Singularity models of pinched solutions of mean curvature flow in higher codimension

2019/10/09 by Naff, Keaton
#Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1910.03968

Abstract

We consider ancient solutions to the mean curvature flow in ℝⁿ⁺¹ (n ≥ 3) that are weakly convex, uniformly two-convex, and satisfy derivative estimates |∇ A| ≤ γ₁ |H|², |∇² A| ≤ γ₂ |H|³. We show that such solutions are noncollapsed. As an application, in arbitrary codimension, we consider compact n-dimensional (n ≥ 5) solutions to the mean curvature flow in ℝ^N that satisfy the pinching condition |H| > 0 and |A|² < c(n) |H|², c(n) = min\(1)/(n-2), (3(n+1))/(2n(n+2))\. We conclude that any blow-up model at the first singular time must be a codimension one shrinking sphere, shrinking cylinder, or translating bowl soliton.

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