3-球面上Hopf环的Willmore流
The Willmore flow of Hopf-tori in the $3$-sphere
AI总结：
本文研究了3球面上Hopf环的Willmore流，证明了流线在全局范围内存在并收敛到共形变换的Clifford环。
摘要：
本文作者研究了经典Willmore流的流线，这些流线从一个光滑参数化的Hopf环在3球面中的初始浸入开始移动。证明了任何这样的Willmore流线在全局范围内存在，特别是不会产生奇点，并且在每种C^m范数下都收敛到某个光滑的Willmore-Hopf环。此外，如果初始浸入F_0的Willmore能量小于或等于阈值8π²/√2，则Willmore流的唯一流线在每种C^m范数下，经过时间依赖的光滑重参数化后完全收敛到一个共形变换的Clifford环。证明的关键工具包括Hopf纤维π:S³→S²相对于L²梯度作用于光滑Hopf环和光滑闭合正则曲线的等变性，Lojasiewicz-Simon梯度不等式的一个特定版本，以及关于弹性能量泛函的Euler-Lagrange方程的光滑、弧长参数化解的分类和描述，以Jacobi椭圆函数和椭圆积分形式表达，追溯到80年代。
英文摘要：
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 $\mathbb{S}^3$. 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 $\frac{8π^2}{\sqrt{2}}$, 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 $π:\mathbb{S}^3 \longrightarrow \mathbb{S}^2$ w.r.t. the effect of the $L^2$-gradient of the Willmore energy applied to smooth Hopf-tori in $\mathbb{S}^3$ and to smooth closed regular curves in $\mathbb{S}^2$, 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.
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