Work

Relative Thom conjectures, symplectic and beyond

Year: 2026

Type: article

Abstract: We establish a criterion that ensures a bounded almost complex curve in a bounded almost complex 4-manifold minimizes genus among 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 nonzero Ozsváth-Szabó 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.

Source: Proceedings of the National Academy of Sciences

DOI: 10.1073/pnas.2536898123

Authors: Matthew Hedden, Katherine Raoux

Institutions: Michigan State University, University of Arkansas at Fayetteville

Language: English

Topic: Geometric and Algebraic Topology

Subfield: Geometry and Topology

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Funders: National Science Foundation, Simons Foundation

Awards: A Heegaard Floer theoretic approach to 4-dimensional genus questions, G5584918989, Topology and Geometry at the Interface of Dimensions 3 and 4