论文标题
最少$ 2 $ -SPHERES和最佳叶子，以$ 3 $ -SPHERES（带有任意度量）
Minimal $2$-Spheres and Optimal Foliations in $3$-Spheres with Arbitrary Metric
论文作者
论文摘要
在本文中，我们证明了$ 3 $ -SPHERE具有任意的Riemannian Metric，要么至少包含两个嵌入式最小$ 2 $ -SPHERES，要么以$ 2 $ -SPHERES的价格承认最佳叶子。这概括了Haslhofer-Ketover（DukeMath。J.，2019年）的最新结果，在此额外假设是，该指标是通用的，在此建立了最佳叶子和最小$ 2 $ -SPHERES的存在。鉴于Wang-Zhou的最新示例，其中一些非笨拙的指标的Min-Max产生了较高的多重性，我们的结果在某种意义上是敏锐的。
In this paper, we prove that the $3$-sphere endowed with an arbitrary Riemannian metric either contains at least two embedded minimal $2$-spheres or admits an optimal foliation by $2$-spheres. This generalizes recent results by Haslhofer-Ketover (Duke Math. J. 2019), where the existence of optimal foliations and minimal $2$-spheres has been established under the additional assumption that the metric is generic. In light of recent examples by Wang-Zhou, where min-max for some non-bumpy metrics on the 3-sphere produces higher multiplicities, our results are in a certain sense sharp.
