Target-local Gromov compactness

TL;DR

This paper establishes a localized version of Gromov's compactness theorem for pseudo-holomorphic curves, accommodating sequences with unbounded boundary components and degenerating targets by treating curves as submanifolds.

Contribution

It introduces a novel local compactness result for pseudo-holomorphic curves applicable to complex degenerations and unbounded boundary components, using minimal surface techniques.

Findings

Proves local Gromov compactness in target symplectic manifolds.

Handles sequences with unbounded free boundary components.

Applicable to degenerating target manifolds with unbounded geometry.

Abstract

We prove a version of Gromov's compactness theorem for pseudo-holomorphic curves which holds locally in the target symplectic manifold. This result applies to sequences of curves with an unbounded number of free boundary components, and in families of degenerating target manifolds which have unbounded geometry (e.g. no uniform energy threshold). Core elements of the proof regard curves as submanifolds (rather than maps) and then adapt methods from the theory of minimal surfaces.