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Bounding Lagrangian widths via geodesic paths

Part of: Symplectic geometry, contact geometry

Published online by Cambridge University Press:  17 September 2014

Matthew Strom Borman  and
Mark McLean
Matthew Strom Borman
Affiliation:
Department of Mathematics, Stanford University, Stanford, CA 94305, USA email borman@math.uchicago.edu
Mark McLean
Affiliation:
Institute of Mathematics, University of Aberdeen, Aberdeen AB24 3FX, UK email mmclean@abdn.ac.uk
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Abstract

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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.

Keywords

symplectic manifoldLagrangian manifoldLagrangian widthLagrangian packingsymplectic capacitysymplectic cohomologywrapped Floer cohomologygeodesic flow

MSC classification

Primary: 53D12: Lagrangian submanifolds; Maslov index 53D25: Geodesic flows 53D40: Floer homology and cohomology, symplectic aspects
Type
Research Article
Information
Compositio Mathematica , Volume 150 , Issue 12 , December 2014 , pp. 2143 - 2183
Copyright
© The Author(s) 2014 

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