Cylindrical Estimates for High Codimension Mean Curvature Flow

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 $|A{|}^2\le c_n|H{|}^2,c_n=\min \{\frac{4}{3n},\frac{1}{n-2}\}$. 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.

References in corpus (1)

• Ricci flow with surgery on three-manifolds