相对汤姆猜想，辛结构与更广泛领域
Relative Thom Conjectures, symplectic and beyond
AI总结：
该研究通过建立准则证明辛表面在特定条件下最小化亏格，并探讨了接触结构紧致性与辛几何的关系。
摘要：
我们建立了一个准则，确保在有界几乎复4流形中，有界几乎复曲线在所有共享其同调类的光滑曲面中最小化亏格。一个直接推论证实了相对辛汤姆猜想，并且进一步得出来自结 Floer 同调的障碍，以限制一个辛表面在辛填充中的边界。我们的结果适用于配备平面场的结，这些平面场没有辛填充；例如，我们证明了在任何具有非零 Ozsvath-Szabo 不变的接触3流形的增厚中，辛表面对于其边界的切亏格最小化。我们猜测这种现象恰好发生在接触结构为紧致时，这将意味着紧致性可以被视为辛几何的概念。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
