### 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.

Original language | English (US) |
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Pages (from-to) | 2143-2183 |

Number of pages | 41 |

Journal | Compositio Mathematica |

Volume | 150 |

Issue number | 12 |

DOIs | |

Publication status | Published - Dec 19 2014 |

Externally published | Yes |

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### Keywords

- Lagrangian manifold
- Lagrangian packing
- Lagrangian width
- geodesic flow
- symplectic capacity
- symplectic cohomology
- symplectic manifold
- wrapped Floer cohomology

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

*Compositio Mathematica*,

*150*(12), 2143-2183. https://doi.org/10.1112/S0010437X14007465