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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
Abstract

We prove a new quantitative version of the Alexandrov theorem which states that if the mean curvature of a regular set in n+1 is close to a constant in the Ln 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 2 and 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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