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arxiv: 2010.12293 · v4 · submitted 2020-10-23 · 🧮 math.SG · math.AP· math.DG

Special Lagrangian webbing

Jake P. Solomon , Amitai M. Yuval This is my paper

Pith reviewed 2026-05-24 14:18 UTC · model grok-4.3

classification 🧮 math.SG math.APmath.DG
keywords special Lagrangian cylinderspositive LagrangiansCalabi-Yau manifoldsMaslov indexLagrangian geodesicsC^{1,1} regularitycylindrical transform
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The pith

Families of imaginary special Lagrangian cylinders exist near transverse Maslov index 0 or n intersection points of positive Lagrangians in Calabi-Yau manifolds.

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

The paper constructs families of imaginary special Lagrangian cylinders near transverse intersection points of positive Lagrangian submanifolds that have Maslov index 0 or n. These cylinders produce geodesics connecting open positive Lagrangian submanifolds near the intersection points. The work also supplies a method to establish C^{1,1} regularity of such geodesics at non-smooth loci and shows that this regularity persists for spheres under small endpoint perturbations, yielding the first non-isometry-invariant examples in arbitrary dimension. The constructions serve as a step toward building geodesics of closed positive Lagrangian submanifolds without relying on perturbation.

Core claim

In a general Calabi-Yau manifold, families of imaginary special Lagrangian cylinders can be built near transverse Maslov index 0 or n intersection points of positive Lagrangian submanifolds. These cylinders imply the existence of geodesics of open positive Lagrangian submanifolds near the intersections. The cylindrical transform is the central tool used throughout. A separate method proves C^{1,1} regularity of positive Lagrangian geodesics at non-smooth points and shows that C^{1,1} geodesics of positive Lagrangian spheres continue to exist after small perturbations of the endpoints. The same approach gives the first examples of C^{1,1} positive Lagrangian geodesics in arbitrary dimension.

What carries the argument

The cylindrical transform, which converts the problem of finding special Lagrangian cylinders into a more tractable equation near the intersection points.

If this is right

  • Geodesics of open positive Lagrangian submanifolds exist near the relevant intersection points.
  • The cylinder construction provides a first step toward non-perturbative geodesics of closed positive Lagrangian submanifolds.
  • C^{1,1} geodesics of positive Lagrangian spheres persist under small perturbations of their endpoints.
  • C^{1,1} solutions to the positive Lagrangian geodesic equation exist in arbitrary dimension without isometry invariance.
  • Geodesics of positive Lagrangian linear subspaces exist in complex vector spaces when the Maslov index is 0 or n.

Where Pith is reading between the lines

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

  • If the cylinders can be extended globally or glued across multiple intersections, they may produce closed geodesics on compact positive Lagrangians.
  • The regularity method might extend to geodesics in other classes of Lagrangians or in non-Calabi-Yau symplectic manifolds.
  • The local existence result suggests that Maslov index 0 or n intersections behave like regular points for the geodesic flow on the space of positive Lagrangians.

Load-bearing premise

The cylindrical transform applies directly at the transverse intersection points without extra conditions beyond those stated for positive Lagrangians.

What would settle it

A concrete local model calculation in a Calabi-Yau manifold showing that no imaginary special Lagrangian cylinder exists near a chosen transverse Maslov index 0 intersection point of two positive Lagrangians.

Figures

Figures reproduced from arXiv: 2010.12293 by Amitai M. Yuval, Jake P. Solomon.

Figure 1
Figure 1. Figure 1: A family of imaginary special Lagrangian cylinders (Zs)s emanating from a Maslov zero intersection point of La￾grangian submanifolds Λ0,Λ1, as in Theorem 1.1. In [48, Theorem 1.6] it is shown that a cone-smooth geodesic (Λt)t∈[0,1] connect￾ing positive Lagrangian spheres Λ0,Λ1, that intersect transversally at two points persists under small perturbations of Λ0,Λ1. The following theorem shows that if the ge… view at source ↗
read the original abstract

We construct families of imaginary special Lagrangian cylinders near transverse Maslov index $0$ or $n$ intersection points of positive Lagrangian submanifolds in a general Calabi-Yau manifold. Hence, we obtain geodesics of open positive Lagrangian submanifolds near such intersection points. Moreover, this result is a first step toward the non-perturbative construction of geodesics of closed positive Lagrangian submanifolds. Also, we introduce a method for proving $C^{1,1}$ regularity of geodesics of positive Lagrangians at the non-smooth locus. This method is used to show that $C^{1,1}$ geodesics of positive Lagrangian spheres persist under small perturbations of endpoints, improving the regularity of a previous result of the authors. In particular, we obtain the first examples of $C^{1,1}$ solutions to the positive Lagrangian geodesic equation in arbitrary dimension that are not invariant under isometries. Along the way, we study geodesics of positive Lagrangian linear subspaces in a complex vector space, and prove an a priori existence result in the case of Maslov index $0$ or $n.$ Throughout the paper, the cylindrical transform introduced in previous work of the authors plays a key role.

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

1 major / 1 minor

Summary. The manuscript constructs families of imaginary special Lagrangian cylinders near transverse Maslov index 0 or n intersection points of positive Lagrangian submanifolds in a general Calabi-Yau manifold, yielding geodesics of open positive Lagrangian submanifolds near such points. It introduces a method for proving C^{1,1} regularity of positive Lagrangian geodesics at non-smooth loci, shows that C^{1,1} geodesics of positive Lagrangian spheres persist under small endpoint perturbations (improving prior work), obtains the first examples of C^{1,1} solutions in arbitrary dimension that are not isometry-invariant, and proves an a priori existence result for geodesics of positive Lagrangian linear subspaces in complex vector spaces when the Maslov index is 0 or n. The cylindrical transform from the authors' previous work is used throughout.

Significance. If valid, the local cylinder constructions and resulting geodesics advance the study of the geometry of positive Lagrangian submanifolds, while the C^{1,1} regularity method and non-isometry-invariant examples in arbitrary dimensions are concrete contributions. The linear-space existence result and the step toward non-perturbative closed geodesics are also of interest.

major comments (1)
  1. [Abstract and sections on cylindrical transform application] Abstract (final sentence) and the sections applying the cylindrical transform: the central cylinder construction and geodesic conclusions depend on the cylindrical transform from the authors' prior work. The manuscript must explicitly verify that the transform's hypotheses (including any requirements on disjointness, regularity, or preservation of positivity/Maslov index) hold at the transverse Maslov index 0 or n intersection points; without this, the families and geodesics do not follow from the stated hypotheses.
minor comments (1)
  1. [Introduction] The term 'imaginary special Lagrangian' and the precise definition of 'positive Lagrangian submanifold' should be recalled or referenced at the first use in the introduction for readers unfamiliar with the prior work.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for identifying this important point about the application of the cylindrical transform. We address the major comment below and will incorporate the requested verification in the revised version.

read point-by-point responses

  1. Referee: [Abstract and sections on cylindrical transform application] Abstract (final sentence) and the sections applying the cylindrical transform: the central cylinder construction and geodesic conclusions depend on the cylindrical transform from the authors' prior work. The manuscript must explicitly verify that the transform's hypotheses (including any requirements on disjointness, regularity, or preservation of positivity/Maslov index) hold at the transverse Maslov index 0 or n intersection points; without this, the families and geodesics do not follow from the stated hypotheses.

    Authors: We agree that the manuscript relies on the cylindrical transform from our prior work for the cylinder constructions and geodesic conclusions, and that explicit verification of the transform's hypotheses is required at the transverse Maslov index 0 or n intersection points. In the revised manuscript we will add a new subsection (immediately after the statement of the main local existence theorems) that checks each hypothesis in turn: (i) the relevant positive Lagrangian submanifolds are disjoint away from the intersection point, (ii) the regularity and positivity conditions are preserved under the local perturbation used to construct the cylinders, and (iii) the Maslov index remains 0 or n. This verification will be carried out using the transversality assumption and the local model near the intersection point already developed in Sections 3 and 4. The addition will make the dependence on the prior transform fully rigorous without altering the statements or proofs of the main results. revision: yes

Circularity Check

2 steps flagged

Cylindrical transform from authors' prior work is load-bearing for all main constructions

specific steps
  1. self citation load bearing [abstract (final sentence)]
    "Throughout the paper, the cylindrical transform introduced in previous work of the authors plays a key role."

    The construction of families of imaginary special Lagrangian cylinders near transverse Maslov index 0 or n intersection points, and the resulting geodesics of open positive Lagrangian submanifolds, are stated to depend on this transform. The transform is introduced in prior work by the same authors (Solomon-Yuval), with no indication that its applicability at the intersection points or preservation of positivity/Maslov conditions has been independently verified or machine-checked outside the self-citation chain.

  2. self citation load bearing [abstract (regularity paragraph)]
    "This method is used to show that C^{1,1} geodesics of positive Lagrangian spheres persist under small perturbations of endpoints, improving the regularity of a previous result of the authors."

    The C^{1,1} regularity result and first examples in arbitrary dimension are presented as an improvement on the authors' own prior result, with the method itself tied to the cylindrical transform from that prior work. The central claim of persistence under perturbation therefore reduces to self-citation without external falsifiability or independent confirmation stated.

full rationale

The abstract states that the cylindrical transform from previous work by the same authors plays a key role throughout the paper. The central results—construction of imaginary special Lagrangian cylinders near Maslov 0/n intersections, resulting geodesics, and the C^{1,1} regularity method—are presented as following from this transform. The paper also improves on a previous result of the authors. No independent verification, external benchmarks, or machine-checked properties of the transform at the relevant intersection points are indicated. This reduces the derivation chain to self-citation for the load-bearing step.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Abstract provides no explicit free parameters, new axioms, or invented entities. The work assumes standard Calabi-Yau structure and transverse intersections with given Maslov index; the cylindrical transform is imported from prior author work rather than derived here.

axioms (2)
  • domain assumption Positive Lagrangian submanifolds exist and intersect transversely with Maslov index 0 or n in a general Calabi-Yau manifold
    Invoked as the setting for the cylinder construction (abstract, first sentence).
  • domain assumption The cylindrical transform from previous work applies at the intersection points
    Stated as playing a key role throughout (abstract, final sentence).

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