Lagrangian isotopies and symplectic function theory

TL;DR

This paper introduces two new invariants for Lagrangian submanifolds in symplectic manifolds, linking topological and analytical aspects, and provides partial computations for specific Lagrangian tori.

Contribution

It defines and analyzes two related invariants of Lagrangian submanifolds, connecting topological and symplectic function theory, with partial explicit calculations.

Findings

Invariants measure Lagrangian isotopies and flux.

Partial computations for certain Lagrangian tori.

Links between topological and analytical invariants.

Abstract

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.