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 norm and the shadow metrics introduced by Biran, Cornea and Shelukhin [BCS18]. The proof relies on a version of the monotonicity lemma, applied on a carefully-chosen metric ball.

Hausdorff limits of submanifolds of symplectic and contact manifolds

“…Proof: By the author's previous work [Cha21], every dF,F ′ -converging sequence in L e Λ,ε (M ) also converges in the Hausdorff metric to the same limit. If we are given a volume bound V > 0, then every Hausdorff-converging sequence in the associated subset of L e Λ,ε (M ) also converges in d H to the same limit by Theorem 2.…”

“…Indeed, the main motivation behind the study of Hausdorff-converging sequences is its importance when studying sequences converging in metrics coming from symplectic topology (c.f. [Cha21]). Therefore, there is another rigidity question that crops up: does Theorem A of [Cha21] holds for non-Lagrangian submanifolds.…”

“…[Cha21]). Therefore, there is another rigidity question that crops up: does Theorem A of [Cha21] holds for non-Lagrangian submanifolds. Of course, such a question makes no sense for most metrics coming from symplectic topology.…”

“…In this subsection, we explore the relation between the ε-tameness condition of the author's previous work [Cha21] and the condition of having bounded volume.…”

“…This exploration was started in the author's previous work [Cha21]. As noted in said work, there is however an obvious problem with introducing the Hausdorff metric: in full generality, there is no relation between the Hausdorff metric and the metrics coming from symplectic topology.…”