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A lower bound for the radius of Weinstein's Lagrangian tubular neighborhood

Hikaru Yamamoto

TL;DR

The paper provides a quantitative lower bound for Weinstein’s Lagrangian tubular neighborhood radius, denoted $r_W(L)$, by introducing $r_W^{imm}(L)$ for immersions and, in the compact embedded case, a bound using the embedding constant $\mathrm{emb}(L)$ and the injectivity radius. The method is constructive: the authors implement a Moser-trick deformation between the canonical form $\tilde{\omega}$ on $T^{\bot}L$ and the pullback $F^{*}\omega$ on shrinking tubular neighborhoods, with explicit, step-by-step estimates ensuring nondegeneracy and the existence of a global diffeomorphism $\Theta$ with $\Theta(p)=p$ and $\Theta^{*}\omega=\tilde{\omega}$. They develop a robust Sasaki/almost-Kähler framework, derive precise bounds for the scaling map, the time-dependent vector field, and the generated flow, and show that the resulting radius scales as $r_W^{imm}(L)\ge 10^{-100}/B$ (and $r_W(L)\ge 10^{-100}/B_*$ in the embedding case), where $B$ and $B_*$ aggregate curvature and second fundamental form data. These results provide explicit, geometry-driven control over the Weinstein neighborhood size, enabling rigorous and quantitative applications to Lagrangian deformations via closed 1-forms and related constructions.

Abstract

For an immersed Lagrangian submanifold $L$ in a Kähler manifold $(M,ω)$, there exists a symplectic local diffeomorphism from a tubular neighborhood of the image of the zero section in the normal bundle $T^{\bot}L$ of $L$, equipped with a canonical symplectic form $\tildeω$, to $(M,ω)$ whose restriction to $L$ is the identity map by Weinstein's Lagrangian tubular neighborhood theorem, where the image of the zero section in $T^{\bot}L$ is identified with $L$. In this paper, we give a lower bound for the supremum of the radii of tubular neighborhoods that have such a symplectic diffeomorphism into $(M,ω)$ from below by a constant explicitly given in terms of up to second derivatives of the Riemannian curvature tensor of $M$ and the second fundamental form of $L$. We also give a similar lower bound in the case where $L$ is compact and embedded.

A lower bound for the radius of Weinstein's Lagrangian tubular neighborhood

TL;DR

The paper provides a quantitative lower bound for Weinstein’s Lagrangian tubular neighborhood radius, denoted , by introducing for immersions and, in the compact embedded case, a bound using the embedding constant and the injectivity radius. The method is constructive: the authors implement a Moser-trick deformation between the canonical form on and the pullback on shrinking tubular neighborhoods, with explicit, step-by-step estimates ensuring nondegeneracy and the existence of a global diffeomorphism with and . They develop a robust Sasaki/almost-Kähler framework, derive precise bounds for the scaling map, the time-dependent vector field, and the generated flow, and show that the resulting radius scales as (and in the embedding case), where and aggregate curvature and second fundamental form data. These results provide explicit, geometry-driven control over the Weinstein neighborhood size, enabling rigorous and quantitative applications to Lagrangian deformations via closed 1-forms and related constructions.

Abstract

For an immersed Lagrangian submanifold in a Kähler manifold , there exists a symplectic local diffeomorphism from a tubular neighborhood of the image of the zero section in the normal bundle of , equipped with a canonical symplectic form , to whose restriction to is the identity map by Weinstein's Lagrangian tubular neighborhood theorem, where the image of the zero section in is identified with . In this paper, we give a lower bound for the supremum of the radii of tubular neighborhoods that have such a symplectic diffeomorphism into from below by a constant explicitly given in terms of up to second derivatives of the Riemannian curvature tensor of and the second fundamental form of . We also give a similar lower bound in the case where is compact and embedded.

Paper Structure

This paper contains 21 sections, 35 theorems, 310 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. Rigorous setting and main theorems
  4. Strategy of the proof and outline of the paper
  5. Strategy of the proof
  6. Outline of the paper
  7. Nondegeneracy of the path of symplectic forms
  8. Almost Kähler structure on the normal bundle of L
  9. Levi--Civita connection on the normal bundle of L
  10. Lower bound of the pull-backed symplectic form
  11. Estimates of the second derivative of F
  12. Estimates for the scaling map
  13. Estimates for the time-dependent vector field
  14. Bound for the norm of the vector field
  15. Bound for the norm of the first derivative of the vecotr field
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $L$ be a compact Lagrangian submanifold in a symplectic manifold $(M,\omega)$. Then, there are open neighborhoods $\mathcal{U}$ of $L$ in $T^{\ast}L$, where $L$ is identified with the image of the zero-section of $T^{\ast}L$, $\mathcal{V}$ of $L$ in $M$ and a diffeomorphism $\Theta : \mathcal{U}

Theorems & Definitions (72)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 2.1
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • ...and 62 more