A Generalized Bernstein's Problem

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. )

• what is GMT? the only one I know is a timezone :) – Dima Pasechnik Nov 27 '18 at 8:00
• Geometric Measure Theory >_< – ZHans Wang Nov 27 '18 at 13:17
• 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? – T_M Nov 27 '18 at 13:59
• Since we always require the hypersurfaces to be area-minimizing, we can control the volume in each compact subset by the volume of spheres – ZHans Wang Nov 28 '18 at 1:12