Convergence and Riemannian bounds on Lagrangian submanifolds

Abstract

We consider collections of Lagrangian submanifolds of a given symplectic manifold which respect uniform bounds of curvature type coming from an auxiliary Riemannian metric. We prove that, for a large class of metrics on these collections, convergence to an embedded Lagrangian submanifold implies convergence to it in the Hausdorff metric. This class of metrics includes well-known metrics such as the Lagrangian Hofer metric, the spectral metric and the shadow metrics introduced by Biran et al. [Lagrangian shadows and triangulated categories, Astérisque 426 (2021) 1–128]. The proof relies on a version of the monotonicity lemma, applied on a carefully-chosen metric ball.