2017/01/07 by Siao-Hao Guo, Guo, Siao-Hao, Nataša Šešum +1
Mathematics · #53C44 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.1701.01835
openalex publication_date 2017/01/07 · openalex created_date 2017/01/26 · openalex updated_date 2026/07/28
J.J.L. Vel\acuteazquez in 1994 used the degree theory to show that there is a perturbation of Simons' cone, starting from which the mean curvature flow develops a type II singularity at the origin. He also showed that under a proper time-dependent rescaling of the solution around the origin, the rescaled flow converges in the C0 sense to a minimal hypersurface which is tangent to Simons' cone at infinity. In this paper, we prove that the rescaled flow actually converges locally smoothly to the minimal hypersurface, which appears to be the singularity model of the type II singularity. In addition, we show that the mean curvature of the solution blows up near the origin at a rate which is smaller than that of the second fundamental form.