P. Vishwamitra

On L^p Bounds for The Second Fundamental Form in Mean Curvature Flows with Uniformly Bounded Mean Curvature and Its Gradient

It is well known that a smooth mean curvature flow with uniformly bounded second fundamental form on [0,T) - where T<+\infty - can be smoothly extended beyond T. The extension problem asks whether a uniformly bounded mean curvature suffices. In our work, we assume uniform bounds on the mean curvature and its gradient to provide local L^p estimates for the second fundamental form for every p\geq 4. For p>n+2, via a Moser iteration we obtain point-wise curvature bounds, implying a smooth extension. In my talk, I will give an exposition on the history of the problem, describe the setting of these results, and present the main integral estimate in the proof.