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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

We consider various constructions of monotone Lagrangian submanifolds of Pn, S2 × S2, and quadric hypersurfaces of Pn. In S2 × S2 and P2 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 P2 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 Pn which can be understood either in terms of the geodesic flow on TSn 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.

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Lagrangian submanifolds, Hamiltonian displaceability
Mathematical Subject Classification 2010
Primary: 53D12
Received: 30 April 2014
Revised: 13 March 2015
Accepted: 15 April 2015
Published: 23 February 2016
Joel Oakley
Department of Mathematics
Belhaven University
Jackson, MS 39202
Michael Usher
Department of Mathematics
University of Georgia
Athens, GA 30602