Relative Thom Conjectures, symplectic and beyond

TL;DR

This paper proves a criterion for when almost complex curves minimize genus in 4-manifolds, confirming the relative symplectic Thom conjecture and linking knot Floer homology obstructions to symplectic surface bounding.

Contribution

It introduces a new criterion for genus minimization of almost complex curves, confirming the relative symplectic Thom conjecture and connecting knot Floer homology to symplectic bounding obstructions.

Findings

Almost complex curves minimize genus under the established criterion.

The relative symplectic Thom conjecture is affirmed.

Knot Floer homology provides obstructions to symplectic surface bounding.

Abstract

We establish a criterion that ensures a bounded almost complex curve in a bounded almost complex 4-manifold minimizes genus amongst all smooth surfaces that share its homology class and the transverse link on its boundary. An immediate corollary affirms the relative symplectic Thom conjecture and, moreover, yields obstructions coming from knot Floer homology to a link bounding a symplectic surface in a symplectic filling. Our results are applicable to knots in manifolds equipped with plane fields that admit no symplectic fillings; for instance, we show that symplectic surfaces in a thickening of any contact 3-manifold with non-zero Ozsvath-Szabo invariant minimize slice genus for their boundary. We conjecture that this phenomenon occurs precisely when the contact structure is tight, which would imply that tightness can be viewed as a symplecto-geometric notion.