Hopf Reduction and Multiplicity-Resolved Endpoints for the Classical Willmore Flow of Hopf Tori
Abstract
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 (∂tγ)⊥=−4∇L2ℰ(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 L2-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 4π2 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.
Related
- The Willmore Energy Landscape of Spheres and Avoidable Singularities of the Willmore Flow
- Willmore energy and minimal fillings
- Eventual Monotonicity of the Willmore Energy under Mean Curvature Flow
- A proof of the Willmore-type conjecture in ℂP2
- A new localization approach for Willmore flows: Asymptotic analysis for the Dirichlet problem
BibTeX
@misc{ahmedou2026hopf,
title = {{Hopf Reduction and Multiplicity-Resolved Endpoints for the Classical Willmore Flow of Hopf Tori}},
author = {Mohameden Ahmedou and Ruben Jakob},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.26997},
eprint = {2609.26997},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.26997},
}
RIS
TY - UNPB
TI - Hopf Reduction and Multiplicity-Resolved Endpoints for the Classical Willmore Flow of Hopf Tori
AU - Mohameden Ahmedou
AU - Ruben Jakob
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.26997
UR - https://arxiv.org/abs/2609.26997
AB - 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 (∂_tγ)^⊥=−4∇_L^2ℰ(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.
ER -