发表机构
Justus-Liebig-Universität Gießen; Technion–Israel Institute of Technology（吉森大学; 以色列理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究在一致定量非脐性等条件下，证明了三维球面中莫比乌斯不变威尔莫尔流有限时间变分曲面极限的唯一性，还将结果推广至无穷时间情形，推进了几何分析中相关流的收敛性研究。
AI 中文摘要
我们证明了在一致定量非脐性条件下，$\u2118^3$中Möbius-invariant Willmore流（莫比乌斯不变威尔莫尔流）有限时间几何端点的唯一性。计重数的varifold（变分曲面）无需重新参数化或Möbius重整化，即可收敛到唯一的整二维varifold。一个内蕴输运估计给出了固定域上诱导面积测度的定量全变差收敛性，以及其推前映射的有界Lipschitz柯西控制。结合Allard紧性与可求长性，这将子序列紧性提升为全轨迹varifold收敛。极限具有$L^2$中的广义欧氏平均曲率，且满足自然的端低下半连续界。对于初始能量至多$8π$的有限最大轨迹，Jakob的子序列二择性变为与序列无关：极限为零，或具有单位密度且亏格为0或1的嵌入Lipschitz支集。对于相同能量界下的Hopf-torus（霍普夫环面）轨迹，非脐性自动满足，且锚定的等速剖面在$W^{2,2}$中弱收敛，在$W^{1,2}$和所有$α<\frac{1}{2}$的$C^{1,α}$中强收敛。在无穷时间下，附加有限耗散长度条件时，相同方法可得到唯一极限。
英文摘要
We prove uniqueness of finite-time geometric endpoints for the Möbius-invariant Willmore flow in $\mathbb{S}^3$ under uniform quantitative nonumbilicity. The multiplicity-counting varifolds converge, without reparametrization or Möbius renormalization, to a unique integral two-varifold. An intrinsic transport estimate gives quantitative total-variation convergence of the induced area measures on the fixed domain and bounded-Lipschitz Cauchy control of their pushforwards. Together with Allard compactness and rectifiability, this upgrades subsequential compactness to full-trajectory varifold convergence. The limit has generalized Euclidean mean curvature in $L^2$ with the natural endpoint lower-semicontinuity bound. For finite maximal trajectories with initial energy at most $8π$, Jakob's subsequential alternative becomes sequence independent: the limit is zero, or it has unit density and embedded Lipschitz support of genus zero or one. For Hopf-torus trajectories under the same energy bound, nonumbilicity is automatic and the anchored constant-speed profiles converge weakly in $W^{2,2}$ and strongly in $W^{1,2}$ and $C^{1,α}$ for every $α<\frac{1}{2}$. At infinite time, the same method yields a unique limit under an additional finite-dissipation-length condition.
