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Eventual Monotonicity of the Willmore Energy under Mean Curvature Flow
Abstract
We establish the eventual monotonicity of the Willmore deficit ∫ΣH2 dμ−16π along the mean curvature flow of smooth closed embedded strictly convex surfaces in ℝ3, thereby complementing Huisken's exponential convergence with an eventual monotonicity property. As a consequence, the Hawking mass is eventually nondecreasing along the contracting mean curvature flow, in contrast with the usual inclusion monotonicity expected of the Hawking mass. We also discuss totally umbilical solutions in three-dimensional space forms, where the monotonicity directions of the Willmore energy and the Hawking mass depend on the sign of the ambient sectional curvature.
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BibTeX
@misc{chen2026eventual,
title = {{Eventual Monotonicity of the Willmore Energy under Mean Curvature Flow}},
author = {Yuyue Chen and Shiying Wu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.18378},
eprint = {2609.18378},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.18378},
}
RIS
TY - UNPB
TI - Eventual Monotonicity of the Willmore Energy under Mean Curvature Flow
AU - Yuyue Chen
AU - Shiying Wu
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2609.18378
UR - https://arxiv.org/abs/2609.18378
AB - We establish the eventual monotonicity of the Willmore deficit ∫_ΣH^2 dμ−16π along the mean curvature flow of smooth closed embedded strictly convex surfaces in ℝ^3, thereby complementing Huisken's exponential convergence with an eventual monotonicity property. As a consequence, the Hawking mass is eventually nondecreasing along the contracting mean curvature flow, in contrast with the usual inclusion monotonicity expected of the Hawking mass. We also discuss totally umbilical solutions in three-dimensional space forms, where the monotonicity directions of the Willmore energy and the Hawking mass depend on the sign of the ambient sectional curvature.
ER -