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

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