Lower semi-continuity of Lagrangian volume

TL;DR

This paper investigates the lower semi-continuity of the volume of monotone Lagrangian submanifolds under Hamiltonian isotopies, establishing conditions under which volume remains lower semi-continuous with respect to Hofer and - distances.

Contribution

It proves volume lower semi-continuity for Lagrangian submanifolds in specific geometric settings, expanding understanding of volume behavior in symplectic topology.

Findings

Volume is -lower semi-continuous when the volume form comes from a Ka4hler metric with many isometries.

Volume is -lower semi-continuous for Lagrangian tori with respect to any compatible metric.

In both cases, volume is Hofer lower semi-continuous.

Abstract

We study lower semi-continuity properties of the volume, i.e., the surface area, of a closed Lagrangian manifold with respect to the Hofer- and $\gamma$-distance on a class of monotone Lagrangian submanifolds Hamiltonian isotopic to each other. We prove that volume is $\gamma$-lower semi-continuous in two cases. In the first one the volume form comes from a K\"ahler metric with a large group of Hamiltonian isometries, but there are no additional constraints on the Lagrangian submanifold. The second one is when the volume is taken with respect to any compatible metric, but the Lagrangian submanifold must be a torus. As a consequence, in both cases, the volume is Hofer lower semi-continuous.