On certain Lagrangian submanifolds of S2 ×S2 and ℂPn

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)

Joel Oakley and Michael Usher

Abstract

Keywords

Mathematical Subject Classification 2010

References

Publication

Authors