Lagrangian non-squeezing and a geometric inequality

TL;DR

This paper establishes a geometric inequality linking symplectic embeddings of the unit codisc bundle to bounds on the length of the shortest closed geodesic on a Riemannian manifold, revealing a deep connection between symplectic geometry and Riemannian geometry.

Contribution

It proves a new inequality relating symplectic embedding properties to geodesic lengths, extending non-squeezing principles to a geometric inequality context.

Findings

If the unit codisc bundle symplectically embeds into a cylinder, the shortest closed geodesic length is at most half the area of the unit disc.

The result connects symplectic embedding constraints with Riemannian geometric properties.

Provides a new perspective on the interplay between symplectic topology and Riemannian geometry.

Abstract

We prove that if the unit codisc bundle of a closed Riemannian manifold embeds symplectically into a symplectic cylinder of radius one then the length of the shortest nontrivial closed geodesic is at most half the area of the unit disc.