Extend Mean Curvature Flow with Finite Integral Curvature

Abstract

In this note, we first prove that the solution of mean curvature flow on a finite time interval $[0,T)$ can be extended over time $T$ if the space-time integration of the norm of the second fundamental form is finite. Secondly, we prove that the solution of certain mean curvature flow on a finite time interval $[0,T)$ can be extended over time $T$ if the space-time integration of the mean curvature is finite. Moreover, we show that these conditions are optimal in some sense.