Solutions to Mean Curvature Flow with Uniform Bounds on the Mean Curvature and Its Gradient

Abstract

In the setting of a complete, smooth properly immersed mean curvature flow, we assume uniformly bounded $|H|$ and $|\nabla H|$ on $M^n\times[0,T)$ and some bounded initial geometry to get local spatial $L^p$ estimates for the second fundamental form with $p\in[4,\infty)$. For $p>n+2$, this leads us to a local space time $L^\infty$ bound for the second fundamental form which allows us to smoothly extend the flow $F:M^n\times [0,T) \rightarrow \mathbb{R}^{n+1}$ past the singular time $T<+\infty$ for a short time.