Picture below is from Huisken, Gerhard, Asymptotic behavior for singularities of the mean curvature flow, J. Differ. Geom. 31, No.1, 285-299 (1990). ZBL0694.53005.
I have many questions in this proof.
First, what is $\mathscr H^n(\widetilde M_n \cap B_R(0))$ ? I can't find the definition in this paper.
Second, $\widetilde A$ is second fundamental form. Why it is uniformly bounded , then the immersion can be locally written as graph of $C^\infty$ function ?
Third, why picking a diagonal sequence ,we can get a smooth limit ? In fact , I don't know the diagonal is about what it is diagonal .