Bellettini’s Sheeting Theorem and Schoen–Simon Regularity

Geometric meaning of \(g\).

For the codimensional one case, one can use \(|\nu-e_{n+1}||\nu+e_{n+1}|\) as the distance between \(T_x M\) and \(P\). We have

This agrees with the standard definition of tilt excess up to a constant multiple.

Proof. Choose \((g-k)^+ \phi\) as a test function in the stability inequality, together with

and

we can finish the proof for this lemma for the \(\phi\) with compact support in the regular part of \(M\). The general case can be obtained by a standard cut-off argument near the singular set of \(M\).

The proof for \(\varepsilon\)-regularity theorem for the tilt is similar to the proof of the previous \(\varepsilon\)-regularity theorem for \(|A|\), as we only need to repeat the de Giorgi iteration process.

The proof of the sheeting theorem is as follows.

Note that by the 1st variation formula, (by choosing \(X = \phi^{2}x_{n+1}e_{n+1}\)) we have

So

Then, given any \(\delta\gt{}0\), we can choose \(\varepsilon\) small enough such that \(g\lt{}\delta\) in \(B_1^n \times \mathbb{R}\cap \text{reg}(M)\) by the \(\varepsilon\)-regularity theorem for the tilt. In particular, each small regular region of \(M\cap B_1^n \times \mathbb{R}\) can be written as a graph over a domain in \(P\) with small gradient. Then, by the connectedness of \(M\cap B_1^n \times \mathbb{R}\) away from the singular set, we know \(M\cap B_1^n \times \mathbb{R}\backslash \pi^{-1}(\Sigma)\) can be written as a graph of smooth multiple valued function (locally, it is the union of smooth graphs) over \(B_1^n \backslash \Sigma\). This can be viewed as a covering map from \(M\cap B_1^n \times \mathbb{R}\backslash \pi^{-1}(\Sigma)\) to \(B_1^n \backslash \Sigma\).

In addition, if \(\mathcal{H}^{n-2}(\text{sing}(M))=0\), then \(B_1^n \backslash \Sigma\) is simply connected. So the covering map is trivial, and hence \(M\cap B_1^n \times \mathbb{R}\) is a union of minimal graphs over \(B_1^n(0)\backslash \Sigma\). Again by the removable singularity theorem, each graph can be extended to a smooth minimal graph over \(B_1^n(0)\). ◻

Proof. By Allard’s compactness theorem, up to a subsequence we have \(|M_i|\to V\) in the varifold sense.

Let \(S=\mathrm{sing}\|V\|\) be the embedded singular point set of \(V\). By Federer’s dimension-reduction argument, it suffices to prove the following lemma.

The proof of this lemma is similar to that of Lemma 4.3.7. We construct the iterated tangents of \(V\) at \(x_0 \in S\) as \(\{ \boldsymbol{C}_1,\cdots , \boldsymbol{C}_N \}\) with the following:

1. \(x_j \in \mathrm{sing}\|\boldsymbol{C}_j\|\backslash \mathcal{S}(\boldsymbol{C}_j)\) for each \(1\leq j\leq N-1\) and \(\boldsymbol{C}_{j+1} \in \mathrm{VarTan}(\boldsymbol{C}_j,x_j)\).

2. Each \(\boldsymbol{C}_j\) is not smoothly embedded (i.e., \(\mathrm{sing}\|\boldsymbol{C}_j\|\neq \emptyset\)).

3. \(\mathrm{dim}(\mathcal{S}(\boldsymbol{C}_{j+1}))\gt{}\mathrm{dim}(\mathcal{S}(\boldsymbol{C}_{j}))\) for each \(j=1,2,\cdots ,N-1\).

4. \(\boldsymbol{C}_N=\boldsymbol{C}'\times \mathbb{R}^{\mathrm{dim}(\mathcal{S}(\boldsymbol{C}_N))}\) where \(\boldsymbol{C}'\backslash \left\{ 0 \right\}\) is a smooth embedded cone after a suitable rotation in \(\mathbb{R}^{n+1}\).

5. For each \(1\leq j\leq N\), we can find a sequence of points \(\left\{ y_k \right\}\) with \(y_k\to x_0\), a sequence of positive real numbers \(\left\{ r_k \right\}\) with \(r_k\to 0^+\) as \(k\to \infty\), such that \(\eta_{y_k,r_k}(M_k)\) converges to \(\boldsymbol{C}_j\) in the sense of varifolds and the convergence is smooth away from the singular set of \(\boldsymbol{C}_j\) by sheeting theorem.

We need to show that \(\boldsymbol{C}'\) has dimension at least \(7\). If the dimension of \(\boldsymbol{C}'\) is one, then \(\boldsymbol{C}_N\) is the sum of distinct half-hyperplanes with multiplicity.

For simplicity, we denote

We denote \(\hat{M}_k=\eta_{y_k,r_k}(M_k)\). By the sheeting theorem, for any \(\tau\gt{}0\) and all large \(k\), the set

decomposes into \(q=\sum_{i=1}^{N_1} q_i\) connected components. Each component is a smooth graph over the corresponding half-hyperplane in \(\boldsymbol{C}_N\).

Now, for \(\mathcal{H}^{n-1}\)-a.e. point \(y \in \mathbb{R}^{n-1}\cap B_1^{n-1}\), Sard’s theorem and \(\mathcal{H}^{n-2}(\mathrm{sing}\,\hat{M}_k)=0\) imply that \(\hat{M}_k \cap B_1^{2} \times \{ y \}\) consists of \(q\) embedded curves for large \(k\).

Choose two pieces \(N_1,N_2 \subset \hat{M}_k\backslash T_\tau(\boldsymbol{C}_N)\) such that \(N_i\) is a graph over a domain in \(\{ (\cos \theta_i r, \sin \theta_i r, y): r \geq \tau, y \in \mathbb{R}^{n-1} \}\) and \(N_1,N_2\) are connected by a curve \(\gamma\) as above.

Note that \(\left\vert \nu(\gamma \cap \{ r=\tau \}\cap N_1)-\nu(\gamma \cap \{ r=\tau \}\cap N_2) \right\vert \geq \frac{1}{2}|\sin(\theta_1 - \theta_2)|\) for \(k\) large enough. So the integral

Now, we integrate over all such curves \(\gamma\) for \(\mathcal{H}^{n-1}\)-a.e. point \(y \in \mathbb{R}^{n-1}\cap B_1^{n-1}\), we have

But on the other hand, using Cauchy-Schwarz inequality, we have

for some \(C=C(n,\Lambda)\) independent of \(\tau\) and \(k\) by the monotonicity formula and the stability inequality. Since \(\tau\) is arbitrary, this is a contradiction.

Hence, we know \(\boldsymbol{C}'\) has dimension at least \(2\). But the fifth condition implies that \(\boldsymbol{C}'\) is a smooth embedded stable cone away from \(\{0\}\), and hence \(\boldsymbol{C}'\) has dimension at least \(7\). ◻