Blow Up of Compact Mean Curvature Flow Solutions with Bounded Mean Curvature

2024/03/25 by Liu, Zichang · 1 citation
#Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2403.16515

Abstract

In 1994, Velázquez constructed a countable family of complete hypersurfaces flowing in ℝ^2N (N≥ 4) by mean curvature, each of which develops a type II singularity at the origin in finite time. Later Guo and Sesum showed that for a non-empty subset of Velázquez's solutions, the mean curvature blows up near the origin, at a rate smaller than that of the second fundamental form; recently Stolarski proved another subset of these solutions has bounded mean curvature up to the singular time. In this paper, we follow their arguments to construct compact mean curvature flow solutions in ℝⁿ (n≥ 8) with bounded mean curvature.

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• Mean curvature flow converging to an minimizing cone and its Hardt-Simon foliation

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