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Cylindrical Estimates for High Codimension Mean Curvature Flow (1805.11808v1)

Published 30 May 2018 in math.DG and math.AP

Abstract: We study high codimension mean curvature flow of a submanifold $\mathcal{M}n$ of dimension $n$ in Euclidean space $\mathbb{R}{n+k}$ subject to the quadratic curvature condition $ |A|{2}\leq c_n |H|{2}, c _n = \min{ \frac{4}{3n} , \frac{1}{n-2}}$. This condition extends the notion of two-convexity for hypersurfaces to high codimension submanifolds. We analyse singularity formation in the mean curvature flow of high codimension by directly proving a pointwise gradient estimate. We then show that near a singularity the surface is quantitatively cylindrical.

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