Entropy and round-point convergence for the σk-curvature flow
Abstract
We prove that every smooth closed strictly convex hypersurface in ℝn+1 contracts to a round point under the σk-curvature flow for 2≤ k≤ n−1, without additional pinching or symmetry assumptions. After rescaling about the extinction point to mean width two, the hypersurfaces converge smoothly and exponentially in normalized time to the unit sphere. The proof is based on an entropy which is invariant under translations and dilations and admits a representation by weighted boundary areas of orthogonal projections. A strengthened Codazzi inequality yields its monotonicity. An upper bound for the entropy controls the inradius relative to the mean width, while its dissipation controls the spatial variation of the normalized speed. We also obtain two-sided curvature bounds at the natural extinction scale.
Related
- Convergence of a vector-valued Allen-Cahn system to Brakke's multiphase mean curvature flow
- Convergence of isoparametric BGN methods for mean curvature flow of closed surfaces with high-order finite elements
- Dynamic Ritz projection of mean curvature flow and optimal bf L2 convergence of parametric FEM
- Convergence of the original BGN method for mean curvature flow
- Convergence of cohomogeneity-one Lagrangian mean curvature flow in positive Kähler-Einstein manifolds
BibTeX
@misc{luo2026entropy,
title = {{Entropy and round-point convergence for the $σ_k$-curvature flow}},
author = {Tianci Luo and Yong Wei and Ruijia Zhang and Rong Zhou},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.03249},
eprint = {2611.03249},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.03249},
note = {CC0},
}
RIS
TY - UNPB
TI - Entropy and round-point convergence for the σ_k-curvature flow
AU - Tianci Luo
AU - Yong Wei
AU - Ruijia Zhang
AU - Rong Zhou
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.03249
UR - https://arxiv.org/abs/2611.03249
AB - We prove that every smooth closed strictly convex hypersurface in ℝ^n+1 contracts to a round point under the σ_k-curvature flow for 2≤ k≤ n−1, without additional pinching or symmetry assumptions. After rescaling about the extinction point to mean width two, the hypersurfaces converge smoothly and exponentially in normalized time to the unit sphere. The proof is based on an entropy which is invariant under translations and dilations and admits a representation by weighted boundary areas of orthogonal projections. A strengthened Codazzi inequality yields its monotonicity. An upper bound for the entropy controls the inradius relative to the mean width, while its dissipation controls the spatial variation of the normalized speed. We also obtain two-sided curvature bounds at the natural extinction scale.
ER -