Proof of 3.4 proposition in Huisken's Asymptotic behavior for singularities of the mean curvature flow

Question

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 .

Answer 1

1. $\mathscr{H}^n$ is the $n-$dimensional Hausdorff measure.
2. You can write any hypersurface $M$ as a graph over a tangent plane $T_p M$ locally (by the implicit function theorem), but once some tangent becomes vertical with respect to this plane it fails to be graphical. You can get a lower bound on the distance from $x$ at which this occurs from an upper bound on curvature $|A| < C$ by comparing to the two spheres with $|A|=C$ that contact $T_pM$ at $p$; so if the bound on curvature is uniform then the "graphical radius" is uniform.
3. Take a sequence $R_n \to \infty$. Starting with $R_1$, we get a sequence of times $s^{(1)}_i \to \infty$ on which $M_{s^{(1)}_i} \cap B_{R_1}$ converges. In the next step, choose a subsequence of times $s_i^{(2)} \subset s_i^{(1)}$ that does the same thing for $R_2$, so that you get the convergence result for all $R \le R_2$ on $s_i^{(2)}$. Repeating this indefinitely and taking the diagonal subsequence $t_i = s_i^{(i)}$ you get a sequence of times with convergence for every $R$.