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Suppose $\Sigma\subset \mathbb{R}^n$ be an area-minimizing hypersurfaces, the standard results in GMT tell that when $3\leq n\leq 7$, $\Sigma$ is a hyperplane; while for $n\geq 8$, there are examples of non-trivial $\Sigma$.

My question is, can we obtain the triviality of $\Sigma$ in higher dimensions when supposing further that the second fundamental form $A_{\Sigma}$ of $\Sigma$ vanish at some point?

( This problem is equivalent to the following, $\ \ $ suppose $\Sigma_k$ area-minimizing in $\mathbb{B}_1^n$, $0\in \Sigma_k$ and $\Sigma_k \to \Sigma_0$ with $0$ a singular point of $\Sigma_0$, then is it true that $|A_{\Sigma_k}|_0 \to +\infty$ ? $\ \ $ It's seemingly true, but I have no idea for a proof. )

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  • $\begingroup$ what is GMT? the only one I know is a timezone :) $\endgroup$ – Dima Pasechnik Nov 27 '18 at 8:00
  • $\begingroup$ Geometric Measure Theory >_< $\endgroup$ – ZHans Wang Nov 27 '18 at 13:17
  • $\begingroup$ I can't see how the two statements are equivalent; maybe you can elaborate on that in the question? In particular, when you re-formulate it in the parentheses, do you have area bounds or not? $\endgroup$ – T_M Nov 27 '18 at 13:59
  • $\begingroup$ Since we always require the hypersurfaces to be area-minimizing, we can control the volume in each compact subset by the volume of spheres $\endgroup$ – ZHans Wang Nov 28 '18 at 1:12

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