On certain Lagrangian submanifolds of S2 ×S2 and ℂPn

Oakley, Joel; Usher, Michael

doi:10.2140/agt.2016.16.149

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On certain Lagrangian submanifolds of $S^2\times S^2$ and $\mathbb{C}\mathrm{P}^n$

Joel Oakley and Michael Usher

Algebraic & Geometric Topology 16 (2016) 149–209

DOI: 10.2140/agt.2016.16.149

Abstract

We consider various constructions of monotone Lagrangian submanifolds of $ℂ P^{n}$ , $S^{2} \times S^{2}$ , and quadric hypersurfaces of $ℂ P^{n}$ . In $S^{2} \times S^{2}$ and $ℂ P^{2}$ we show that several different known constructions of exotic monotone tori yield results that are Hamiltonian isotopic to each other, in particular answering a question of Wu by showing that the monotone fiber of a toric degeneration model of $ℂ P^{2}$ is Hamiltonian isotopic to the Chekanov torus. Generalizing our constructions to higher dimensions leads us to consider monotone Lagrangian submanifolds (typically not tori) of quadrics and of $ℂ P^{n}$ which can be understood either in terms of the geodesic flow on $T^{*} S^{n}$ or in terms of the Biran circle bundle construction. Unlike previously known monotone Lagrangian submanifolds of closed simply connected symplectic manifolds, many of our higher-dimensional Lagrangian submanifolds are provably displaceable.

Keywords

Lagrangian submanifolds, Hamiltonian displaceability

Mathematical Subject Classification 2010

Primary: 53D12

References

Publication

Received: 30 April 2014

Revised: 13 March 2015

Accepted: 15 April 2015

Published: 23 February 2016

Authors

Joel Oakley
Department of Mathematics Belhaven University Jackson, MS 39202 USA

Michael Usher
Department of Mathematics University of Georgia Athens, GA 30602 USA

Volume 16, issue 1 (2016)