$\mathcal{C}^0$-rigidity of Lagrangian submanifolds and punctured holomorphic discs in the cotangent bundle

TL;DR

This paper proves the $$-rigidity of the area spectrum and Maslov class for Lagrangian submanifolds using punctured pseudoholomorphic discs in cotangent bundles, revealing new symplectic geometric properties.

Contribution

It establishes the $$-rigidity of key invariants of Lagrangian submanifolds via the existence of punctured pseudoholomorphic discs with boundary on the zero section.

Findings

Rigidity of the area spectrum for Lagrangian submanifolds

Rigidity of the Maslov class for Lagrangian submanifolds

Applications of punctured discs in symplectic geometry

Abstract

Our main result is the $\mathcal{C}^0$-rigidity of the area spectrum and the Maslov class of Lagrangian submanifolds. This relies on the existence of punctured pseudoholomorphic discs in cotangent bundles with boundary on the zero section, whose boundaries represent any integral homology class. We discuss further applications of these punctured discs in symplectic geometry.