Bounding Lagrangian widths via geodesic paths

TL;DR

This paper introduces a new method to bound the width of Lagrangians using Floer cohomology and geodesic paths, establishing an energy-capacity inequality and applying it to non-positively curved Lagrangians.

Contribution

It develops a wrapped Floer capacity based on geodesic paths to bound Lagrangian widths, linking Floer theory with geometric constraints.

Findings

Width of Lagrangian bounded by four times its displacement energy.

Established energy-capacity inequality for Lagrangians in Liouville manifolds.

Applied method to Lagrangians with non-positive sectional curvature.

Abstract

The width of a Lagrangian is the largest capacity of a ball that can be symplectically embedded into the ambient manifold such that the ball intersects the Lagrangian exactly along the real part of the ball. Due to Dimitroglou Rizell, finite width is an obstruction to a Lagrangian admitting an exact Lagrangian cap in the sense of Eliashberg-Murphy. In this paper we introduce a new method for bounding the width of a Lagrangian Q by considering the Lagrangian Floer cohomology of an auxiliary Lagrangian L with respect to a Hamiltonian whose chords correspond to geodesic paths in Q. This is formalized as a wrapped version of the Floer-Hofer-Wysocki capacity and we establish an associated energy-capacity inequality with the help of a closed-open map. For any orientable Lagrangian Q admitting a metric of non-positive sectional curvature in a Liouville manifold, we show the width of Q is bounded above by four times its displacement energy.