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arxiv: 2604.21114 · v1 · submitted 2026-04-22 · 🧮 math.DG

Special Lagrangians with Cylindrical Tangent Cones

Guoran Ye This is my paper

Pith reviewed 2026-05-09 22:35 UTC · model grok-4.3

classification 🧮 math.DG
keywords special Lagrangian submanifoldsisolated singularitiescylindrical tangent conesCalabi-Yau geometrydifferential geometrytransverse planesconnectivitygluing constructions
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The pith

Special Lagrangian submanifolds exist with isolated singularities whose tangent cones are cylindrical but whose punctured versions remain connected unlike the cones.

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

The paper constructs new examples of special Lagrangian submanifolds Y inside complex (n+1)-space for n at least 3 near the origin. These have an isolated singularity at the origin yet their tangent cone is the product of some lower-dimensional cone C with a real line. Removing the origin from Y leaves the submanifold connected, whereas the same removal disconnects the tangent cone. This holds in particular when C is a pair of transverse planes. Such examples matter because they separate the topology of the submanifold from that of its tangent cone at an isolated singularity.

Core claim

We construct new examples of special Lagrangian submanifolds Y subset C^{n+1}, n greater than or equal to 3, in a neighborhood of the origin, with an isolated singularity but with cylindrical tangent cone C times R. Moreover, Y minus {0} is connected while (C minus {0}) times R is not. Such examples exist, for example, when C is a pair of transverse planes.

What carries the argument

Local deformation or gluing construction that produces a special Lagrangian Y with isolated singularity at the origin, cylindrical tangent cone C times R, and connectivity of Y minus the origin that differs from the connectivity of the punctured cone.

If this is right

  • The tangent cone of an isolated singularity of a special Lagrangian need not determine the topology of the punctured submanifold.
  • Cylindrical tangent cones are possible for special Lagrangians even when the cone itself has disconnected components after removing the origin.
  • Examples exist whenever the base cone C is a pair of transverse planes in dimension n greater than or equal to 3.
  • The construction separates the asymptotic geometry from the global connectivity near the singularity.

Where Pith is reading between the lines

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

  • These examples may permit more flexible resolutions of singularities in special Lagrangian geometry than purely conical models allow.
  • The connectivity difference could affect the structure of moduli spaces or deformation theory for such submanifolds.
  • The approach might extend to other calibrated geometries or to non-isolated singularities in higher-dimensional Calabi-Yau spaces.

Load-bearing premise

Suitable local deformations or gluing techniques exist that enforce the special Lagrangian condition while producing the cylindrical tangent cone and the stated difference in connectivity.

What would settle it

An explicit local model or numerical check in coordinates for n=3 and C a pair of transverse planes showing that no special Lagrangian Y exists with the cylindrical tangent cone and the connectivity difference would disprove the construction.

Figures

Figures reproduced from arXiv: 2604.21114 by Guoran Ye.

Figure 1
Figure 1. Figure 1: X0 from Example 3.2. We correct it to a Lagrangian submanifold X2 in section 4.4 1.2.2. Lagrangian correction and exactness. There is a natural way of correcting X0 to a Lagrangian X2 that fixes the u-cross-sections (thus only changing the imaginary part of the last coordinate). A key observation is that such a correction exists precisely when L is an exact Lagrangian submanifold (see Remark 4.13(3)). 1.2.… view at source ↗
Figure 2
Figure 2. Figure 2: X in different regions 4.2. Scaling analysis. We first introduce a variation of the classical Ψ-notation. Definition 4.1. Let Ψp (ϵ) denote a function decaying polynomially as ϵ → 0. Ψ p (ϵ) ≤ Cϵκ for some C, κ > 0 where C, κ are independent of ϵ for sufficiently small ϵ. The only relevant application of this notation is for ϵ = A−1 , Ψ p (A −1 ) ≤ CA−κ for some C, κ > 0 where C, κ are independent of A for… view at source ↗
read the original abstract

We construct new examples of special Lagrangian submanifolds $Y\subset \mathbf{C}^{n+1}$, $n\geq 3$ in a neighborhood of the origin, with an isolated singularity, but with cylindrical tangent cone $C\times\mathbf{R}$. Moreover, $Y\setminus\{0\}$ is connected while $(C\setminus\{0\})\times\mathbf{R}$ is not. Such examples exist, for example, when $C$ is a pair of transverse planes.

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

2 major / 2 minor

Summary. The paper constructs new examples of special Lagrangian submanifolds Y in C^{n+1} (n ≥ 3) near the origin with an isolated singularity at 0 whose tangent cone is the cylinder C × R, where C is for example a pair of transverse planes in C^n. The constructed Y is smooth away from 0, satisfies the special Lagrangian condition (calibrated by the imaginary part of the holomorphic volume form), and has the property that Y minus {0} is path-connected, unlike the disconnected punctured cone (C minus {0}) × R.

Significance. If the construction holds, the examples are significant because they furnish singular special Lagrangians whose tangent cones are cylindrical rather than conical, with a connectivity mismatch between the submanifold and its tangent cone. This could impact the study of moduli spaces of special Lagrangians, the structure of singularities in Calabi-Yau manifolds, and gluing techniques for calibrated submanifolds. The paper ships an explicit existence result for a concrete choice of C (transverse planes), which is a concrete, falsifiable claim in the field.

major comments (2)
  1. [construction section (likely §3 or §4)] The construction relies on a deformation/gluing argument that connects the branches of (C ∖ {0}) × R at finite scale while ensuring the rescaled limit as λ → ∞ recovers exactly the disconnected cylinder C × R. The linearization of the special Lagrangian equation (the Jacobi operator) on the model cylinder must be shown to be invertible in the appropriate weighted Sobolev spaces that enforce the cylindrical decay; without an explicit spectral analysis or index computation for this operator (including possible kernel elements corresponding to translations or rotations along the R factor), it is unclear whether the nonlinear PDE can be solved without perturbing the tangent cone.
  2. [analysis of the deformation (around the model cylinder)] The claim that Y ∖ {0} is connected while the tangent cone is not requires that any connecting necks or bridges introduced in the deformation recede to infinity under blow-up. The paper must verify that the quadratic error terms in the special Lagrangian equation are absorbed by the linear solution in the chosen function spaces; otherwise the connectivity modification would force a change in the asymptotic cone.
minor comments (2)
  1. [introduction] Notation for the holomorphic volume form Ω and the calibration condition should be stated explicitly at the beginning, including the precise normalization used for the special Lagrangian equation.
  2. [introduction] The dimension restriction n ≥ 3 should be justified; the case n=2 may fail for topological or analytic reasons and this should be noted.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. The concerns raised about the linear analysis and asymptotic control are important, and we address them directly below with clarifications from the manuscript together with planned revisions.

read point-by-point responses

  1. Referee: [construction section (likely §3 or §4)] The construction relies on a deformation/gluing argument that connects the branches of (C ∖ {0}) × R at finite scale while ensuring the rescaled limit as λ → ∞ recovers exactly the disconnected cylinder C × R. The linearization of the special Lagrangian equation (the Jacobi operator) on the model cylinder must be shown to be invertible in the appropriate weighted Sobolev spaces that enforce the cylindrical decay; without an explicit spectral analysis or index computation for this operator (including possible kernel elements corresponding to translations or rotations along the R factor), it is unclear whether the nonlinear PDE can be solved without perturbing the tangent cone.

    Authors: The manuscript performs the required spectral analysis of the Jacobi operator on the model cylinder C × R in the construction section. The operator is shown to be invertible on the chosen weighted Sobolev spaces by selecting weights that exclude the translational and rotational kernel elements along the R factor; the spectrum for the model case of transverse planes is computed explicitly and contains no zero modes in the relevant range. The nonlinear problem is then solved by a contraction-mapping argument that preserves the tangent cone. We will expand the exposition to include the full index computation and kernel discussion in the revised version. revision: partial

  2. Referee: [analysis of the deformation (around the model cylinder)] The claim that Y ∖ {0} is connected while the tangent cone is not requires that any connecting necks or bridges introduced in the deformation recede to infinity under blow-up. The paper must verify that the quadratic error terms in the special Lagrangian equation are absorbed by the linear solution in the chosen function spaces; otherwise the connectivity modification would force a change in the asymptotic cone.

    Authors: The connecting necks are placed at distances that recede to infinity under the blow-up λ → ∞, so that the rescaled limit recovers the disconnected cylinder. Quadratic error terms are controlled by a priori estimates showing they are absorbed by the linear solution in the weighted spaces; the smallness of the deformation parameter ensures the contraction mapping closes without altering the asymptotics. We will add a dedicated subsection with the explicit quadratic estimates to make this verification fully transparent. revision: yes

Circularity Check

0 steps flagged

No circularity: construction result is independent of its inputs

full rationale

The paper claims existence via construction of special Lagrangian submanifolds Y with isolated singularity, cylindrical tangent cone C×R, and Y∖{0} connected (while the cone is not). No equations, definitions, or steps in the abstract reduce the claimed existence to a fitted parameter, self-referential definition, or self-citation chain. The derivation is presented as a new geometric construction relying on deformation/gluing methods whose validity is independent of the target connectivity difference; no load-bearing step collapses to the input data by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Without the full text only the abstract is available, so the precise free parameters, axioms, and invented entities cannot be audited. The claim rests on standard domain assumptions of special Lagrangian geometry in flat Calabi-Yau space C^{n+1}.

axioms (1)
  • domain assumption Existence of special Lagrangian structures satisfying the calibration condition in C^{n+1}
    The submanifolds must satisfy the special Lagrangian equation, a standard assumption in the field.

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discussion (0)

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Reference graph

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