An $\epsilon$-regularity Theorem For The Mean Curvature Flow

Abstract

In this paper, we will derive a small energy regularity theorem for the mean curvature flow of arbitrary dimension and codimension. It says that if the parabolic integral of $|A{|}^2$ around a point in space-time is small, then the mean curvature flow cannot develop singularity at this point. As an application, we can prove that the 2-dimensional Hausdorff measure of the singular set of the mean curvature flow from a surface to a Riemannian manifold must be zero.

References in corpus (1)

• Long-time Existence and Convergence of Graphic Mean Curvature Flow in Arbitrary Codimension

Cited by in corpus (2)

• Mean curvature flow of surfaces in a hyperkähler $4$-manifold
• Quantitative Stratification and the Regularity of Mean Curvature Flow