AI 中文总结
该研究构造了黎曼3-球中含4参数的真嵌入零亏格曲面族，作为Marques-Neves典范曲面族的自由边界类比，还应用其得到Almgren-Pitts 4-宽度上界，并证明特定紧致3-流形中至少存在3个自由边界极小环面。
AI 中文摘要
我们在黎曼3-球中构造了一个含4个参数的真嵌入零亏格曲面族，该曲面族至多包含两个边界分支。此构造是Marques-Neves在Willmore猜想证明中得到的3-球上含5个参数的典范曲面族的自由边界类比。在欧氏单位球中，该族将临界悬链面实现为Simon-Smith min-max极限。作为应用，我们给出了Almgren-Pitts 4-宽度ω₄(𝔹³)的上界，其值等于临界悬链面的面积。此外，我们证明了在任意具有非负Ricci曲率且边界严格凸的紧致3-流形中，至少存在3个自由边界极小环面。
英文摘要
We construct a $4$-parameter family of properly embedded genus-zero surfaces with at most two boundary components in a Riemannian $3$-ball. The construction is a free boundary analog of the canonical $5$-parameter family of surfaces on the $3$-sphere by Marques-Neves in the proof of Willmore conjecture. In the Euclidean unit ball, this family realizes the critical catenoid as a Simon-Smith min-max limit. As applications, we give an upper bound of Almgren-Pitts $4$-width $ω_4(\mathbb{B}^3)$ by the area of the critical catenoid. Moreover, we prove the existence of at least three free boundary minimal annuli in any compact $3$-manifold with nonnegative Ricci curvature and strictly convex boundary.
Comments73 pages, 1 figure, minor modifications, comments are welcome!
