单位球内具有Legendrian毛细管边界的紧致拉格朗日自相似子流形的分类
Classification of compact Lagrangian self-similar submanifolds with Legendrian capillary boundary in the unit ball
AI总结：
该论文分类单位球内满足H+εX^⊥=0且边界为Legendrian子流形的紧致拉格朗日自相似子流形，证明其边界连通性结论并给出不同复维下的具体解族。
AI中文摘要：
我们对复欧氏空间C^n（n≥2）的闭单位球内满足H+εX^⊥=0（ε∈{-1,0,1}）、边界为单位球面上的Legendrian子流形且各连通分支具有常接触角的光滑紧致连通拉格朗日浸入X进行分类。我们证明该边界至多有两个连通分支：当边界连通时，X是赤道拉格朗日n-圆盘的微分同胚；当边界有两个连通分支时，X可全局分解为X(s,p)=γ(s)ψ(p)，其中ψ是单位球内的紧致极小Legendrian浸入，γ是具有唯一径向极小值的Anciaux轮廓，两个接触角互补。在复二维情形下，所有非圆盘解为ε=0时拉格朗日悬链线段的有限覆盖，或ε=±1时旋转Anciaux环面的有限覆盖；在更高复维情形下，迭代Calabi悬链构造出极小Legendrian链具有非平凡拓扑的族。
英文摘要：
We classify smooth compact connected Lagrangian immersions $X$ in the closed unit ball of $\C^n$, $n\ge2$, satisfying $H+\varepsilon X^\perp=0$, $\varepsilon\in\{-1,0,1\}$, with Legendrian boundary on the unit sphere and constant contact angle on each connected component. We prove that the boundary has at most two connected components. When the boundary is connected, $X$ is a diffeomorphism onto an equatorial Lagrangian $n$-disk. When the boundary has two components, $X$ splits globally as $X(s,p)=γ(s)ψ(p)$, where $ψ$ is a compact minimal Legendrian immersion in the unit sphere and $γ$ is an Anciaux profile with a unique radial minimum. The two contact angles are supplementary. In complex dimension two, every non-disk solution is a finite cover of a Lagrangian catenoid segment for $\varepsilon=0$ or of a rotational Anciaux annulus for $\varepsilon=\pm1$. In higher complex dimensions, iterated Calabi suspensions produce families whose minimal Legendrian links have nontrivial topology.
