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

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.