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

TL;DR

This paper establishes explicit lower bounds for the radius of Weinstein's Lagrangian tubular neighborhoods in Kähler manifolds, depending on curvature and second fundamental form, improving understanding of local symplectic geometry around Lagrangian submanifolds.

Contribution

It provides the first explicit quantitative lower bounds for the size of Weinstein's Lagrangian tubular neighborhoods based on geometric data.

Findings

Lower bounds depend on second derivatives of curvature tensor and second fundamental form.

Bounds are explicit and applicable to both immersed and embedded Lagrangian submanifolds.

Results enhance understanding of local symplectic geometry around Lagrangian submanifolds.

Abstract

For an immersed Lagrangian submanifold $L$ in a K\"ahler manifold $(M,\omega)$, 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 $\tilde{\omega}$, to $(M,\omega)$ 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 $(M,\omega)$ 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.