Hausdorff limits of submanifolds of symplectic and contact manifolds

Abstract

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.