Quantitative Alexandrov theorem and asymptotic behavior of the volume preserving mean curvature flow

Julin, Vesa; Niinikoski, Joonas

doi:10.2140/apde.2023.16.679

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Quantitative Alexandrov theorem and asymptotic behavior of the volume preserving mean curvature flow

Vesa Julin and Joonas Niinikoski

Vol. 16 (2023), No. 3, 679–710

DOI: 10.2140/apde.2023.16.679

Abstract

We prove a new quantitative version of the Alexandrov theorem which states that if the mean curvature of a regular set in $ℝ^{n + 1}$ is close to a constant in the $L^{n}$ sense, then the set is close to a union of disjoint balls with respect to the Hausdorff distance. This result is more general than the previous quantifications of the Alexandrov theorem, and using it we are able to show that in $ℝ^{2}$ and $ℝ^{3}$ a weak solution of the volume preserving mean curvature flow starting from a set of finite perimeter asymptotically convergences to a disjoint union of equisize balls, up to possible translations. Here by a weak solution we mean a flat flow, obtained via the minimizing movements scheme.

Keywords

mean curvature flow, large time behavior, constant mean curvature, minimizing movements

Mathematical Subject Classification

Primary: 35J93, 35K93, 53C45

Milestones

Received: 1 July 2020

Accepted: 8 July 2021

Published: 25 May 2023

Authors

Vesa Julin
Department of Mathematics and Statistics University of Jyvaskyla Jyvaskyla Finland

Joonas Niinikoski
Department of Mathematics and Statistics University of Jyvaskyla Jyvaskyla Finland

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Vol. 16, No. 3, 2023