Key Steps in the Regularity Proofs

Up to a subsequence, we can assume \(|M_k|\) or \(V_k\) converges to \(V\) in the varifold sense. Pick any point \(x \in \mathrm{spt}\|V\|\). We need to analyze the tangent cone of \(V\) at \(x\). There are several cases:

• Hyperplanes. We need to develop a sheeting theorem to show the regularity and smooth convergence.

• Classical Cones. We need a minimum distance theorem (embedded case) or a decomposition theorem (immersed case) to show the regularity and smooth convergence.

• \(\boldsymbol{C}\times \mathbb{R}^m\) where \(\boldsymbol{C}\) is a stable cone with isolated singularity. Classification of stable cones.

• other cones. We need dimension reduction argument to reduce to the previous case.