Compactness for holomorphic curves with switching Lagrangian boundary conditions

TL;DR

This paper establishes a compactness theorem for holomorphic curves with boundary on immersed Lagrangians with self-intersections, showing that intersection counts are uniformly bounded by energy, advancing understanding in symplectic geometry.

Contribution

It introduces a compactness result for holomorphic curves with switching boundary conditions on immersed Lagrangians, a novel extension in symplectic topology.

Findings

Holomorphic curves with boundary on immersed Lagrangians have uniformly bounded intersections.

The number of intersections is controlled by the Hofer energy.

The paper provides new tools for studying Lagrangian intersections in symplectic geometry.

Abstract

We prove a compactness result for holomorphic curves with boundary on an immersed Lagrangian submanifold with clean self-intersection. As a consequence, we show that the number of intersections of such holomorphic curves with the self-intersection locus is uniformly bounded in terms of the Hofer energy.