I am interested in a boundary version of this question About hypersurfaces in R^n+1 with bounded 2nd fundamental form.
The question is as follows. Let $\Sigma^k\subset \Bbb R^n$ be a submanifold with boundary. Suppose $x\in \partial \Sigma$ and $|A|\le Cr^{-1}$ on intrinsic ball $B_{2r}^\Sigma(x)$. Can $B_r^\Sigma(x)$ be written as a graph over $T_x\Sigma$? It looks like the same proof as the interior case will show that this is true.
The second question is a motivation for the first one (as Otis Chodosh mentioned in the overflow question.) Let $\Omega\subset \Bbb R^n$ be a bounded open smooth domain. If a sequence of minimal submanifolds $\Sigma_i\subset \Omega$, $\partial\Sigma_i\subset \partial \Omega$ satisfies $|A_i|\le C$ on $\bar \Omega$ and $||\Sigma_i||(\bar \Omega)\le C$, then is there a subsequence $\Sigma_i\to \Sigma$ smoothly on $\bar \Omega$? Obviously, we only need to concern the boundary convergence.
In fact, I have an example in mind which seems to disprove question 2. Let $\Sigma$ be the catenoid and $\Sigma_i=i^{-1}\Sigma$. Then $\Sigma_i\to 2P$ where $P$ is the plane $x_3=0$ and the convergence is smooth away from 0. Thus there exists $r_i\to0$ such that $|A_{\Sigma_i}|=1$ on $\partial B_{r_i}(0)$. Now we take $p_i\in \partial B_{r_i}(0)\cap \Sigma_i$ and let $x_i$ be such that $B_1(x_i)$ is tangent to $B_{r_i}(0)$ at $p_i$. We consider a connected component of $B_1(x_i)\cap \Sigma_i$ containing $p_i$ and denote it by $M_i$. We also translate so that $x_i$ is at the origin. Then $|A_{M_i}|\le 1$ and $|A_{M_i}|(p_i-x_i)=1$. And we have the area bound from catenoid. If question 2 were true, then $M_i\to M$ smoothly on $\bar B_1$. However, $M_i\to P\cap B_1(0)$ which is flat. I feel that this is contradicting the smooth convergence and $|A_{M_i}|(p_i-x_i)=1$.
So I am asking whether this is a counter example to question 2, if yes, how does the interior proof fails for boundary case, how to understand the curvature disappearing in the example and is there a weak statement so that we do have convergence on all of $\bar\Omega$
