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
本文通过将三维球面中 Hopf 环面的经典 Willmore 流精确约化为球面弹性流，结合已知收敛定理，证明了任意初始能量下流线的完全光滑收敛，并给出了未重参数化轨迹的定量收敛结果，包括端点密度与覆盖重数的关系。
AI 中文摘要
我们在三维球面中建立了经典 Willmore 流对 Hopf 环面到球面弹性流的精确参数化约化。在 Willmore 流时间中，约化为 $(\partial_t\gamma)^{\perp}=-4\nabla_{L^2}\mathcal E(\gamma)$；其全局重构使用拉回圆丛归一化，并且不需要沿流线保持简单性。将已知的球面弹性流的全局子收敛与 Pozzetta 的完全收敛定理相结合，则对任意初始能量下的每个简单参数化 Hopf 环面，在模域微分同胚下，流线完全光滑收敛。然后我们获得未重参数化轨迹的定量结论：有限的法向 $L^2$ 度量长度、拉回面积测度的全变差收敛、图像测度的有界 Lipschitz 收敛以及显式尾部估计。完整 varifold 端点的密度等于极限弹性轮廓的覆盖重数，并且 Pinkall 的标记平坦格收敛。特别地，一个能量从上方任意接近 $4\pi^2$ 的简单初始 Hopf 浸入可以光滑收敛到 Clifford 环面的某个重数为二的重参数化，尽管其在模空间中的极限共形类恰好是平方类。
英文摘要
We establish an exact parametrized reduction of the classical Willmore flow of Hopf tori in the round three-sphere to the spherical elastic flow. In Willmore-flow time the reduction is $(\partial_tγ)^{\perp}=-4\nabla_{L^2}\mathcal E(γ)$; its global reconstruction uses a pullback-circle-bundle normalization and does not require preservation of simplicity along flow lines. Combining the known global subconvergence of the spherical elastic flow with Pozzetta's full convergence theorem then yields full smooth convergence of flow lines, modulo domain diffeomorphisms, for every simply parametrized Hopf torus at arbitrary initial energy. We then obtain quantitative conclusions for the unreparametrized trajectory: finite normal $L^2$-metric length, total-variation convergence of the pullback area measures, bounded-Lipschitz convergence of the image measures, and explicit tail estimates. The density of the full varifold endpoint equals the covering multiplicity of the limiting elastic profile, and Pinkall's marked flat lattices converge. In particular, a simple initial Hopf immersion with energy arbitrarily close to $4π^2$ from above can smoothly converge to some parametrization of the Clifford torus with multiplicity two, although its limiting conformal class in moduli space is exactly the square class.
