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arXiv · 2006.05227

Convexity Estimates for High Codimension Mean Curvature Flow

Stephen Lynch·Huy The Nguyen
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Abstract

We consider the evolution by mean curvature of smooth $n$-dimensional submanifolds in $\mathbb{R}^{n+k}$ which are compact and quadratically pinched. We will be primarily interested in flows of high codimension, the case $k\geq 2$. 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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Stephen Lynch, Huy The Nguyen. 2020-06-09. Convexity Estimates for High Codimension Mean Curvature Flow. https://arxiv.org/abs/2006.05227

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