Sharp dynamical isoperimetric principle for mean curvature flow
Abstract
In this paper, we establish a sharp dynamical isoperimetric principle for weak mean curvature flow. We prove that, among weak mean curvature flows with fixed area of initial smooth closed hypersurfaces, the standard smoothly shrinking spherical mean curvature flow uniquely maximizes the extinction time. More precisely, for the level set flow Kt starting from the boundary of a smooth bounded domain Ω⊂ℝn+1, as well as for the integral Brakke flow {μt} with initial Radon measure μ0=ℋnllcorner∂Ω, we establish the corresponding optimal extinction time estimates Text(Ω), TBext(Ω) ≤ 1/2n ((P( Ω))/|𝕊n|)2/n, where P(Ω) is the perimeter of Ω representing the area of ∂ Ω, and the equality holds if and only if Ω is a round ball and the flow is the standard multiplicity-one smoothly self-shrinking round sphere. In particular, we obtain the sharp Lp-estimates for the arrival time function of a smooth bounded mean convex domain. In addition, we also establish the sharp isoperimetric inequality for the parabolic measure of the space-time track filling X of outward minimizing level set flow starting from the boundary of a smooth bounded domain Ω⊂ ℝn+1: ℋparn+2(X) ≤ π/(2n(n+2)2|𝕊n|2/n) P(Ω)(n+2)/n, where the equality holds if and only if Ω is a round ball and the flow is standard smoothly shrinking round sphere.
Related
- Isoperimetric and total curvature inequalities in Cartan-Hadamard manifolds with nullity
- The Sharp Isoperimetric Inequality in the First Heisenberg Group
- The sharp isoperimetric inequality for varifolds in codimension two
- The Cartan-Hadamard isoperimetric inequality under controlled variation of curvature
- Discrete Isoperimetric Inequalities via Curvature
BibTeX
@misc{choi2026sharp,
title = {{Sharp dynamical isoperimetric principle for mean curvature flow}},
author = {Beomjun Choi and Wenkui Du and Seung Chul Park and Junseo Youn},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.10461},
eprint = {2610.10461},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.10461},
}
RIS
TY - UNPB
TI - Sharp dynamical isoperimetric principle for mean curvature flow
AU - Beomjun Choi
AU - Wenkui Du
AU - Seung Chul Park
AU - Junseo Youn
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.10461
UR - https://arxiv.org/abs/2610.10461
AB - In this paper, we establish a sharp dynamical isoperimetric principle for weak mean curvature flow. We prove that, among weak mean curvature flows with fixed area of initial smooth closed hypersurfaces, the standard smoothly shrinking spherical mean curvature flow uniquely maximizes the extinction time. More precisely, for the level set flow K_t starting from the boundary of a smooth bounded domain Ω⊂ℝ^n+1, as well as for the integral Brakke flow {μ_t} with initial Radon measure μ_0=ℋ^nllcorner∂Ω, we establish the corresponding optimal extinction time estimates T_ext(Ω), T^B_ext(Ω) ≤ 1/2n ((P( Ω))/|𝕊^n|)^2/n, where P(Ω) is the perimeter of Ω representing the area of ∂ Ω, and the equality holds if and only if Ω is a round ball and the flow is the standard multiplicity-one smoothly self-shrinking round sphere. In particular, we obtain the sharp L^p-estimates for the arrival time function of a smooth bounded mean convex domain. In addition, we also establish the sharp isoperimetric inequality for the parabolic measure of the space-time track filling X of outward minimizing level set flow starting from the boundary of a smooth bounded domain Ω⊂ ℝ^n+1: ℋ_par^n+2(X) ≤ π/(2n(n+2)^2|𝕊^n|^2/n) P(Ω)^(n+2)/n, where the equality holds if and only if Ω is a round ball and the flow is standard smoothly shrinking round sphere.
ER -