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.

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.

We study sequences of immersions respecting bounds coming from Riemannian geometry and apply the ensuing results to the study of sequences of submanifolds of symplectic and contact manifolds. This allows us to study the subtle interaction between the Hausdorff metric and the Lagrangian Hofer and spectral metrics. In the process, we get proofs of metric versions of the nearby Lagrangian conjecture and of the Viterbo conjecture on the spectral norm. We also get C 0 -rigidity results in the vein of the Gromov-Eliashberg theorem for a vast class of important submanifolds of symplectic and contact manifoldseven when no Riemannian bounds are present.

In this paper, we introduce coisotropic characteristic classes in order to study coisotropic immersions in C n , and prove that they actually are the isotropic classes introduced by Lalonde [7]. We conclude with some remarks on the differences between the h-principles for coisotropic immersions, and the one for isotropic immersions that the coisotropic classes fail to capture. Résumé Dans cet article, nous introduisons des classes caractéristiques coisotropes afin d'étudier les immersions coisotropes dans C n et démontrons qu'elles sont en réalité les classes isotropes introduites par Lalonde [7]. Nous concluons par une discussion sur les différences entre le h-principe pour les immersions isotropes et celui pour les immersions coisotropes que les classes coisotropes n'arrivent pasà détecter.

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