In this article, the author investigates flow lines of the classical Willmore flow, which start to move in a smooth parametrization of a Hopf-torus in \(\textbf{S}^3\). We prove that any such flow line of the Willmore flow exists globally, in particular does not develop any singularities, and subconverges to some smooth Willmore-Hopf-torus in every \(C^{m}\)-norm. Moreover, if in addition the Willmore energy of the initial immersion \(F_0\) is required to be smaller than or equal to the threshold \(\frac{8\pi ^2}{\sqrt{2}}\), then the unique flow line of the Willmore flow, starting to move in \(F_0\), converges fully to a conformally transformed Clifford torus in every \(C^{m}\)-norm, up to time dependent, smooth reparametrizations. Key instruments for the proofs are the equivariance of the Hopf-fibration \(\pi :\textbf{S}^3 \longrightarrow \textbf{S}^2\) w.r.t. the effect of the \(L^2\)-gradient of the Willmore energy applied to smooth Hopf-tori in \(\textbf{S}^3\) and to smooth closed regular curves in \(\textbf{S}^2\), a particular version of the Lojasiewicz–Simon gradient inequality, and a well-known classification and description of smooth, arc-length parametrized solutions of the Euler–Lagrange equation of the elastic energy functional in terms of Jacobi elliptic functions and elliptic integrals, dating back to the 80s.

在本文中，作者研究了经典 Willmore 流的流线，该流线开始在\(\textbf{S}^3\)中的 Hopf 环面的平滑参数化中移动。我们证明任何这样的 Willmore 流的流线都全局存在，特别是不会产生任何奇点，并且在每个\(C^{m}\) -范数中子收敛到一些平滑的 Willmore-Hopf-torus 。此外，如果另外要求初始浸入的威尔莫尔能量\(F_0\)小于或等于阈值\(\frac{8\pi ^2}{\sqrt{2}}\)，则威尔莫尔流的独特流线，开始在\(F_0\)中移动，在每个\(C^{m}\) -范数中完全收敛到共形变换的克利福德环面，直到时间相关的、平滑的重新参数化。证明的关键工具是 Hopf 纤维化的等方差\(\pi :\textbf{S}^3 \longrightarrow \textbf{S}^2\)与\(L^2\)梯度的影响应用于平滑\(\textbf{S}^3\)中的 Hopf-tori和平滑\(\textbf{S}^2\)中的闭合正则曲线的 Willmore 能量，这是 Lojasiewicz-Simon 梯度的特定版本不等式，以及弹性能量泛函欧拉-拉格朗日方程的平滑、弧长参数化解的众所周知的分类和描述，用雅可比椭圆函数和椭圆积分表示，其历史可以追溯到 80 年代。