Commentarii Mathematici Helvetici

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Volume 93, Issue 4, 2018, pp. 829–882
DOI: 10.4171/CMH/451

Published online: 2018-11-20

Lagrangian isotopies and symplectic function theory

Michael Entov[1], Yaniv Ganor[2] and Cedric Membrez[3]

(1) Technion - Israel Institute of Technology, Haifa, Israel
(2) Tel Aviv University, Israel
(3) Tel Aviv University, Israel

We study two related invariants of Lagrangian submanifolds in symplectic manifolds. For a Lagrangian torus these invariants are functions on the first cohomology of the torus.

The first invariant is of topological nature and is related to the study of Lagrangian isotopies with a given Lagrangian flux. More specifically, it measures the length of straight paths in the first cohomology that can be realized as the Lagrangian flux of a Lagrangian isotopy.

The second invariant is of analytical nature and comes from symplectic function theory. It is defined for Lagrangian submanifolds admitting fibrations over a circle and has a dynamical interpretation.

We partially compute these invariants for certain Lagrangian tori.

Keywords: Symplectic manifold, Lagrangian submanifold, Lagrangian isotopy, Poisson bracket, symplectic function theory

Entov Michael, Ganor Yaniv, Membrez Cedric: Lagrangian isotopies and symplectic function theory. Comment. Math. Helv. 93 (2018), 829-882. doi: 10.4171/CMH/451