular reasoning.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The central existence theorems are proved by assembling external results rather than by fitting data. The fillable family uses Bedford-Klingenberg disc fillings and Duval-Sibony/Gayet convexity criteria; the non-fillable family uses Slapar's Stein neighborhood theorem and Lin's exact Lagrangian caps as the topological cap. Two of the ingredients, [DR16] and [DR24], are self-citations of the first author's earlier work and are load-bearing for the knotted examples. There are no invented physical entities and no empirically fitted constants. The sufficiently small or large choices epsilon, C, and B are construction parameters, not fitted values.

free parameters (3)

epsilon (ambient surgery parameter) = sufficiently small
Controls the cylinder C_epsilon in Lemma 2.1 and the persistence of holomorphic disc families. Existence is asserted for all sufficiently small epsilon, with no explicit numeric value.
C (weight in plurisubharmonic function) = sufficiently large
In Theorem 3.1, phi = C*tilderho + psi is made strictly plurisubharmonic on C2 minus H by taking C >> 0.
B (symplectization cutoff) = arbitrarily large
In the proof of Theorem 6.2, the trivial Lagrangian cylinder is taken over [1, B] with B arbitrarily large after rescaling, to allow insertion of a knotted Lagrangian concordance.

assumptions (6)

standard math Bedford-Klingenberg [BK91]: a generic sphere in the boundary of a rationally convex domain with only good hyperbolic tangencies admits a holomorphic disc filling diffeomorphic to a ball.
Invoked in Lemma 2.3 to fill each genus-0 component Sigma_j_i, and in Lemma 2.8 for pre-Lagrangian tori.
standard math Slapar [Sla04, Theorem 2]: surfaces with only flat hyperbolic complex tangencies admit a Stein neighborhood basis with a plurisubharmonic function vanishing exactly on the surface.
Central to Theorem 3.1 and Theorem 3.3; the paper's local tangency model is engineered to be flat in Slapar's sense.
standard math Duval-Sibony [DS95] and Gayet [Gay00, Lemme 1]: rational convexity follows from the existence of a global plurisubharmonic function with prescribed vanishing properties, and totally real rationally convex submanifolds are Lagrangian for a global Kahler form.
Provides the rational convexity criterion used in Theorem 3.1 and Lemma 2.9.
domain assumption Lin [Lin16]: existence of exact Lagrangian genus-2 caps of the relevant Legendrian unknot, and the associated exact Lagrangian cobordism machinery.
The non-fillable examples in Theorem 5.1 cap off with Lin's exact Lagrangian genus-2 cap; if this construction fails, Theorem 5.1 and Corollary 5.5 fail.
domain assumption Dimitroglou Rizell [DR16] and [DR24]: Legendrian ambient surgery and knotted Lagrangian concordances realizing arbitrary knot groups, cited without proof.
Used to build Lagrangian cobordism pieces and to produce knotted rationally convex surfaces in Theorem 6.2; these are self-citations of the first author's prior work.
standard math The local model {v = (Re u)^2 - (Im u)^2} is a flat hyperbolic complex tangency and can be approximated by good hyperbolic tangencies in the sense of [GS12].
Used in Section 4.1 to make the surface Lagrangian for a degenerate Kahler form, and in Lemma 2.3 to apply Bedford-Klingenberg fillings.

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Pith. "Pith review of Rationally Convex Surfaces with hyperbolic complex tangencies." pith.science (2026). https://pith.science/paper/XW2VIJU7

@misc{pith2026250203357,
  author       = {Pith},
  title        = {Pith review of: Rationally Convex Surfaces with hyperbolic complex tangencies},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XW2VIJU7}},
  note         = {Machine review of arXiv:2502.03357}
}

read the original abstract

We construct the first examples of rationally convex surfaces in the complex plane with hyperbolic complex tangencies. In fact, we give two very different types of rationally convex surfaces: those that admit analytic fillings by handle-bodies, and those that do not have any compact Riemann surfaces attached at all. The fillable examples all live in the round sphere and are unknotted, while the non-fillable examples can moreover be produced in several different smooth isotopy classes.

Figures

Figures reproduced from arXiv: 2502.03357 by the authors.

**Figure 1.** The characteristic distribution ξ ∩ T Cϵ with the stable and unstable manifolds of h± shown in blue. Here θ is an S 1 -valued coordinate on the cylinder, which can be taken to coincide with the ambient coordinate ±y near h± [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗

**Figure 2.** The surface Σj i is a sphere with four discs removed. The filling produced by Bedford–Klingenberg yields the handle-body that bounds the surface in the picture, which contains the parts of the disc families T ± i ⊂ Ti [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** With our orientation convention for the characteristic foliation, we obtain the following. Lemma 2.5. The orientation on the surface T g Cl, which coincides with dt ∧ dθ at any boundary point of a smooth holomorphic disc in the above moduli space as described above, coincides with η ∧ dθ, where η is a one-form for which χ = ker η, an such that η induces orientation of the characteristic distribution according to our… view at source ↗

Figures from the paper (10 more)

**Figure 3.** Figure 3: The model a single incoming edge at a hyperbolic point, with two outgoing edges. Note that the hyperbolic point is negative here. (Using the terminology from [BK91], the discs approach the hyperbolic point “from the outside.”) T + i T − i T − i h− h+ h− h+ T + i T − i…

**Figure 4.** Figure 4: The a priori possibilities of the graphs that corresponds to the moduli space of the discs produced by [BK91] and that fill a component Σj i ⊂ Σi . Configuration (a) is shown on the left, where the disc family T j,+ i form the two incoming edges at h+, while the disc …

**Figure 5.** Figure 5: The filling of T g Cl obtained by gluing together the partial fillings shown on the left in [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]

**Figure 6.** Figure 6: The Legendrian unknot Λ−1,−2 of rot = −1 and tb = −2 depicted in a contact Darboux ball (R 2 xy × Rz, dz − ydx). On top we see the front projection, and on the bottom the so-called Lagrangian projection. Next we give a description of the local model in terms of the coo…

**Figure 7.** Figure 7: The image of the local model Σ0 under the standard momentum map. The projection of Σ0 coincides with the line with slope 1/2 for ∥z1∥ 2 ≥ 1/3, while it is tangent to the horizontal axis precisely at the origin. 1 ∥z1∥ 2 ∥z2∥ 2 2/3 1/3 2/3 1 1/3 Σ0 [PITH_FULL_IMAGE:fig…

**Figure 8.** Figure 8: A foliation by convex spheres whose normals are colinear with (2f(r 2 1 ), r2 1 ) along Σ0 = {r 2 2 = f(r 2 1 )}, and which coincide with the standard concentric round spheres near the boundary of the unit ball. Note that the image of Σ0 is the graph of the parabola {r…

**Figure 9.** Figure 9: Bottom to middle: A Legendrian isotopy consisting of four Reidemeister one-moves in the front. Middle to top: two Legendrian ambient surgeries performed at the pairs of cusp-edges that face each other produces a link of two standard Legendrian unknots shown on top. Th…

**Figure 6.** Figure 6: It should be noted that the existence of [PITH_FULL_IMAGE:figures/full_fig_p019_6.png]

**Figure 10.** Figure 10: The exact Lagrangian surface with conical singularities produced in the proof of Theorem 5.1. In {τ ≤ 0} we see two Lagrangian cones over the Legendrian unknots Λ±1,−2 contained in disjoint Darboux balls. In {τ < 0} these Darboux balls are connected by a Weinstein han…

**Figure 11.** Figure 11: The Legendrian unknot Λ0,−2(g−1)−1 for g = 4 (shown on the left) is Lagrangian cobordant to a union of two unlinked Legendrian unknots Λ0,−3 (shown in the middle) by an exact Lagrangian handle-attachment cobordism that corresponds to the Legendrian ambient surgery alo…

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.

[1] [1]

Bedford and W

E. Bedford and W. Klingenberg. On the envelope of holomorphy of a 2 -sphere in C ^2 . J. Amer. Math. Soc. , 4(3):623--646, 1991

work page 1991

[2] [2]

L. Chen, G. Crider-Phillips, B. Reinoso, J. M. Sabloff, and L. Yao. Non-orientable lagrangian fillings of legendrian knots. Preprint, https://arxiv.org/abs/2203.16605 [math.SG], 2022

work page Pith review arXiv 2022

[3] [3]

Cieliebak and Y

K. Cieliebak and Y. Eliashberg. From Stein to Weinstein and back. Symplectic geometry of affine complex manifolds. Colloquium Publications. American Mathematical Society 59. Providence, RI: American Mathematical Society (AMS). 354 p. \ 78.00 , 2012

work page 2012

[4] [4]

Chekanov

Y. Chekanov. Differential algebra of L egendrian links. Invent. Math. , 150(3):441--483, 2002

work page 2002

[5] [5]

Duval and D

J. Duval and D. Gayet. Riemann surfaces and totally real tori. Comment. Math. Helv. , 89(2):299--312, 2014

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Dimitroglou Rizell

G. Dimitroglou Rizell. Legendrian ambient surgery and L egendrian contact homology. J. Symplectic Geom. , 14(3):811--901, 2016

work page 2016

[7] [7]

Lagrangian Approximation of Totally Real Concordances

G. Dimitroglou Rizell. Lagrangian approximation of totally real concordances. Preprint, https://arxiv.org/abs/2408.16614 [math.SG], 2024

work page Pith review arXiv 2024

[8] [8]

Dimitroglou Rizell, E

G. Dimitroglou Rizell, E. Goodman, and A. Ivrii. Lagrangian isotopy of tori in S^2 S^2 and C P^2 . Geom. Funct. Anal. , 26(5):1297--1358, 2016

work page 2016