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arxiv: 2311.18510 · v3 · pith:G5ZRHTVNnew · submitted 2023-11-30 · 🧮 math.SG

Contact action functional, calculus of variation and canonical generating function of Legendrian submanifolds

Yong-Geun Oh , Seungook Yu This is my paper

Pith reviewed 2026-05-24 05:38 UTC · model grok-4.3

classification 🧮 math.SG
keywords contact geometryLegendrian submanifoldsgenerating functionsaction functionalCarnot path spaceone-jet bundleHamiltonian isotopy
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The pith

The contact action functional on the Carnot path space generates Legendrian submanifolds in the one-jet bundle.

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

The paper establishes a contact geometry version of Weinstein's observation that the classical action functional generates Hamiltonian isotopes of the zero section. By defining the action functional on the space of horizontal curves in the contact distribution that are Hamiltonian-translated, called the Carnot path space, the authors provide a generating function for Legendrian submanifolds. This leads to a canonical construction similar to Laudenbach-Sikorav's finite dimensional approximation. The work sets up a framework for spectral invariants in the Legendrian setting.

Core claim

We formulate a contact analogue on the one-jet bundle J¹B of Weinstein's observation which reads the classical action functional on the cotangent bundle is a generating function of any Hamiltonian isotope of the zero section. We do this by identifying the correct action functional which is defined on the space of Hamiltonian-translated (piecewise smooth) horizontal curves of the contact distribution, which we call the Carnot path space. Then we give a canonical construction of the Legendrian generating function which is the Legendrian counterpart of Laudenbach-Sikorav's canonical construction of the generating function of Hamiltonian isotope of the zero section on the cotangent bundle which

What carries the argument

The Carnot path space, the space of Hamiltonian-translated piecewise smooth horizontal curves of the contact distribution, on which the action functional is defined to serve as a generating function for Legendrian submanifolds.

If this is right

  • The construction yields a Legendrian generating function for any contact Hamiltonian isotope of the zero section.
  • A finite dimensional approximation of the action functional produces the generating function.
  • The setup enables a Floer theoretic construction of spectral invariants for Legendrian submanifolds in a sequel paper.

Where Pith is reading between the lines

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

  • The same finite dimensional approximation technique may apply to other contact manifolds beyond the one-jet bundle.
  • Results on spectral invariants for Legendrians could be compared directly with those for Lagrangians in the cotangent bundle.

Load-bearing premise

The action functional must be defined on the Carnot path space of Hamiltonian-translated horizontal curves for the generating function property to hold.

What would settle it

A calculation showing that the critical points of the constructed functional on the Carnot path space fail to recover the points of a Hamiltonian isotope of the zero section in the one-jet bundle would falsify the claim.

read the original abstract

In the present paper, we formulate a contact analogue on the one-jet bundle $J^1B$ of Weinstein's observation which reads the classical action functional on the cotangent bundle is a generating function of any Hamiltonian isotope of the zero section. We do this by identifying the correct action functional which is defined on the space of Hamiltonian-translated (piecewise smooth) horizontal curves of the contact distribution, which we call the Carnot path space. Then we give a canonical construction of the Legendrian generating function which is the Legendrian counterpart of Laudenbach-Sikorav's canonical construction of the generating function of Hamiltonian isotope of the zero section on the cotangent bundle which utilizes a finite dimensional approximation of the action functional. Motivated by this construction, we develop a Floer theoretic construction of spectral invariants for the Legendrian submanifolds in the sequel [OY] which is the contact analog to the construction given in [Oh97, Oh99] for the Lagrangian submanifolds in the cotangent bundle.

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 claims to formulate a contact analogue of Weinstein's observation on the one-jet bundle J¹B: it identifies an action functional defined on the Carnot path space (the space of Hamiltonian-translated piecewise-smooth horizontal curves of the contact distribution) that serves as a generating function for any Legendrian isotope of the zero section. It then gives a canonical construction of the Legendrian generating function via finite-dimensional approximation of this functional, mirroring Laudenbach–Sikorav, and motivates a Floer-theoretic construction of spectral invariants for Legendrians in the sequel [OY].

Significance. If the central variational claim holds, the work would supply a direct contact-geometric counterpart to the classical action-functional generating-function theory on cotangent bundles, enabling canonical constructions and Floer-theoretic invariants for Legendrians that parallel the Lagrangian results of Oh and others. The explicit analogy to established symplectic constructions is a strength that could facilitate transfer of techniques between the two geometries.

major comments (2)
  1. [Abstract / construction of the action functional] The central claim requires that critical points of the action functional on the Carnot path space recover Legendrian submanifolds (or their Hamiltonian isotopies) without extraneous Reeb or curvature contributions. No explicit first-variation computation or Euler–Lagrange characterization is supplied to confirm this under the horizontal constraint and piecewise-smooth topology; this verification is load-bearing for the subsequent canonical construction and the Floer theory in [OY].
  2. [Canonical construction via finite-dimensional approximation] The finite-dimensional approximation step is asserted to be the Legendrian counterpart of Laudenbach–Sikorav, yet the manuscript does not detail how the Carnot-path-space constraint is preserved (or approximated) in the finite-dimensional model; without this, it is unclear whether the resulting generating function satisfies the required Legendrian condition.
minor comments (2)
  1. [Introduction] Define the Carnot path space and its topology at the first appearance rather than deferring the precise identification to later sections.
  2. [Definition of the action functional] Clarify the precise relation between the contact Hamiltonian isotopy and the horizontal curves used in the path space; the current phrasing leaves open whether the Reeb component is fully quotiented out.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. The two major comments identify places where additional explicit verification would strengthen the presentation. We address each point below and will incorporate the requested details in a revised version.

read point-by-point responses

  1. Referee: [Abstract / construction of the action functional] The central claim requires that critical points of the action functional on the Carnot path space recover Legendrian submanifolds (or their Hamiltonian isotopies) without extraneous Reeb or curvature contributions. No explicit first-variation computation or Euler–Lagrange characterization is supplied to confirm this under the horizontal constraint and piecewise-smooth topology; this verification is load-bearing for the subsequent canonical construction and the Floer theory in [OY].

    Authors: We agree that an explicit first-variation computation is necessary to confirm the correspondence. In the revised manuscript we will insert a new subsection (placed immediately after the definition of the Carnot path space) that computes the first variation of the action functional. The calculation will be performed under the horizontal constraint and with respect to the piecewise-smooth topology, showing that stationary points correspond exactly to Legendrian submanifolds (or their Hamiltonian isotopies) with no extraneous Reeb or curvature terms. This addition will also serve as the foundation for the finite-dimensional approximation and the Floer-theoretic construction in the sequel. revision: yes

  2. Referee: [Canonical construction via finite-dimensional approximation] The finite-dimensional approximation step is asserted to be the Legendrian counterpart of Laudenbach–Sikorav, yet the manuscript does not detail how the Carnot-path-space constraint is preserved (or approximated) in the finite-dimensional model; without this, it is unclear whether the resulting generating function satisfies the required Legendrian condition.

    Authors: We acknowledge that the preservation of the Carnot-path-space constraint in the finite-dimensional model requires further explanation. In the revised version we will expand the section on the canonical construction to include a precise description of the approximation scheme: we will specify the finite-dimensional subspaces of piecewise-smooth horizontal curves, the projection maps that respect the contact distribution, and the manner in which the Legendrian condition is maintained at each stage of the approximation. This will make the parallel with Laudenbach–Sikorav explicit while adapting it to the contact setting, thereby confirming that the resulting generating function is indeed Legendrian. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation presents independent construction of contact analogue.

full rationale

The paper's central step is the identification of the action functional on the Carnot path space (Hamiltonian-translated piecewise-smooth horizontal curves) as the contact analogue of Weinstein's observation, followed by a canonical finite-dimensional approximation mirroring Laudenbach-Sikorav. This is presented as a new formulation on J¹B, with the subsequent Legendrian generating function construction derived from it. References to [Oh97, Oh99] concern independent prior Lagrangian results by one author; the sequel [OY] is motivated by the present work rather than presupposed. No quoted equation or step reduces the claimed generating property to a fitted input, self-definition, or load-bearing self-citation chain. The derivation chain is self-contained as a direct variational and approximation construction without the enumerated circular patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only; no explicit free parameters, axioms, or invented entities can be extracted beyond standard background in contact and symplectic geometry.

pith-pipeline@v0.9.0 · 5708 in / 1050 out tokens · 24424 ms · 2026-05-24T05:38:02.043098+00:00 · methodology

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