关于4维空间中接触类型超曲面的研究
On contact type hypersurfaces in 4-space
AI总结：
本文研究了标准辛R⁴中接触类型超曲面的3维闭流形拓扑限制，利用Heegaard Floer同调证明Brieskorn齐次球面无法嵌入R⁴，并给出了无法成为辛凸域边界的接触3流形实例。
摘要：
我们考虑标准辛R⁴中接触类型超曲面所限制的3维闭流形的拓扑特性。利用来自Heegaard Floer同调的障碍，我们证明没有Brieskorn齐次球面可以嵌入R⁴，这一结果对Gompf和Kollár的猜想有影响。这特别意味着没有有理凸域在C²中的边界能同胚于Brieskorn球面。我们还给出了无限多个既能被Stein域所边界又不能被辛凸域所边界的接触3流形实例；特别是我们找到了在C²中无法保持其边界接触结构而成为Weinstein域的Stein域。
英文摘要：
We consider constraints on the topology of closed 3-manifolds that can arise as hypersurfaces of contact type in standard symplectic $R^4$. Using an obstruction derived from Heegaard Floer homology we prove that no Brieskorn homology sphere admits a contact type embedding in $R^4$, a result that has bearing on conjectures of Gompf and Kollár. This implies in particular that no rationally convex domain in $C^2$ has boundary diffeomorphic to a Brieskorn sphere. We also give infinitely many examples of contact 3-manifolds that bound Stein domains but not symplectically convex ones; in particular we find Stein domains in $C^2$ that cannot be made Weinstein with respect to the ambient symplectic structure while preserving the contact structure on their boundaries.
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