AI 中文总结
本文证明辛4流形上满足同调条件的稳定超曲面可被闭特征等分布的超曲面C∞逼近，结合KAM理论与定量闭合引理，并推广到三维流形上的体积保持向量场。
AI 中文摘要
我们证明，辛 $4$-流形的任何稳定超曲面，若辛形式的同调类限制为有理类的倍数，则可以被 $C^\infty$-逼近为（可能不稳定的）超曲面，其闭特征等分布。该同调条件因 Herman 的著名例子而是必要的。证明将 KAM 理论与 Reeb 流及保面积映射的近期定量闭合引理相结合。作为进一步应用，我们证明闭三维流形上的任何可测度体积保持向量场都可以被具有等分布周期轨道的体积保持向量场 $C^\infty$-逼近。
英文摘要
We show that any stable hypersurface of a symplectic $4$-manifold, on which the cohomology class of the symplectic form restricts to a multiple of a rational class, can be $C^\infty$-approximated by (possibly unstable) hypersurfaces whose closed characteristics equidistribute. The cohomological condition is necessary due to a famous example of Herman. The proof combines KAM theory with recent quantitative closing lemmas for Reeb flows and area-preserving maps. As a further application, we prove that every geodesible volume-preserving vector field on a closed three-manifold can be $C^\infty$-approximated by volume-preserving vector fields with equidistributed periodic orbits.
Comments40 pages, comments welcome
