AI 中文总结
本研究针对闭定向3-流形的叶状结构实现问题，利用Gabai的缝合流形理论，提出从任意亏格Heegaard图构造叶状结构的三步方法，证明了经向缝合把手体的分解性质，完成了适用于任意亏格的叶状结构构造。
AI 中文摘要
每个闭定向3-流形都同时具有Heegaard分裂和可定向余维1叶状结构。我们研究叶状结构实现问题：直接从任意亏格的Heegaard图构造叶状结构。利用Gabai的缝合流形理论，我们引入经向缝合把手体的概念，并证明这类把手体可通过基本缝合分解运算分解为一个Reeb分支和若干乘积圆盘。我们提出三步构造：(i)从给定Heegaard图构造两个经向缝合把手体；(ii)通过反向圆盘分解沿相容圆盘区域将二者粘合；(iii)用一个经向缝合把手体和若干Reeb分支填充剩余空腔。该构造适用于任意亏格的Heegaard图。
英文摘要
Every closed orientable 3-manifold admits both a Heegaard splitting and a coorientable codimension-one foliation. We address the foliation realization problem: constructing a foliation directly from a Heegaard diagram of arbitrary genus. Using Gabai's sutured manifold theory, we introduce the notion of a meridional sutured handlebody and prove that every such handlebody decomposes, via basic sutured decomposition operations, into a Reeb component together with product disks. We then present a three-step construction: (i) building two meridional sutured handlebodies from a given Heegaard diagram, (ii) gluing them along compatible disk regions by reversing disk decompositions, and (iii) filling the remaining cavities with a meridional sutured handlebody and some Reeb components. The construction applies to Heegaard diagrams of any genus.
Comments15 page, 16 figures
