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arxiv: 2606.24683 · v1 · pith:LICKXZZGnew · submitted 2026-06-23 · 🧮 math.DG

Lagrangian Submanifolds with Legendrian Boundary in the Unit Ball

Dong Gao , Hui Ma , Zeke Yao This is my paper

Pith reviewed 2026-06-25 22:35 UTC · model grok-4.3

classification 🧮 math.DG
keywords Lagrangian submanifoldsLegendrian boundaryself-similar submanifoldsrigidity theoremsunit ball in complex spaceLiouville formunique continuation
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The pith

Any compact exact Lagrangian self-similar submanifold with connected Legendrian boundary in the unit ball must be an equatorial n-disk.

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

The paper proves a rigidity theorem for compact Lagrangian submanifolds inside the complex unit ball that meet the boundary sphere in a Legendrian submanifold. Under the assumptions of exactness and connectedness of the boundary, or alternatively a sign condition on the contact angle, any self-similar such submanifold must coincide with an equatorial n-dimensional disk. This classification extends earlier results limited to two dimensions and to minimal surfaces by covering higher dimensions and the larger class of self-similar Lagrangians. The argument proceeds by applying unique continuation to the Liouville form at the boundary rather than using holomorphic sections.

Core claim

Any compact exact Lagrangian self-similar submanifold with connected Legendrian boundary in the unit ball must be an equatorial n-disk. The same conclusion holds without the exactness assumption provided the cosine of the contact angle has constant sign; this covers in particular the case of Legendrian free boundary. The proof employs the Liouville form and boundary unique continuation for differential forms.

What carries the argument

The Liouville form on the ball combined with boundary unique continuation applied to differential forms, which forces the submanifold to be flat under the self-similarity and boundary conditions.

If this is right

  • The result includes minimal Lagrangian submanifolds as the special case where the self-similar factor vanishes.
  • Prior two-dimensional minimal rigidity theorems now hold in all dimensions.
  • The Legendrian boundary condition alone is sufficient for rigidity, without requiring an additional capillary condition.
  • Non-disk Lagrangian self-similar submanifolds with Legendrian boundary exist only if the boundary fails to be connected or if the sign condition on the contact angle is violated.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The unique continuation technique on the Liouville form may extend to rigidity questions for Lagrangians in other Kähler manifolds with boundary.
  • One could investigate whether the connectedness assumption on the boundary can be relaxed while preserving the conclusion.
  • The examples constructed with capillary boundary suggest that capillary conditions allow more flexibility than pure Legendrian ones.
  • Similar rigidity might hold for other geometric flows or solitons in symplectic geometry.

Load-bearing premise

The Legendrian boundary must be connected and the submanifold must satisfy either exactness or a fixed sign condition on the cosine of the contact angle.

What would settle it

The existence of a compact exact Lagrangian self-similar submanifold in the unit ball, with connected Legendrian boundary, that is not an equatorial disk would disprove the main theorem.

Figures

Figures reproduced from arXiv: 2606.24683 by Dong Gao, Hui Ma, Zeke Yao.

Figure 1
Figure 1. Figure 1: A closed solution of (4.5) for n = 2, p = 1, and q = 3. The curve is numerically plotted with r(0) ≈ 3.339, ϕ(0) = π/2, and α(0) = 0. Let F : R × S 1 → C 2 , F(s, u) = γ 2 1,3 (s)ψ1(u) be the corresponding Lagrangian self-shrinker immersion. We will show that the closure of each connected component of F(R × S 1 ) ∩ B 4 is an embedded Lagrangian self-shrinker annulus with Legendrian capillary boundary [PIT… view at source ↗
Figure 2
Figure 2. Figure 2: r 4 sin(4ϕ) = 1 2 . The following identification was already noted by Anciaux in [1]. We include the explicit verifi￾cation here for later use. Claim 1. The image of F c is precisely Mn c , i.e. F c (R × S n−1 ) = Mn c . Proof of the claim. Indeed, for p ∈ S n−1 , we have F c (s, p) = r(s)e iϕ(s)p. Writing x = r(s) cos(ϕ(s))p and y = r(s) sin(ϕ(s))p. we see from (4.11) that |x|y = |y|x, Im (|x| + i|y|) n … view at source ↗
read the original abstract

We study compact Lagrangian submanifolds in the unit ball $\mathbb B^{2n}\subset\mathbb C^n$ with Legendrian boundary. We prove that any compact exact Lagrangian self-similar submanifold with connected Legendrian boundary must be an equatorial $n$-disk. The same rigidity holds, without exactness, for Legendrian boundary under a fixed sign assumption on the cosine of the contact angle; in particular, it holds for Legendrian free boundary. These results extend the two dimensional minimal rigidity theorems of Li-Wang-Weng and Luo-Sun to higher dimensions and to the Lagrangian self-similar setting, which includes the minimal case. Notably, the Legendrian capillary condition in Li-Wang-Weng's theorem is weakened to the Legendrian boundary condition. Our proof uses the Liouville form and boundary unique continuation for differential forms, rather than holomorphic differential techniques. Finally, we construct non-disk-type Lagrangian self-similar examples with Legendrian capillary boundary.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper proves a rigidity result stating that any compact exact Lagrangian self-similar submanifold with connected Legendrian boundary in the unit ball B^{2n} ⊂ C^n must be an equatorial n-disk. The same conclusion holds without the exactness assumption provided there is a fixed sign condition on the cosine of the contact angle (including the free boundary case). The argument relies on the Liouville form together with boundary unique continuation for differential forms, presented as an alternative to holomorphic differential methods. The results extend 2D minimal rigidity theorems to higher dimensions and the self-similar setting. The manuscript also constructs non-disk Lagrangian self-similar examples with Legendrian capillary boundary.

Significance. If the central claims hold, the work supplies a new technical route to rigidity statements in Lagrangian geometry that avoids holomorphic differentials and applies directly to self-similar submanifolds. The explicit weakening of the capillary condition to a plain Legendrian boundary, together with the counterexamples for the capillary case, sharpens the understanding of which boundary conditions force rigidity. The method is described as using standard tools (Liouville form and unique continuation), which, if correctly implemented, would constitute a clean contribution.

minor comments (2)
  1. [Abstract] Abstract: the statements of the two main theorems are clear, but the abstract does not indicate the range of dimensions n for which the results hold or whether n=1 is included; adding this would improve precision.
  2. [Abstract] Abstract: references to the 2D theorems of Li-Wang-Weng and Luo-Sun are given by name only; full bibliographic details should appear at first mention in the introduction.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their report and for recognizing the potential significance of our alternative approach to rigidity results via the Liouville form and boundary unique continuation. The referee's recommendation is listed as uncertain, but no specific major comments or technical objections were provided in the report. We therefore offer a brief overall response below and note that we are happy to address any further points the referee may wish to raise.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper proves a rigidity result for compact exact Lagrangian self-similar submanifolds with connected Legendrian boundary using the Liouville form and boundary unique continuation for differential forms. This is an independent analytic argument extending prior 2D results (Li-Wang-Weng, Luo-Sun) via a different method, without fitted parameters, self-definitional reductions, or load-bearing self-citations. Assumptions (exactness or fixed-sign contact angle, connected boundary) are stated explicitly and the counterexamples for the capillary case are constructed separately. No step reduces to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard facts from symplectic and contact geometry (Lagrangian condition, Legendrian boundary, Liouville form properties) plus the boundary unique continuation principle for forms; no free parameters or invented entities are mentioned.

axioms (2)
  • standard math Standard properties of the Liouville form on the unit ball and its restriction to Lagrangian submanifolds
    Invoked to obtain the integral identity or vanishing used in the rigidity proof
  • standard math Boundary unique continuation for differential forms
    Used to conclude that the form vanishes identically from boundary data

pith-pipeline@v0.9.1-grok · 5693 in / 1372 out tokens · 20911 ms · 2026-06-25T22:35:15.587865+00:00 · methodology

discussion (0)

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Reference graph

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