Second fundamental form blows up at minimal hypersurface singularity

Question

I have seen the claim that "$A$ bounded on an area minimizing current implies no singular set" in a couple of papers by Lohkamp, but with no reference (see https://arxiv.org/abs/1805.02180 e.g.).

Precisely, consider $T$ an i.m.r. $(n-1)$-current in a Riemannian $n$-manifold $M$. Suppose $T$ is locally mass minimizing. If, on the regular set of $T$, the second fundamental form $A$ is uniformly bounded up to the singular set, then the singular set is empty. Alternatively, if $x_j\in \mathrm{reg}(T)$ and $x_j\to x\in \mathrm{sing}(T)$, then $|A(x_j)|\to\infty$.

Lohkamp says this a folklore theorem and gives no proof. Is there a reference for this out there or is the proof really simple?

Answer 1

Well I know that you can rule out higher multiplicity planes; that argument is in Federer somewhere. So we just have to show that every tangent cone is a plane.

Here's a sketch that comes to mind: If the curvature is bounded by say $C$ near a singular point $x_0$, then consider taking a tangent cone at $x_0$. You would rescale a ball $B_{\epsilon_j}(x_0)$ to size $1$. Then the curvature on the regular part of the rescaled current would be $C\epsilon_j$. So when you take the tangent cone, the regular part would converge smoothly to flat pieces. The closure of the regular part of the cone has to be the whole cone. But flat pieces can't meet or intersect in any way other than all be part of the same plane.