Mean curvature flow with pinched curvature integral

Abstract

If Σ is an n-dimensional noncompact self-shrinker and the second fundamental form of Σ is $L^p$ integrable for $p\ge n$, we show that Σ is asymptotic to a regular cone. We also prove long-time existence of the mean curvature flow starting from complete manifolds with bounded curvature and small total curvature.