Convexity Estimates for High Codimension Mean Curvature Flow

2020/06/09 by Lynch, Stephen, Nguyen, Huy The
#Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2006.05227

Abstract

We consider the evolution by mean curvature of smooth n-dimensional submanifolds in ℝ^n+k which are compact and quadratically pinched. We will be primarily interested in flows of high codimension, the case k≥ 2. We prove that our submanifold is asymptotically convex, that is the first eigenvalue of the second fundamental form in the principal mean curvature direction blows up at a strictly slower rate than the mean curvature vector. We use this convexity estimate to show that at a singular time of the flow, there exists a rescaling that converges to a smooth codimension-one limiting flow which is convex and moves by translation.

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