On the conditions of extending mean curvature flow

Abstract

The authors consider a family of smooth immersions F (·, t): M ⁿ → ℝⁿ⁺¹ of closed hypersurfaces in ℝⁿ⁺¹ moving by the mean curvature flow $\frac{{\partial F(p,t)}} {{\partial t}} = - H(p,t) \cdot \nu (p,t)$ for t ∈ [0, T). They show that if the norm of the second fundamental form is bounded above by some power of mean curvature and the certain subcritical quantities concerning the mean curvature integral are bounded, then the flow can extend past time T. The result is similar to that in [6–9].