We will discuss local estimates for the evolutions of strictly convex hypersur- faces by Gauss curvature. We will address geometric cut-off functions and an ap- plication of the Euler’s formula to the Pogorelov type estimate.

2. The uniqueness of fully nonlinear evolutions of complete non-compact hypersurfaces.

It is well-known that the heat equation ut “ uxx, defined on Rˆr0,Tq, does not have a unique solution even for the trivial initial data u0pxq “ 0. However, we can observe that the Mean curvature flow has the unique solution for the trivial initial data; a hyperplane. We will discuss the comparison principle and Jensen’s approximate solutions to show the uniqueness of the complete convex solution of fully nonlinear flows. Special emphasis will be given to the Mean curvature flow.