Uniqueness of Finite-Time Varifold Limits for the Möbius-Invariant Willmore Flow
Abstract
We prove uniqueness of finite-time geometric endpoints for the Möbius-invariant Willmore flow in 𝕊3 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 L2 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 W2,2 and strongly in W1,2 and C1,α for every α<1/2. At infinite time, the same method yields a unique limit under an additional finite-dissipation-length condition.
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BibTeX
@misc{ahmedou2026uniqueness,
title = {{Uniqueness of Finite-Time Varifold Limits for the Möbius-Invariant Willmore Flow}},
author = {Mohameden Ahmedou and Ruben Jakob},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.00501},
eprint = {2609.00501},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.00501},
}
RIS
TY - UNPB
TI - Uniqueness of Finite-Time Varifold Limits for the Möbius-Invariant Willmore Flow
AU - Mohameden Ahmedou
AU - Ruben Jakob
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.00501
UR - https://arxiv.org/abs/2609.00501
AB - We prove uniqueness of finite-time geometric endpoints for the Möbius-invariant Willmore flow in 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<1/2. At infinite time, the same method yields a unique limit under an additional finite-dissipation-length condition.
ER -