Legendrian non-squeezing via microsheaves

TL;DR

This paper demonstrates that certain Legendrian submanifolds in pre-quantizations exhibit rigidity preventing their squeezing into smaller regions, extending known results from dimension 3 to higher dimensions using microsheaf categories.

Contribution

It generalizes Legendrian non-squeezing results to higher dimensions via microsheaves, providing a new proof method and broadening the understanding of Legendrian isotopy rigidity.

Findings

Legendrian lifts cannot be squeezed into arbitrarily small cylinders.

Normal rulings provide a new proof of non-squeezing in dimension 3.

Microsheaves are used to establish rigidity in high dimensions.

Abstract

We show that Legendrian pre-quantization lifts of many non-exact Lagrangian submanifolds in $\mathbb{C}^n$ retain some quantitative rigidity from the symplectic base. In particular, they cannot be moved by Legendrian isotopy into an arbitrarily small pre-quantized cylinder. This is a high dimensional generalization of results of Dimitroglou Rizell--Sullivan in dimension 3. In this setting, we give a new proof of non-squeezing using normal rulings, and in high dimension, we obtain our results using a category of (micro)sheaves associated to a Legendrian submanifold of pre-quantizations.