Convergence and Riemannian bounds on Lagrangian submanifolds

TL;DR

This paper establishes conditions under which collections of Lagrangian submanifolds in symplectic manifolds converge in the Hausdorff sense, using various metrics including Hofer and spectral norms, with implications for symplectic topology.

Contribution

It introduces a framework linking convergence in specific metrics to Hausdorff convergence for Lagrangian submanifolds, expanding understanding of metric structures in symplectic geometry.

Findings

Convergence in certain metrics implies Hausdorff convergence for Lagrangian submanifolds.

Includes metrics like Lagrangian Hofer, spectral norm, and shadow metrics.

Uses a monotonicity lemma on metric balls to prove results.

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 norm and the shadow metrics introduced by Biran, Cornea and Shelukhin arXiv:1806.06630. The proof relies on a version of the monotonicity lemma, applied on a carefully-chosen metric ball.