6533b873fe1ef96bd12d56ca

RESEARCH PRODUCT

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

Vesa JulinJoonas Niinikoski

subject

Mathematics - Differential Geometrymean curvature flowMathematics - Analysis of PDEsDifferential Geometry (math.DG)FOS: Mathematicsminimizing movements35J93 53C44 53C45constant mean curvaturelarge time behaviorAnalysis of PDEs (math.AP)

description

We prove a new quantitative version of the Alexandrov theorem which states that if the mean curvature of a regular set in Rn+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 R2 and R3 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. peerReviewed

yearjournalcountryeditionlanguage
2020-05-28
https://dx.doi.org/10.48550/arxiv.2005.13800