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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 .

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  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 $R$.
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  • $\begingroup$ Thanks very much. About the 3, why there is a sequence of times $s^{(1)}_i \to \infty$ on which $M_{s^{(1)}_i} \cap B_{R_1}$ converges ? Seemly, it is from the compactness theorem in [8]. But the dimension and center of gravity conditions is not meet. I have add the compactness theorem of [8] in my question. Thanks. $\endgroup$ – lanse7pty May 15 '17 at 7:53
  • $\begingroup$ @lanse7pty: I don't know - try translating each $M_s$ to satisfy the centre of mass condition. I think this compactness theorem should hold in any dimension, with the hypotheses changed to bounded $\mathscr{H}^n$-measure instead of bounded area and probably $p>n$ rather than $p>2$. A quick google turns up arxiv.org/pdf/1006.5697.pdf which seems relevant. $\endgroup$ – Anthony Carapetis May 15 '17 at 8:03

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