Hausdorff limits of submanifolds of symplectic and contact manifolds

TL;DR

This paper investigates the limits of sequences of submanifolds in symplectic and contact manifolds, revealing new rigidity results and metric versions of key conjectures in symplectic topology.

Contribution

It introduces new methods to analyze the Hausdorff limits of submanifolds respecting Riemannian bounds, leading to proofs of metric versions of the nearby Lagrangian and Viterbo conjectures.

Findings

Proofs of metric versions of the nearby Lagrangian conjecture

Proofs of the Viterbo conjecture on the spectral norm

C^0-rigidity results for submanifolds in 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 for a vast class of important submanifolds of symplectic and contact manifolds in the presence of Riemannian bounds. Likewise, we get a Lagrangian generalization of results of Hofer and Viterbo on simultaneous $C^0$ and Hofer/spectral limits~ -- ~even without any such bounds.