Regularity and Compactness Theorems | Gaoming Wang

Regularity and Compactness Results for Stable Minimal Hypersurfaces

Mon, 29 Jun 2026 00:00:00 +0000

The first result is the generalized Bernstein theorem from Schoen–Simon–Yau.

Theorem 4.1.1

Suppose \(M \subset \mathbb{R}^{n+1}\) is a complete, stable, minimal hypersurface without boundary and with at most (intrinsic) Euclidean volume growth. Then, if \(n \leq 5\), \(M\) must be an affine hyperplane.

Note that the intrinsic Euclidean volume growth condition is weaker than the extrinsic Euclidean volume growth condition. This is also a key condition for the Stable Bernstein Theorem without area growth condition.

Theorem 4.1.2

Let \(\{M_k\}\) be a sequence of embedded, stable, orientable minimal hypersurfaces in \(B_2^{n+1}(0)\) with the following properties:

\(0\in \bar{M}_k\) for each \(k\).
\(\mathcal{H}^{n-2}(\mathrm{sing}M_k)=0\) for each \(k\).
\(\mathcal{H}^n(M_k \cap B_2^n(0))\leq \Lambda\) for some constant \(\Lambda\gt{}0\) independent of \(k\).

Then, up to a subsequence, \(M_k\) converges in the varifold sense to a stable minimal hypersurface \(M\) in \(B_2^{n+1}(0)\), which is smooth except for a closed singular set of Hausdorff dimension at most \(n-7\).

In addition, for \(n=7\), the singular set is discrete. \(M_k\) converges to a stable minimal hypersurface \(M\) in the varifold sense means that the varifold \(|M_k|\) converges to the varifold \(|M|\) in the sense of measures of weak limit.

Recall that a closed set \(S\) is of Hausdorff dimension at most \(k\) if for any \(\varepsilon\gt{}0\), we have \(\mathcal{H}^{k+\varepsilon}(S)=0\).

Corollary 4.1.3

Suppose \(M \subset \mathbb{R}^{n+1}\) is a complete, stable, embedded minimal hypersurface without boundary and with at most extrinsic Euclidean volume growth. Then, if \(n \leq 6\), \(M\) must be an affine hyperplane.

The dimension \(n\leq 6\) is sharp, as we already proved that Simons’ cone is stable in \(\mathbb{R}^8\).

Bellettini’s work completes the corresponding area-growth statement in the borderline dimension \(n=6\) for stable immersions. To state the result in a form that is independent of whether one measures volume intrinsically or extrinsically, we also record the comparison theorem of Florit–Simon.

Theorem 4.1.4

Let \(\Sigma^d\hookrightarrow \mathbb{R}^N\) be a complete, connected, smooth minimal immersion, and let \(p\in \Sigma\). Define the intrinsic and extrinsic area densities by

\[ \mathbf{M}_R^{\mathrm{int}}(\Sigma,p):=\frac{|B_R^\Sigma(p)|}{R^d}, \qquad \mathbf{M}_R^{\mathrm{ext}}(\Sigma,p):=\frac{\mathrm{Area}(\Sigma\cap B_R^{\mathbb{R}^N}(p))}{R^d}, \]

where \(B_R^\Sigma(p)=\{x\in \Sigma: d_\Sigma(x,p)\lt{}R\}\), and the extrinsic area is counted with multiplicity. Then \(\mathbf{M}_R^{\mathrm{int}}(\Sigma,p)\) and \(\mathbf{M}_R^{\mathrm{ext}}(\Sigma,p)\) are monotone nondecreasing in \(R\), and

\[ \lim_{R\to\infty}\mathbf{M}_R^{\mathrm{int}}(\Sigma,p) =\lim_{R\to\infty}\mathbf{M}_R^{\mathrm{ext}}(\Sigma,p)\in[\omega_d,\infty], \]

where \(\omega_d=|B_1^{\mathbb{R}^d}|\). In particular, \(\Sigma\) has bounded intrinsic area density if and only if it has bounded extrinsic area density; in either case, the immersion is proper.

Corollary 4.1.5

Let \(2\leq n\leq6\), and let \(\Sigma^n\hookrightarrow \mathbb{R}^{n+1}\) be a complete, connected, two-sided, stable minimal immersion. If \(\Sigma\) has Euclidean area growth, equivalently

\[ |B_R^\Sigma(p)|\leq \Lambda R^n \quad\text{for some }p\in\Sigma,\ \Lambda\lt{}\infty,\text{ and all }R\gt{}0, \]

\[ \mathrm{Area}(\Sigma\cap B_R^{\mathbb{R}^{n+1}}(p))\leq \Lambda R^n \quad\text{for some }p\in\mathbb{R}^{n+1},\ \Lambda\lt{}\infty,\text{ and all }R\gt{}0, \]

then \(\Sigma\) is an affine hyperplane.

For \(2\leq n\leq5\), this is the Schoen–Simon–Yau area-growth stable Bernstein theorem. The new borderline case is \(n=6\): Bellettini proves the classification under extrinsic Euclidean area growth, and Theorem 4.1.4 converts intrinsic area growth into the same extrinsic hypothesis. Thus the area-growth version of the immersed stable Bernstein theorem is settled in the full sharp range \(2\leq n\leq6\).

A natural question is whether the compactness conclusion in Theorem 4.1.2 remains true if we assume \(\mathcal{H}^{n-1}(\mathrm{sing}M_k)=0\) instead of \(\mathcal{H}^{n-2}(\mathrm{sing}M_k)=0\). The answer is yes, by the following deep result of Wickramasekera.

Theorem 4.1.6

Suppose \(V_i\) is a sequence of stationary integral \(n\)-varifolds in \(B_2^{n+1}(0)\) and \(V_i\) also satisfies the following conditions:

\(0\in \mathrm{spt}\|V_i\|\).
\(\|V_i\|(B_2^{n+1}(0))\leq \Lambda\) for some constant \(\Lambda\gt{}0\) independent of \(i\).
(Stability) Each \(V_i\) is stable in \(B_2^{n+1}(0)\) on its regular set, i.e., for any \(\phi \in C_c^1(\mathrm{reg}V_i)\),

\[ \int_{\mathrm{reg}V_i} |A_i|^2 \phi^2 d\|V_i\| \leq \int_{\mathrm{reg}V_i} |\nabla \phi|^2 d\|V_i\|. \]

(Alpha-Structural Hypothesis) There exists \(\alpha\in (0,1)\) such that for each \(i\), no point of \(\mathrm{spt}\|V_i\|\cap B_1^{n+1}(0)\) has a neighborhood in which \(\mathrm{spt}\|V_i\|\) is the union of three or more embedded \(C^{1,\alpha}\) hypersurfaces-with-boundary meeting only along their common boundary.

Then, up to a subsequence, \(V_i\) converges in the varifold sense to a stationary integral \(n\)-varifold \(V_\infty\) in \(B_2^{n+1}(0)\), which is stable and whose singular set in \(B_2^{n+1}(0)\) has Hausdorff dimension at most \(n-7\).

The theorem is formulated for stationary integral varifolds, so no orientability assumption is part of the statement. In applications to stable immersions, two-sidedness is imposed separately when one writes the stability inequality on the regular set.

The next conjecture concerns compactness for stable minimal immersions with singular sets.

The class of branched two-sided stable minimal \(n\)-dimensional immersions with the singular set of locally finite \((n-2)\)-measure is compact under varifold convergence.

The first result in this direction is the following density-\(2\) regularity theorem.

Theorem 4.1.7

Let \(\delta \in (0,1)\). Suppose \(M_k\) is a sequence of orientable stable minimal hypersurfaces immersed in \(B_2^{n+1}(0)\) such that:

\(0\in \bar{M}_k\).
\(\|M_k\|(B_2^{n+1}(0))\leq (3-\delta)\omega_n 2^n\).
\(\mathcal{H}^{n-2}(\mathrm{sing}M_k)=0\).

The general case, with non-optimal singular set dimension, is proved by Hong-Li-Wang [HLW24].

Theorem 4.1.8

Suppose \(M_k\) is a sequence of orientable stable minimal hypersurfaces immersed in \(B_2^{n+1}(0)\) such that:

\(0\in \bar{M}_k\).
\(\|M_k\|(B_2^{n+1}(0))\leq \Lambda\).
\(\sup_k\operatorname{dim}_{\mathcal{H}}(\mathrm{sing}M_k)\lt{} n-4+\frac{4}{n}\).

Minter–Xiao subsequently proved the optimal non-branched version of this regularity and compactness theorem.

Theorem 4.1.9

Let \(n\geq2\). Suppose \(M_k\) is a sequence of two-sided stable minimal hypersurfaces smoothly and properly immersed in \(B_1^{n+1}(0)\) such that:

\(0\in \bar M_k\) for each \(k\);
\(\sup_k\mathcal H^n(M_k\cap B_1^{n+1}(0))\lt{}\infty\);
\(\mathcal H^{n-2}(\mathrm{sing}M_k)=0\) for each \(k\), where

\[ \mathrm{sing}M_k:=B_1^{n+1}(0)\cap(\bar M_k\setminus M_k). \]

Then, after passing to a subsequence, \(M_k\) converges as varifolds to a stationary integral varifold \(V\) in \(B_1^{n+1}(0)\). Moreover, there is a relatively closed set

\[ S\subset \mathrm{spt}\,\|V\|\cap B_1^{n+1}(0), \qquad \dim_{\mathcal H}S\leq n-7, \]

such that \(V\) is represented on \(B_1^{n+1}(0)\setminus S\) by a proper, two-sided, stable minimal immersion, and \(M_k\) converges locally smoothly to this immersion away from \(S\). In particular, \(S=\emptyset\) for \(2\leq n\leq6\), while \(S\) is discrete for \(n=7\).

The hypothesis \(\mathcal H^{n-2}(\mathrm{sing}M_k)=0\) is the optimal non-branched assumption. It improves Theorem 4.1.8, where the non-immersed singular set is required to have Hausdorff dimension strictly smaller than \(n-4+\frac4n\). The branch point case is not included here: when the non-immersed singular set has positive \(\mathcal H^{n-2}\)-measure, branch points may occur and the corresponding compactness theory remains a separate problem.

Key Steps in the Regularity Proofs

Mon, 29 Jun 2026 00:00:00 +0000

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.

Regularity in the Immersed Setting

Mon, 29 Jun 2026 00:00:00 +0000

One of the key ingredients in the proof of Theorem 4.1.8 is the following \(\varepsilon\)-regularity theorem.

Theorem 4.3.1

Let \(n\geq 3\). Suppose \(M^n\) is a two-sided stable minimal hypersurface immersed in \(B^{n+1}_4(0)\) and the singular set of \(M\) satisfies \(\bar{n}:=\mathrm{dim}(\mathrm{sing}M)\lt{} n-2-\frac{2(n-2)}{n}\). Additionally, assume \(\mathcal{H}^n(M\cap B^{n+1}_4(0))\leq \Lambda\) for some \(\Lambda \in (0,+\infty)\). Then, for any \(\alpha \in (\frac{n-2}{n},\min \left\{ \frac{n-\bar{n}-2}{2},1 \right\})\), there exists \(\varepsilon=\varepsilon(n,\bar{n},\alpha,\Lambda) \in (0,1)\) such that if

\[ \int_{ B^{n+1}_2(0)\cap M} |A|^{2\alpha}\leq \varepsilon, \]

then

\[ \sup_{B^{n+1}_{\frac{1}{2}}(0)\cap M}|A|^{2\alpha}\leq C \int_{ B^{n+1}_2(0)\cap M} |A|^{2\alpha} \]

for some constant \(C=C(n,\bar{n},\Lambda,\alpha)\).

The above result relies on the following weak (intrinsic) Caccioppoli inequality.

Lemma 4.3.2

For any \(\alpha \in (\frac{n-2}{n},\min \left\{ \frac{n-\bar{n}-2}{2} ,1\right\})\), and any locally Lipschitz function \(\phi\) supported in \(B^{n+1}_3(0)\), we have

\begin{equation} \tag{4.3.1} \begin{aligned} \int_{M\cap \{u\gt{}k\}} \left( 1-\frac{k}{u} \right){}&|\nabla u|^2\phi^2\leq C\int_{M\cap \{u\gt{}k\}} (u-k)^2|\nabla \phi|^2\nonumber\\+{}&Ck^2\int_{ M\cap \{u\gt{}k\}} \left( (u -k)^{\frac{2}{\alpha}}+k^{\frac{2}{\alpha}} \right) \phi^2, \label{eq:lemWeakCacci} \end{aligned} \end{equation}

where the constant \(C=C(n,\bar{n}, \Lambda,\alpha)\). Here \(u=|A|^\alpha\).

Proof. We first show that (4.3.1) holds for bounded locally Lipschitz \(\phi\) with compact support in \(B^{n+1}_4(0)\), vanishing in a neighborhood of \(\mathrm{sing}M \cap B^{n+1}_4(0)\), and for any \(\alpha \in (\frac{n-2}{n},1)\).

For such \(\phi\) and \(\alpha\), choose \(\varphi = (|A|^\alpha-k)^+ \phi\) for \(k\geq 0\). One checks that \(((|A|^\alpha-k)^+)^2 \in C^1(M)\cap W^{2,\infty}_{\mathrm{loc}}(M)\). Hence we can insert \(\varphi\) into the stability inequality.

We observe that

\[ \begin{aligned} \frac{1}{2}\Delta(|A|^\alpha-k)^2={}&\alpha\left( 1-\frac{k}{|A|^{\alpha}} \right)|A|^{2\alpha-2}|A|\Delta|A|\\ +&\alpha \left( \left( 1-\frac{k}{|A|^\alpha} \right)(\alpha-1)+\alpha \right)|A|^{2\alpha-2}|\nabla|A||^2. %+\alpha[(2\alpha-1)|A|^\alpha-k(\alpha-1)]|A|^{\alpha-2}|\nabla|A||^2 \end{aligned} \]

Using Simons’ inequality

\[ |A|\Delta |A|\geq\frac{2}{n}|\nabla|A||^2-|A|^4, \]

we obtain

\[ \begin{aligned} \int_{M} |\nabla \varphi|^2 ={}& \int_{M_{\gt{}k}} \alpha^2|A|^{2\alpha-2}|\nabla|A||^2\phi^2 +(|A|^\alpha-k)^2|\nabla \phi|^2 +\frac{1}{2}\left< \nabla (|A|^\alpha-k)^2, \nabla \phi^2 \right> \\ ={}& \int_{M_{\gt{}k}} \alpha^2|A|^{2\alpha-2}|\nabla|A||^2\phi^2 +(|A|^\alpha-k)^2|\nabla \phi|^2 -\frac{1}{2}\phi^2\Delta(|A|^\alpha-k)^2\\ ={}& \int_{M_{\gt{}k}} (|A|^\alpha-k)^2|\nabla\phi|^2 -\int_{M_{\gt{}k}} \alpha \left( 1-\frac{k}{|A|^\alpha} \right)|A|^{2\alpha-2}|A|\Delta|A|\phi^2\\ &+\alpha(1-\alpha)\left( 1-\frac{k}{|A|^\alpha} \right)|A|^{2\alpha-2}|\nabla|A||^2\phi^2\\ \leq{}& \int_{M_{\gt{}k}} ((|A|^\alpha-k)^2)|\nabla\phi|^2\\ &-\int_{M_{\gt{}k}} \alpha\left( \frac{2}{n}+\alpha-1 \right) \left( 1-\frac{k}{|A|^\alpha} \right)|A|^{2\alpha-2}|\nabla|A||^2\phi^2 +\int_{M_{\gt{}k}} \alpha |A|^{2\alpha+2}\phi^2 \end{aligned} \]

where \(M_{\gt{}k}\) denotes \(M\cap \{|A|^\alpha\gt{}k\}\). On the other hand, by stability, we have

\[ \int_{M} |\nabla\varphi|^2\geq \int_{M} |A|^2\varphi^2=\int_{M_{\gt{}k}} |A|^{2}(|A|^{\alpha}-k)^2\phi^2. \]

Now, we write \(\delta:=\alpha - \frac{n-2}{n}\gt{}0\). Then the stability inequality gives

\[ \begin{aligned} & \delta\int_{M_{\gt{}k}} \alpha \left( 1-\frac{k}{|A|^\alpha} \right)|A|^{2\alpha-2}|\nabla|A||^2\phi^2\\ \leq{}& \int_{M_{\gt{}k}} ((|A|^\alpha-k)^2)|\nabla \phi|^2 +\int_{M_{\gt{}k}} \alpha |A|^{2\alpha}|A|^2\phi^2\\ &-\int_{M_{\gt{}k}} |A|^{2}(|A|^{\alpha}-k)^2\phi^2. \end{aligned} \]

Now, let \(u=|A|^\alpha\). Then,

\begin{equation} \tag{4.3.2} \frac{\delta}{\alpha} \int_{M_{\gt{}k}} \left( 1-\frac{k}{u} \right)|\nabla u|^2\phi^2\leq \int_{M_{\gt{}k}} (u-k)^2|\nabla \phi|^2+\int_{M_{\gt{}k}}u^{\frac{2}{\alpha}}\left( \alpha u^2-(u-k)^2 \right) \phi^2. \label{eq:pfWeakPoincareExtraTerm} \end{equation}

Now we estimate \(u^{\frac{2}{\alpha}}\left( \alpha u^2-(u-k)^2 \right)\).

\begin{equation} \tag{4.3.3} \begin{aligned} \alpha u^2-(u -k)^2={} & -(1-\alpha) (u -k)^2+2\alpha k(u-k)+\alpha k^2 \leq \frac{\alpha^2k^2}{1-\alpha}+\alpha k^2=\frac{\alpha}{1-\alpha} k^{2}. \label{eq:pfYoung} \end{aligned} \end{equation}

where we used Young’s inequality. By the trivial inequality \((x+y)^a\leq 2^a(x^a+y^a)\) for \(x,y\geq 0\), \(a\gt{}0\), we have

\[ u^{\frac{2}{\alpha}}\left(\alpha u^2-(u-k)^2\right)\leq 2^{\frac{2}{\alpha}}\frac{\alpha}{1-\alpha} \left( (u -k)^{\frac{2}{\alpha}}+k^{\frac{2}{\alpha}} \right)k^2. \]

Substituting this into (4.3.2), we obtain (4.3.1).

To complete the proof, we show by approximation that (4.3.1) holds for any bounded locally Lipschitz \(\phi\) supported in \(B_3(0)\), assuming \(\alpha \in (\frac{n-2}{n},\min \left\{ \frac{n-\bar{n}-2}{2},1 \right\})\). Note that \(\phi\) may be non-zero on the singular set of \(M\).

We first derive a preliminary estimate on \(|A|\).

Lemma 4.3.3

If \(\mathcal{H}^{n-2}(\mathrm{sing}(M)\cap B^{n+1}_4(0))=0\), then we have \(|A| \in L^2(B^{n+1}_{\frac{7}{2}}(0)\cap M)\) and

\[ \int_{M\cap B^{n+1}_\rho(x) } |A|^2\leq C\rho^{n-2}, \]

for any \(x \in B^{n+1}_{\frac{7}{2}}(0)\) and \(\rho \in (0,\frac{1}{4})\), where \(C=C(\Lambda)\).

Proof. For each \(\varepsilon\gt{}0\), we choose balls \(\left\{ B^{n+1}_{r_i}(x_i) \right\}_{i=1}^N\) such that \(\mathrm{sing}(M)\cap B^{n+1}_4(0)\subset \bigcup_{i=1}^N B^{n+1}_{r_i}(x_i)\) and \(\sum_{i=1}^N r_i^{n-2}\leq \varepsilon\). We choose \(\zeta_i\) to be a non-negative \(C^1\) function such that \(\zeta_i\) is supported outside of \(B^{n+1}_{r_i}(x_i)\), \(\zeta_i=1\) outside of \(B^{n+1}_{2r_{i}}(x_i)\), and \(|\nabla \zeta_i|\leq \frac{2}{r_i}\). Then, we define \(\zeta_\varepsilon=\min_{1\leq i\leq N}\zeta_i\). We insert \(\zeta_\varepsilon \phi\) into the stability inequality where \(\phi\) is a non-negative locally Lipschitz function with compact support in \(B^{n+1}_4(0)\). Then,

\[ \int_{M\cap B^{n+1}_4(0)} |A|^2\phi^2\zeta_\varepsilon^2\leq 2\int_{M\cap B^{n+1}_4(0)} \left|\nabla \phi\right|^2\zeta_\varepsilon^2+2\int_{M\cap B^{n+1}_4(0)}|\nabla \zeta_\varepsilon|^2\phi^2 \]

by Cauchy-Schwarz inequality. Note that

\[ \begin{aligned} \int_{M\cap B^{n+1}_4(0)} |\nabla \zeta_\varepsilon|^2\phi^2\leq{} & \|\phi\|_{L^\infty(B^{n+1}_4(0))}^2\sum_{i=1}^{N}\int_{M\cap B^{n+1}_{2r_i}(x_i)} |\nabla \zeta_i|^2\\ \leq{}& C\|\phi\|_{L^\infty(B^{n+1}_4(0))}^2\sum_{i=1}^{N}r_i^{n-2}\leq C\|\phi\|_{L^\infty(B^{n+1}_4(0))}^2\varepsilon \end{aligned} \]

which converges to \(0\) as \(\varepsilon\to 0^+\). Then, we have

\[ \int_{M} |A|^2\phi^2\leq 2\int_{M}|\nabla \phi|^2. \]

In particular, it implies \(|A| \in L^2(B^{n+1}_{\frac{7}{2}}(0))\) if we choose \(\phi\equiv 1\) on \(B^{n+1}_{\frac{7}{2}}(0)\). Now, we choose \(\phi\) supported on \(B^{n+1}_{2\rho}(x)\), and equal to \(1\) on \(B^{n+1}_\rho(x)\), with \(|\nabla \phi|\leq \frac{2}{\rho}\). Together with the monotonicity formula, we have

\[ \int_{M\cap B^{n+1}_\rho(x)} |A|^2\leq 2\int_{M\cap B^{n+1}_{2\rho}(x)\backslash B^{n+1}_\rho(x)} \frac{1}{\rho^2}\leq C\rho^{n-2}, \]

for some \(C=C(\Lambda)\). ◻

The remaining part is similar to the proof of the previous lemma. Based on the assumption of \(\bar{n}\) and \(\alpha\), we know \(\mathcal{H}^{n-2-2\alpha}(\mathrm{sing}M)=0\). Therefore, for any \(\varepsilon\gt{}0\), there exist \(B^{n+1}_{r_1}(x_1),B^{n+1}_{r_2}(x_2),\cdots , B^{n+1}_{r_N}(x_N)\) with \(x_i\in B^{n+1}_{\frac{7}{2}}(0)\) and \(0\lt{}r_i\lt{}\frac{1}{4}\) for each \(1\leq i\leq N\), such that

\begin{equation} \tag{4.3.4} \mathrm{sing}M\cap B^{n+1}_3(0)\subset \bigcup_{i=1}^N B^{n+1}_{r_i}(x_i),\quad \text{ and }\quad \sum_{i =1}^{N}r_i^{n-2-2\alpha}\leq \varepsilon. \label{eq:pfSingCover} \end{equation}

We choose \(\zeta_i\) and \(\zeta_\varepsilon\) as in the proof of Lemma 4.3.3. For any locally Lipschitz \(\phi\) with compact support in \(B^{n+1}_3(0)\), \(\zeta_\varepsilon \phi\) vanishes near \(\mathrm{sing}M\), allowing us to use (4.3.1) with \(\zeta_\varepsilon \phi\) in place of \(\phi\). Thus, we have

\begin{equation} \tag{4.3.5} \begin{aligned} {} & \int_{M_{\gt{}k}} \left( 1-\frac{k}{u} \right)|\nabla u|^2\zeta_\varepsilon^2\phi^2\nonumber \\ \leq {}& C\int_{M_{\gt{}k}} (u-k)^2\zeta_\varepsilon^2|\nabla \phi|^2+Ck^2\int_{M_{\gt{}k}} \left( (u -k)^{\frac{2}{\alpha}}+k^{\frac{2}{\alpha}} \right) \zeta_\varepsilon^2\phi^2 + C \int_{M_{\gt{}k}} (u-k)^2|\nabla\zeta_\varepsilon|^2\phi^2, \label{eq:pfWeakPoincareApprox} \end{aligned} \end{equation}

by the Cauchy-Schwarz inequality.

For the first two terms on the right-hand side of (4.3.5), since \(|A|^{2\alpha}\) and \(|A|^2\) are integrable in \(B^{n+1}_3(0)\cap M\) by Lemma 4.3.3, we can let \(\varepsilon\to 0^+\), leading to

\[ C \int_{M_{\gt{}k}} (u-k)^2|\nabla \phi|^2+Ck^2\int_{M_{\gt{}k}} \left( (u -k)^{\frac{2}{\alpha}}+k^{\frac{2}{\alpha}} \right) \phi^2. \]

Then, we need to show

\[ \lim_{\varepsilon\to 0^+} \int_{M_{\gt{}k}} (u-k)^2|\nabla\zeta_\varepsilon|^2\phi^2=0. \]

Applying Lemma 4.3.3, (4.3.4), and Hölder inequality, we obtain

\[ \begin{aligned} {} & \int_{M_{\gt{}k}} (u-k)^2|\nabla \zeta_\varepsilon|^2\phi^2\\ \leq {}& \sum_{i=1}^{N}\|\phi\|_{L^\infty(B^{n+1}_3(0))}^2\int_{M\cap B^{n+1}_{r_i}(x_i)} |A|^{2\alpha}|\nabla \zeta_i|^2\\ \leq{}& \|\phi\|_{L^\infty(B^{n+1}_3(0))}^2\sum_{i=1}^{N}\left( \int_{M\cap B^{n+1}_{r_i}(x_i)} |A|^2 \right)^{\alpha}\left( \int_{M\cap B^{n+1}_{r_i}(x_i)} |\nabla \zeta_i|^{\frac{2}{1-\alpha}} \right)^{1-\alpha}\\ \leq {}& C\|\phi\|_{L^\infty(B^{n+1}_3(0))}^2\sum_{i=1}^{N}r_i^{\alpha(n-2)}r_i^{\left(n-\frac{2}{1-\alpha}\right)(1-\alpha)}=C\|\phi\|_{L^\infty(B^{n+1}_3(0))}^2\sum_{i=1}^{N}r_i^{n-2-2\alpha}\\ \leq {}& C\|\phi\|_{L^\infty(B^{n+1}_3(0))}^2\varepsilon. \end{aligned} \]

Hence, letting \(\varepsilon\to 0^+\), we conclude that (4.3.1) holds for any bounded locally Lipschitz \(\phi\) supported in \(B^{n+1}_3(0)\). ◻

The preceding proof also shows that \(|\nabla u|^2\) is integrable in \(B^{n+1}_3(0)\cap M\) and hence \(u\in W^{1,2}(B^{n+1}_3(0)\cap M)\). We now prove Theorem 4.3.1.

Proof. Consider

\[ k_l=d\left( 1 - \frac{1}{2^{l-1}} \right),\quad \text{and}\quad R_l=\frac{1}{2}+\frac{1}{2^l}, \]

for \(0\lt{}d\leq 1\). \(k_l\) increases to \(d\) and \(R_l\) decreases to \(1/2\) as \(l\to \infty.\) For simplicity, we write \(\Omega_l = M\cap \left\{ u\gt{}k_l \right\}\cap B^{n+1}_{R_l}(0)\).

Applying the previous lemma and noting that

\[ 1-\frac{k_l}{u}\geq 1-\frac{k_l}{k_{l+1}}\geq \frac{1}{2^l}, \]

for any \(u\gt{}k_{l+1}\), we have

\[ \begin{aligned} \frac{1}{2^l}\int_{ M_{\gt{}k_{l+1}}} |\nabla u|^2 \phi^2\leq{}& C\left[ \int_{ M_{\gt{}k_{l}}} (u-k_{l})^2|\nabla \phi|^2\right.\\ &\left.+d^2\int_{ M_{\gt{}k_{l}}} (u -k_{l})^{\frac{2}{\alpha}}\phi^2+ d^{2+\frac{2}{\alpha}}\int_{ M_{\gt{}k_l}}\phi^2 \right] . \end{aligned} \]

Using

\[ |\nabla((u-k_{l+1})\phi)|^2\leq 2|\nabla u|^2\phi^2+2(u-k_{l+1})^2|\nabla \phi|^2, \]

we obtain

\[ \begin{aligned} {} & \int_{ M_{\gt{}k_{l+1}}} |\nabla ((u-k_{l+1})\phi)|^2\\ \leq {}& 2^l C\left[ \int_{ M_{\gt{}k_{l}}} (u-k_{l})^2|\nabla \phi|^2+d^2\int_{ M_{\gt{}k_{l}}} (u -k_{l})^{\frac{2}{\alpha}}\phi^2+ d^{2+\frac{2}{\alpha}}\int_{ M_{\gt{}k_l}}\phi^2 \right] . \end{aligned} \]

Now choose \(\phi\) supported in \(B^{n+1}_{R_l}(0)\), with \(\phi=1\) on \(B^{n+1}_{R_{l+1}}(0)\), \(|\nabla \phi|\leq 2^{l+2}\), and \(0\leq \phi \leq 1\). Together with the Michael–Simon inequality [MS73]

\[ \left( \int_{M} |\varphi|^{\frac{2n}{n-2}} \right)^{\frac{n-2}{n}}\leq C \int_{M} \left|\nabla \varphi\right|^2, \]

for a constant \(C\) only depending on \(n\). Then, we have

\begin{equation} \tag{4.3.6} \begin{aligned} {} & \left( \int_{ \Omega_{l+1}} (u-k_{l+1})^{\frac{2n}{n-2}} \right)^{\frac{n-2}{n}}\nonumber \\ \leq {}& C^l\left[ \int_{ \Omega_l} (u-k_{l})^2+d^2\int_{ \Omega_l} (u -k_{l})^{\frac{2}{\alpha}}+ d^{2+\frac{2}{\alpha}}\mathcal{L}^n(\Omega_l) \right]. \label{eq:pfIterEst} \end{aligned} \end{equation}

Using the fact that when \(u\geq k_l\), we know \(u-k_{l-1}\geq \frac{d}{2^{l-1}}\). Hence, for any \(0\leq \beta\leq \frac{2n}{n-2}\),

\[ \begin{aligned} \int_{ \Omega_l} (u-k_l)^\beta\leq{} & \int_{ \Omega_l}(u-k_l)^\beta \left( \frac{2^{l-1}}{d} \right)^{\frac{2n}{n-2}-\beta}(u-k_{l-1})^{\frac{2n}{n-2}-\beta}\\ \leq{}& \frac{C^l}{d^{\frac{2n}{n-2}-\beta}}\int_{ \Omega_{l-1}} (u-k_{l-1})^{\frac{2n}{n-2}}, \end{aligned} \]

where constant \(C=C(n)\). Note that since \(\frac{2}{\alpha}\lt{} \frac{2n}{n-2}\), we can use the above inequality with \(\beta=0,2\), and \(\frac{2}{\alpha}\) in (4.3.6) to obtain

\begin{equation} \tag{4.3.7} S_{l+1}^{\frac{n-2}{n}}\leq C^l\left( \frac{1}{d^{\frac{2n}{n-2}-2}}+\frac{1}{d^{\frac{2n}{n-2}-\frac{2}{\alpha}-2}}+\frac{1}{d^{\frac{2n}{n-2}-2-\frac{2}{\alpha}}} \right)S_{l-1}, \label{eq:pfIterRaw} \end{equation}

where

\[ S_l:= \int_{\Omega_l} (u-k_{l})^{\frac{2n}{n-2}}. \]

Using \(d\leq 1\), (4.3.7) implies

\begin{equation} \tag{4.3.8} \frac{S_{l+1}}{d^{\frac{2n}{n-2}}}\leq C^l \left( \frac{S_{l-1}}{d^{\frac{2n}{n-2}}} \right)^{\frac{n}{n-2}}, \label{eq:pfIter} \end{equation}

for some \(C=C(n,\bar{n},\Lambda,\alpha)\). Iterating (4.3.8), we obtain

\[ \frac{S_{2l+1}}{d^{\frac{2n}{n-2}}}\leq C^{2+\frac{4(n-2)}{n}+\cdots + 2l (\frac{n-2}{n})^{l-1}}\left( C^2\frac{S_1}{d^{\frac{2n}{n-2}}} \right)^{\left( \frac{n}{n-2} \right)^l}\leq C^{\frac{n^2}{2}}\left( C^2\frac{S_1}{d^{\frac{2n}{n-2}}} \right)^{\left( \frac{n}{n-2} \right)^l}. \]

Hence, if we require

\[ S_1\leq (\varepsilon' d)^{\frac{2n}{n-2}}, \]

for some positive \(\varepsilon'\) only depending on \(n\), \(\bar{n}\), \(\delta\), \(\Lambda\), and \(\alpha\), then we have \(\lim_{l\to \infty} S_{2l+1}=0\). This implies

\[ \left|A\right|^\alpha\leq d \text{ on }B^{n+1}_{\frac{1}{2}}(0). \]

Finally, we need to ensure \(S_1\leq (\varepsilon' d)^{\frac{2n}{n-2}}\).

Using Lemma 4.3.2 with \(k=0\) and a suitable test function, we obtain

\[ \int_{M\cap B^{n+1}_{\frac{3}{2}}(0)} |\nabla u|^2 \leq C\int_{M\cap B^{n+1}_2(0)}u^2 \]

for some \(C=C(n,\bar{n},\Lambda,\alpha)\). Thus, by Michael–Simon’s inequality, we have

\begin{equation} \tag{4.3.9} S_1^{\frac{n-2}{n}}\leq C \int_{M\cap B^{n+1}_1(0)} \left|\nabla (u\varphi)\right|^2 \leq C \int_{M\cap B^{n+1}_2(0)} u^2=C\int_{M\cap B^{n+1}_2(0)} |A|^{2\alpha}. \label{eq:pfMSLastEst} \end{equation}

for \(\varphi\) supported on \(B^{n+1}_{\frac{3}{2}}(0)\), equal to \(1\) on \(B^{n+1}_1(0)\), and \(|\nabla \varphi|\leq 4\), where \(C=C(n,\bar{n},\Lambda,\alpha)\). Now choose \(\varepsilon=\frac{(\varepsilon')^2}{C}\), where \(C\) is the constant in (4.3.9), and set \(d =\sqrt{\frac{1}{\varepsilon}\int_{M\cap B^{n+1}_2(0)} |A|^{2\alpha}} \in (0,1]\) by assumption. Consequently, \(S_1^{\frac{n-2}{n}}\leq (\varepsilon' d)^2\) holds by (4.3.9). For such a choice of \(d\), we know \(\lim_{l\to \infty} S_l=0\), which implies

\[ \sup_{B^{n+1}_{\frac{1}{2}}(0)\cap M}|A|^{2\alpha}\leq d^2 =C \int_{M\cap B^{n+1}_2(0)\cap M} |A|^{2\alpha} \]

for some \(C=C(n,\bar{n},\Lambda,\alpha)\). ◻

With this \(\varepsilon\)-regularity theorem, we can prove the following results.

Proposition 4.3.4

Let \(n\geq 3\), and \(\bar{n}\lt{}n-4+\frac{4}{n}\). Suppose \(M_j\) is a sequence of immersed, two-sided, stable minimal hypersurfaces in \(B^{n+1}_4(0)\) with \(\mathrm{dim}(\mathrm{sing}(M_j)\cap B^{n+1}_4(0))\leq \bar{n}\), and that \(M_j\) converges (as varifolds) to \(q|P\cap B^{n+1}_4(0)|\) as \(j\to \infty\), where \(P\) is a hyperplane and \(q\) is a positive integer. Then,

\[ \lim_{j\to \infty} \sup_{B^{n+1}_{\frac{1}{2}}(0)\cap M_j}|A_{M_j}|=0. \]

Moreover, \(\mathrm{sing} M_j\cap B^{n+1}_{\frac{1}{4}}(0)=\emptyset\) and \(M_j\cap B^{n+1}_{\frac{1}{4}}(0)\) has exactly \(q\) connected components for \(j\) large enough, and each component of \(M_j\cap B^{n+1}_{\frac{1}{4}}(0)\) converges smoothly to \(P\) in \(B^{n+1}_{\frac{1}{4}}(0)\) as smooth immersions.

Proof. We suppose \(P=\left\{ x_{n+1}=0 \right\}\).

We claim that

\[ \int_{ M_j\cap B_2(0)} |A_j|^{2\alpha} \to 0. \]

We also use the following theorem due to Schoen–Simon [SS81].

Theorem 4.3.5

We have the following inequality

\[ \int_{ } |A|^{2}\varphi^{2}\leq C \int_{ } |\nabla \varphi|^{2} (1-(\nu\cdot e_{n+1})^{2}) \]

By the monotonicity formula, we have

\[ \lim_{j\to \infty} \sup_{B^{n+1}_3(0)\cap M_j}|x_{n+1}|=0. \]

Otherwise, we can find a sequence \(p_j \in M_j\) such that \(|p_{j,n+1}|\geq \delta\gt{}0\) for some \(0\lt{}\delta\lt{}1\). By the monotonicity formula, we have

\[ \mathcal{H}^n(M_j\cap B^{n+1}_{\frac{\delta}{2}}(p_j))\geq C \delta^n. \]

Then, we have

\[ q|P\cap B^{n+1}_4(0)|(B_{\delta}(p_0))\geq \limsup_{j \to \infty}V_j(B_{\frac{\delta}{2}}(p_j))\geq C \delta^n, \]

where \(|p_{0,n+1}|\geq \delta\), a contradiction.

Now, we choose \(\varphi^{2}x_{n+1}e_{n+1}\) as a test vector field in the first variation formula where \(\varphi\) is a smooth function supported in \(B^{n+1}_3(0)\). Then, we have

\[ \int_{ } \varphi^{2} |e_{n+1}^{\top}|^{2}=-2\int_{ } x_{n+1}\varphi \nabla \varphi \cdot e_{n+1}^{\top}\leq \frac{1}{2}\int_{ } \varphi^{2} |e_{n+1}^{\top}|^{2}+2\int_{ } |\nabla \varphi|^{2}|x_{n+1}|^{2}. \]

So we have

\[ \int_{ M_j} \varphi^{2}(1-(\nu\cdot e_{n+1})^{2})\leq 4\int_{ M_j} |\nabla \varphi|^{2}|x_{n+1}|^{2}, \]

which goes to zero as \(j\to \infty\). Hence

\[ \int_{ M_j \cap B_2(0)}|A_j|^{2} \to 0. \]

Now, we use Hölder’s inequality to get

\[ \int_{ M_j \cap B_2(0)}|A_j|^{2\alpha}\leq \left( \int_{ M_j \cap B_2(0)}|A_j|^{2} \right)^{\alpha}\left( \mathcal{H}^n(M_j\cap B_2(0)) \right)^{1-\alpha}\to 0. \]

Now we apply the \(\varepsilon\)-regularity theorem (Theorem 4.3.1) to get

\begin{equation} \tag{4.3.10} \lim_{j\to \infty} \sup_{B^{n+1}_{\frac{1}{2}}(0)\cap M_j}|A_{M_j}|=0. \label{eq:pfLimitAconverges} \end{equation}

Next, let us denote \(S_j=P(\mathrm{sing}(M_j))\) where \(P\) is the projection to \(\left\{ x_{n+1}=0 \right\}\). Then the projection \(P\) gives a covering map from \(M_j\cap (B_{\frac{1}{4}}^n(0)\setminus S_j) \times \mathbb{R}\) to \((B_{\frac{1}{4}}^n(0)\backslash S_j) \times \left\{ 0 \right\}\), and the covering degree is \(q\) by (4.3.10) for \(j\) large enough. Since \(B_{\frac{1}{4}}^n(0)\backslash S_j\) is simply connected (because \(\mathrm{dim}(S_j)\leq \bar{n}\)), we know that \(M_j\cap (B_{\frac{1}{4}}^n(0)\setminus S_j) \times \mathbb{R}\) has exactly \(q\) connected components, and each component can be written as a graph of a smooth function over \(B_{\frac{1}{4}}^n(0)\backslash S_j\). By the removable singularity theorem (cf. [DGS65, Sim77]), we know that such a function can be extended to a smooth function on \(B_{\frac{1}{4}}^n\) which solves the minimal surface equation. Hence, \(\mathrm{sing}M_j \cap B_{\frac{1}{4}}^n(0)\times \mathbb{R}=\emptyset\) and it can be decomposed into \(q\) connected components, each of which converges smoothly to \(B_{\frac{1}{4}}^n(0)\times \left\{ 0 \right\}\) as smooth immersions by standard PDE theory. ◻

We consider a flat cone \(\boldsymbol{C}\) in \(\mathbb{R}^{n+1}\) defined as a union of hyperplanes and half-hyperplanes. Explicitly, we write

\[ \boldsymbol{C}:=\sum_{i =1}^{N_1}p_i|P_i|+\sum_{i=1}^{N_2}q_i|H_i|, \]

for \(\{ p_i \}_{i=1}^{N_1}\), \(\{ q_i \}_{i=1}^{N_2}\subset \mathbb{N}\). Here \(\{ P_i \}_{i=1}^{N_1}\) are distinct hyperplanes and \(\{ H_i \}_{i=1}^{N_2}\) are distinct half-hyperplanes such that \(0 \in P_i\) for each \(1\leq i\leq N_1\), \(0 \in \bar{H}_i\) for each \(1\leq i\leq N_2\), and \(H_j\nsubseteq P_i\) for each \(1\leq i\leq N_1\) and \(1\leq j\leq N_2\).

We denote the singular set of \(\boldsymbol{C}\) in the embedded sense as \(T(\boldsymbol{C})\), which is precisely defined as:

\[ T(\boldsymbol{C}):=\left\{ x \in \mathrm{spt}\|\boldsymbol{C}\|: \mathrm{spt}\|\boldsymbol{C}\| \text{ is not part of a hyperplane near }x\right\}. \]

We denote \(T_\tau(\boldsymbol{C})\) as the \(\tau\)-neighborhood of \(T(\boldsymbol{C})\) for \(\tau\gt{}0\).

Proposition 4.3.6

Let \(n\geq 3\), and \(\bar{n}\lt{}n-4+\frac{4}{n}\). Suppose \(M_j\) is a sequence of smooth, immersed, two-sided stable minimal hypersurfaces in \(B^{n+1}_4(0)\) with \(\mathrm{dim}(\mathrm{sing}(M_j)\cap B^{n+1}_4(0))\leq \bar{n}\), such that \(M_j\) converge (as varifolds) to \(\boldsymbol{C}\lfloor(B^{n+1}_4(0))\) as \(j\to \infty\). Then,

\[ \lim_{j\to \infty} \sup_{B^{n+1}_{\frac{1}{2}}(0)\cap M_j}|A_{M_j}|=0. \]

In particular, \(\boldsymbol{C}\) is a sum of hyperplanes with multiplicities, \(\mathrm{sing} M_j\cap B^{n+1}_{\frac{1}{4}}(0)=\emptyset\), and \(M_j\cap B^{n+1}_{\frac{1}{4}}(0)\) converges smoothly to \(\boldsymbol{C}\) in \(B^{n+1}_{\frac{1}{4}}(0)\) as immersions with \(q\) connected components, where \(q=\Theta(\|\boldsymbol{C}\|,0)\).

Proof. For each fixed \(\tau\gt{}0\), we know that \(M_j\cap B^{n+1}_3(0)\) converges smoothly to \(\boldsymbol{C}\lfloor (B^{n+1}_3(0)\backslash T_\tau(\boldsymbol{C}))\) as \(j\to \infty\) by Proposition 4.3.4. For each \(\tau\gt{}0\), by the proof of Proposition 4.3.4, we know

\[ \lim_{j\to \infty} \int_{ B^{n+1}_2(0)\cap M_j\backslash T_\tau(\boldsymbol{C})} |A_{M_j}|^{2\alpha}=0. \]

Now take \(k=0\), and let \(\phi\) be a nonnegative cutoff function supported in \(B^{n+1}_{\frac{3}{2}}(0)\), equal to \(1\) in \(B^{n+1}_1(0)\), with \(|\nabla \phi|\leq 4\). Applying Lemma 4.3.2 together with the Michael–Simon inequality [MS73], we have

\begin{equation} \tag{4.3.11} \begin{aligned} &\int_{ M_j\cap B^{n+1}_1(0)}|A_{M_j}|^{\frac{2\alpha n}{n-2}}\leq C \left( \int_{ M_j\cap B^{n+1}_{\frac{3}{2}}(0)} |A_{M_j}|^{2\alpha} \right)^{\frac{n}{n-2}}\nonumber \\ \leq{}& C \left( \int_{ M_j\cap B^{n+1}_{\frac{3}{2}}(0)} |A_{M_j}|^{2} \right)^{\frac{n\alpha}{n-2}}(\mathcal{H}^n(M_j\cap B^{n+1}_{\frac{3}{2}}(0)))^{\frac{n(1-\alpha)}{n}}, \label{eq:pfSecondUniBound} \end{aligned} \end{equation}

for some \(C=C(n)\). Note that the stability condition and Lemma 4.3.3 imply that the right-hand side of (4.3.11) is uniformly bounded. Hence,

\[ \sup_{j\gt{}0}\int_{ M_j\cap B^{n+1}_1(0)} |A_{M_j}|^{\frac{2\alpha n}{n-2}}\lt{}\infty. \]

By Hölder’s inequality, we have

\[ \begin{aligned} &\int_{ M_j\cap T_\tau(\boldsymbol{C}) \cap B^{n+1}_1(0)}|A_{M_j}|^{2\alpha}\\ \leq{} & \left( \int_{ M_j\cap T_\tau(\boldsymbol{C})\cap B^{n+1}_1(0)}|A_{M_j}|^{\frac{2\alpha n}{n-2}} \right)^{\frac{n-2}{n}}\mathcal{H}^n(M_j\cap T_\tau(\boldsymbol{C})\cap B^{n+1}_1(0))^{\frac{2}{n}}.\\ \end{aligned} \]

By a standard covering argument using the monotonicity formula, we know that

\[ \mathcal{H}^n(M_j\cap T_\tau(\boldsymbol{C}) \cap B^{n+1}_2(0))\leq C\tau, \]

for some \(C=C(n,\Theta(\|\boldsymbol{C}\|,0))\) for \(j\) large enough. Hence, we have

\begin{equation} \tag{4.3.12} \lim_{j\to \infty} \int_{ B^{n+1}_1(0)\cap M_j} |A_{M_j}|^{2\alpha}=\lim_{j\to \infty} \int_{ B^{n+1}_1(0)\cap M_j\cap T_\tau(\boldsymbol{C})} |A_{M_j}|^{2\alpha}\leq C\tau^{\frac{2}{n}}, \label{eq:pfAlphaTau} \end{equation}

for some \(C\lt{}\infty\) which is independent of \(\tau\). Since the left-hand side of (4.3.12) is independent of \(\tau\), and \(\tau\) is arbitrary, we obtain

\[ \lim_{j\to \infty} \int_{ B^{n+1}_1(0)\cap M_j} |A_{M_j}|^{2\alpha}=0. \]

Thus, we apply the \(\varepsilon\)-regularity theorem (Theorem 4.3.1) to conclude that

\[ \lim_{j\to \infty} \sup_{B^{n+1}_{\frac{1}{2}}(0)\cap M_j}|A_{M_j}|=0, \]

which implies that each connected component of \(M_j\cap B^{n+1}_{\frac{1}{2}}(0)\) converges to a hyperplane in the varifold sense. Furthermore, by Proposition 4.3.4, for \(j\) large enough, \(\mathrm{sing}M_j\cap B^{n+1}_{\frac{1}{4}}(0)\) is empty, and \(M_j\cap B^{n+1}_{\frac{1}{4}}(0)\) has exactly \(q\) connected components, each converging smoothly to a hyperplane in \(B^{n+1}_{\frac{1}{4}}(0)\). ◻

We now prove Theorem 4.1.8. By Allard’s compactness theorem (cf. Theorem 3.1.15), we obtain a stationary integral varifold \(V\) in \(B^{n+1}_4(0)\) such that, up to a subsequence, \(|M_j|\) converges to \(V\) in the varifold sense. Let \(S=\mathrm{sing}\|V\|\) be the singular point set of \(V\). We need to analyze the tangent cone \(\boldsymbol{C}\) of \(V\) at \(x_0\in S\cap B^{n+1}_{\frac{1}{2}}(0)\). Indeed, we have the following lemma.

Lemma 4.3.7

For any \(\boldsymbol{C} \in \mathrm{VarTan}(V,x_0)\) for \(x_0 \in S \cap B^{n+1}_{\frac{1}{2}}(0)\), we can write \(\boldsymbol{C}=\boldsymbol{C}'\times \mathbb{R}^{n-p}\) for some \(p\geq 7\).

Proof. For any cone \(\boldsymbol{C}\), we write \(\mathcal{S}(\boldsymbol{C})\) (the spine of \(\boldsymbol{C}\)) to be the linear subspace containing all \(x\in \mathbb{R}^{n+1}\) such that \(\boldsymbol{C}\) is invariant under the translation along the line spanned by \(x\). For any \(x_0 \in S\), we introduce the notion of iterated tangents of \(V\) at \(x_0\) as follows. We say a collection of cones \(\left\{ \boldsymbol{C}_1,\boldsymbol{C}_2,\cdots ,\boldsymbol{C}_N \right\}\) is iterated tangents of \(V\) at \(x_0\) if \(\boldsymbol{C}_1\) is the tangent cone of \(V\) at \(x_0\), and \(\boldsymbol{C}_{j+1}\) is the tangent cone of \(\boldsymbol{C}_j\) at \(x_j \in \mathrm{sing}\|\boldsymbol{C}_j\|\backslash \mathcal{S}(\boldsymbol{C}_j)\) for \(1\leq j\leq N-1\). Moreover, we can choose iterated tangents satisfying the following properties:

Each \(\boldsymbol{C}_j\) is not smoothly immersed (i.e., \(\mathrm{sing}\|\boldsymbol{C}_j\|\neq \emptyset\)).
\(\mathrm{dim}(\mathcal{S}(\boldsymbol{C}_{j+1}))\gt{}\mathrm{dim}(\mathcal{S}(\boldsymbol{C}_{j}))\) for each \(j=1,2,\cdots ,N-1\).
\(\boldsymbol{C}_N=\boldsymbol{C}'\times \mathbb{R}^{\mathrm{dim}(\mathcal{S}(\boldsymbol{C}_N))}\) where \(\boldsymbol{C}'\backslash \left\{ 0 \right\}\) is a smooth immersed cone after a suitable rotation in \(\mathbb{R}^{n+1}\).
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 Proposition 4.3.4 and Proposition 4.3.6.

In particular, the fourth condition implies that the smooth immersed part of \(\boldsymbol{C}_j\) is stable, and the second condition implies \(N\) is a finite number.

The first three conditions are immediate from properties of tangent cones. The only nontrivial part is the last condition, which can be proved by induction on \(j\). Suppose we have found \(y_k,r_k\) such that \(\eta_{y_k,r_k}(M_k)\) converges to \(\boldsymbol{C}_j\) in the sense of varifolds. Then, by the choice of \(\boldsymbol{C}_{j+1}\), we know there exists \(\rho_k\) such that \(\eta_{x_j,\rho_k}(\boldsymbol{C}_j)\) converges to \(\boldsymbol{C}_{j+1}\) in the sense of varifolds. Thus, with \(z_k=y_k+r_k x_j\) and \(s_k=r_k \rho_k\), we have \(\eta_{z_k,s_k}(M_k)\) converging to \(\boldsymbol{C}_{j+1}\) in the varifold sense.

Now, let us determine the dimension of \(\boldsymbol{C}'\). Note that \(\boldsymbol{C}_N\) cannot be a hyperplane by the first condition.

If the dimension of \(\boldsymbol{C}'\) is one, then \(\boldsymbol{C}_N\) is the sum of distinct half-hyperplanes with multiplicity. But by Proposition 4.3.6, we know \(\boldsymbol{C}_N\) can only be a sum of hyperplanes with multiplicity, which contradicts the first condition.

Therefore, we know \(\boldsymbol{C}'\) has dimension at least two. But the fourth condition implies that \(\boldsymbol{C}'\) is a smooth immersed stable cone away from \(\left\{ 0 \right\}\), and hence, \(\boldsymbol{C}'\) has dimension at least \(7\).

Hence, by the second condition, we obtain \(\mathrm{dim}(\boldsymbol{C})\geq n-7\) for any \(\boldsymbol{C}\in \mathrm{VarTan}(V,x_0)\), and the lemma follows. ◻

Now, we are ready to finish the proof of Theorem 4.1.8.

Theorem 4.3.8

Let \(\mathcal{V}\) be the collection of all the limit varifolds defined in Theorem 4.1.8. Then,

\[ \operatorname{dim}(\mathrm{sing}(\|V\|\cap B_1))\leq n-7. \]

In particular, if \(n=7\), then \(\mathrm{sing}(\|V\|\cap B_1)\) is discrete.

Proof. We denote \(F^l=\{ V \in \mathcal{V}: \mathcal{H}^l(\mathrm{sing}\cap B_1)\gt{}0 \}\).

Proposition 4.3.9

For each \(V \in F^l\), there exists \(\boldsymbol{C} \in \mathrm{VarTan}(V,x)\cap F^l\) for \(\mathcal{H}^l-\)a.e. \(x \in \mathrm{sing}(\|V\|)\cap B_1\).

Proof. Recall that we actually have for \(\mathcal{H}^l\)-a.e. \(x \in \mathrm{sing}(\|V\|)\cap B_1\), we have

\[ \limsup_{r \to 0} \frac{\mathcal{H}^l_\infty(\text{sing}\|V\|\cap B_r(x))}{\omega_n r^l}\gt{}0. \]

We choose \(r_i \to 0\) such that

\[ \lim_{i \to \infty} \frac{\mathcal{H}^l_\infty(\text{sing}\|V\|\cap B_{r_i}(x))}{\omega_n r_i^l}\gt{}0. \]

By taking a subsequence, we can assume \((\eta_{x,r_i})_{\#}V\) converges to \(\boldsymbol{C} \in \mathrm{VarTan}(V,x)\).

If \(\mathcal{H}^l_\infty(\mathrm{sing}\,\|\boldsymbol{C}\|)=0\), then for any \(\varepsilon\gt{}0\), we can find a covering of \(\mathrm{sing}\,\|\boldsymbol{C}\|\) by balls \(\{ B_{s_j}(y_j) \}_{j=1}^\infty\) such that

\[ \sum_{j=1}^\infty s_j^l \lt{} \varepsilon. \]

Note that \(\mathrm{sing}\,\|\boldsymbol{C}\|\cap B_1(0)\) is compact, we know \(\mathrm{sing}\,\|(\eta_{x,r_i})_{\#}V\|\cap B_1(0)\) can also be covered by \(\{ B_{s_j}(y_j) \}_{j=1}^\infty\) for \(i\) large enough. Thus, we have

\[ \frac{\mathcal{H}^l_\infty(\mathrm{sing}\,\|V\|\cap B_{r_i}(x))}{r_i^l}\lt{}\varepsilon \]

for \(i\) large enough, which is a contradiction. ◻

Now we apply the above proposition iteratively to obtain a sequence of varifolds \(\{ V_k \}_{k=0}^{K}\) such that:

\(V_0=V\).
\(V_{k+1} \in \mathrm{VarTan}(V_k,x_k)\) for some \(x_k \in \mathrm{sing}(\|V_k\|)\cap B_1 \backslash \mathcal{S}(V_k)\).
\(\operatorname{dim}(\mathcal{S}(V_{k+1}))\gt{}\operatorname{dim}(\mathcal{S}(V_k))\) for each \(0\leq k \leq K-1\).
\(\mathcal{H}^l(\mathrm{sing}(\|V_k\|)\cap B_1)\gt{}0\) for each \(0\leq k \leq K\).
\(V_K=\boldsymbol{C}\times \mathbb{R}^{m}\) where \(\boldsymbol{C}\backslash \{0\}\) is a smooth immersed cone for some \(m\geq 0\).

In particular, the last two conditions imply \(m=l\). By Lemma 4.3.7, we know \(l \leq n-7\).

In the case \(n=7\), we have

\[ \mathcal{H}^\alpha(\mathrm{sing}\,\|V_K\|\cap B_1)=0 \]

for any \(\alpha\gt{}0\) by the above result.

If \(\mathrm{sing}(\|V\|\cap B_1)\) is not discrete, then we can find \(x_j \in \mathrm{sing}(\|V\|\cap B_1)\) such that \(x_j \to x_0 \in \mathrm{sing}(\|V\|\cap B_1)\). Now, up to a subsequence, we can assume \((\eta_{x_0,|x_j-x_0|})_{\#}V\) converges to \(\boldsymbol{C} \in \mathrm{VarTan}(V,x_0)\) and we denote \(\xi = \lim_{j \to \infty} \frac{x_j-x_0}{|x_j-x_0|} \neq 0\). So \(\mathcal{S}(\boldsymbol{C})\) contains the line spanned by \(\xi\). In particular, \(\mathcal{H}^1(\mathrm{sing}(\|\boldsymbol{C}\|)\cap B_1)\gt{}0\) which is a contradiction. Hence, \(\mathrm{sing}(\|V\|\cap B_1)\) is discrete. ◻

Bellettini’s Sheeting Theorem and Schoen–Simon Regularity

Mon, 29 Jun 2026 00:00:00 +0000

Theorem 4.4.1

Suppose \(M\) is a properly immersed, two-sided, stable minimal hypersurface in \(B^{n+1}_4(0)\) with \(\mathcal{H}^{n-2}(\mathrm{sing}(M))\lt{}\infty\) and

\[ \mathcal{H}^n(M\cap B^{n+1}_4(0))\leq \Lambda . \]

Then there exists \(\varepsilon=\varepsilon(n,\Lambda)\gt{}0\) such that if

\[ \int_{M\cap B^{n+1}_4(0)} \text{dist}(x,P)^2 d\mathcal{H}^n \leq \varepsilon \]

for \(P=\{ x_{n+1}=0 \}\), then \(\pi:M \cap B^n_1(0)\times \mathbb{R}\backslash \pi^{-1}(\Sigma)\to B^n_1(0)\backslash \Sigma\) is a smooth projection map, where \(\Sigma\) is the projection of \(\text{sing}(M)\) to \(P\) in \(B^n_1(0)\).

In particular, if \(\mathcal{H}^{n-2}(\text{sing}(M))=0\), then \(\Sigma=\emptyset\) and \(M\cap B^n_1(0)\times \mathbb{R}\) is a union of minimal graphs over \(B^n_1(0)\).

Theorem 4.4.2

Suppose \(M\) is a properly immersed, two-sided, stable minimal hypersurface in \(B^{n+1}_4(0)\) with \(\mathcal{H}^{n-2}(\mathrm{sing}(M))\lt{}\infty\) and

\[ \mathcal{H}^n(M\cap B^{n+1}_4(0))\leq \Lambda . \]

Then, there exists \(\varepsilon=\varepsilon(n,\Lambda)\gt{}0\) such that if

\[ \int_{M\cap B^{n+1}_4(0)} |x_{n+1}|^2 d\mathcal{H}^n \leq \varepsilon, \]

then

\[ g(x)\leq C \left( \int_{M\cap B_{2}(0)} g^{2}(x) \right)^{\frac{4}{4+n}}. \]

Here, \(g(x)=\sqrt{1-(\nu \cdot e_{n+1})^{2}}\) where \(\nu\) is the unit normal vector of \(M\).

Geometric meaning of \(g\).

Definition 4.4.3

The tilt excess of \(M\) in \(B^{n+1}_r(0)\) with respect to \(P=\{ x_{n+1}=0 \}\) is

\[ E_M(r,P):=r^{-n}\int_{M\cap B^{n+1}_r(0)} |T_x M - P|^{2} d\mathcal{H}^n. \]

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

\[ |T_xM-P|^{2}\simeq |\nu-e_{n+1}|^{2}|\nu+e_{n+1}|^{2}=4(1-(\nu \cdot e_{n+1})^{2})=4g^{2}. \]

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

Lemma 4.4.4

For any \(k\in [0,\frac{1}{2n}]\), we have

\[ \frac{1}{2n}\int_{ \{ g\gt{}k \}} |\nabla g|^{2} \left( 1-\frac{k}{g} \right) \phi^{2} \leq \int_{\{ g\gt{}k \} } (g-k) ^{2}|\nabla \phi|^{2}. \]

for any Lipschitz function \(\phi\) supported in \(B^{n+1}_{\frac{3}{2}}(0)\).

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

\[ g\Delta g = - \frac{|\nabla g|^{2}}{1-g^{2}}+|A|^{2}(1-g^{2}) \]

and

\[ \frac{|\nabla g|^{2}}{1-g^{2}}\leq \frac{n-1}{n}|A|^{2}, \]

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

\[ \int_{ } g^{2} \phi^{2} \leq 4 \int_{ } |x_{n+1}|^{2} |\nabla \phi|^{2}. \]

\[ \int_{ B_2} g^{2} \leq C \int_{ B_4} |x_{n+1}|^{2} \leq C \varepsilon \]

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 \(B_1^n(0)\). ◻

Theorem 4.4.5

Let \(\{M_k\}\) be a sequence of embedded, stable, orientable minimal hypersurfaces in \(B_2^{n+1}(0)\) with the following properties:

\(0\in \bar{M}_k\) for each \(k\).
\(\mathcal{H}^{n-2}(\mathrm{sing}M_k)=0\) for each \(k\).
\(\|M_k\|(B_2^{n+1}(0))\leq \Lambda\) for some constant \(\Lambda\gt{}0\) independent of \(k\).

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.

Lemma 4.4.6

For any \(\boldsymbol{C} \in \mathrm{VarTan}(V,x_0)\) for \(x_0 \in S \cap B^{n+1}_{\frac{1}{2}}(0)\), we can write \(\boldsymbol{C}=\boldsymbol{C}'\times \mathbb{R}^{n-p}\) for some \(p\geq 7\).

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:

\(x_j \in \mathrm{sing}\|\boldsymbol{C}_j\|\backslash \mathcal{S}(\boldsymbol{C}_j)\) for each \(1\leq j\leq N-1\) and \(\boldsymbol{C}_{j+1} \in \mathrm{VarTan}(\boldsymbol{C}_j,x_j)\).
Each \(\boldsymbol{C}_j\) is not smoothly embedded (i.e., \(\mathrm{sing}\|\boldsymbol{C}_j\|\neq \emptyset\)).
\(\mathrm{dim}(\mathcal{S}(\boldsymbol{C}_{j+1}))\gt{}\mathrm{dim}(\mathcal{S}(\boldsymbol{C}_{j}))\) for each \(j=1,2,\cdots ,N-1\).
\(\boldsymbol{C}_N=\boldsymbol{C}'\times \mathbb{R}^{\mathrm{dim}(\mathcal{S}(\boldsymbol{C}_N))}\) where \(\boldsymbol{C}'\backslash \left\{ 0 \right\}\) is a smooth embedded cone after a suitable rotation in \(\mathbb{R}^{n+1}\).
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

\[ \boldsymbol{C}_N=\sum_{i=1}^{N_1} q_i \{ (\cos \theta_i r, \sin \theta_i r, y): r \geq 0, y \in \mathbb{R}^{n-1} \}. \]

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

\[ \hat{M}_k \cap B^{n+1}_2(0)\backslash T_\tau(\boldsymbol{C}_N) \]

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

\[ \int_{ \gamma \cap \{ r\lt{}\tau \}} |A|\geq \frac{1}{2}|\sin(\theta_1 - \theta_2)|. \]

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

\[ \int_{ \hat{M}_k \cap B_\tau^{2} \times B_1^{n-1}} |A| \geq C \min_{i\neq j}|\theta_i-\theta_j|. \]

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

\[ \begin{aligned} \int_{ \hat{M}_k \cap B_\tau^{2} \times B_1^{n-1}} |A| &\leq \left( \mathcal{H}^n(\hat{M}_k \cap B_\tau^{2} \times B_1^{n-1}) \right)^{\frac{1}{2}} \left( \int_{ \hat{M}_k \cap B_\tau^{2} \times B_1^{n-1}} |A|^{2} \right)^{\frac{1}{2}} \nonumber \\ &\leq C \sqrt{\tau} \end{aligned} \]

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 \(7\). ◻

Sketch of Wickramasekera’s Regularity Theorem

Mon, 29 Jun 2026 00:00:00 +0000

Theorem 4.5.1

Suppose \(V_i\) is a sequence of stationary integral \(n\)-varifolds in \(B_2^{n+1}(0)\) and \(V_i\) also satisfies the following conditions:

\(0\in \mathrm{spt}\|V_i\|\).
\(\|V_i\|(B_2^{n+1}(0))\leq \Lambda\) for some constant \(\Lambda\gt{}0\) independent of \(i\).
(Stability) Each \(V_i\) is stable in \(B_2^{n+1}(0)\) on its regular set, i.e., for any \(\phi \in C_c^1(\mathrm{reg}V_i)\),

\[ \int_{\mathrm{reg}V_i} |A_i|^2 \phi^2 d\|V_i\| \leq \int_{\mathrm{reg}V_i} |\nabla \phi|^2 d\|V_i\|. \]

(Alpha-Structural Hypothesis) There exists \(\alpha\in (0,1)\) such that for each \(i\), no point of \(\mathrm{spt}\|V_i\|\cap B_1^{n+1}(0)\) has a neighborhood in which \(\mathrm{spt}\|V_i\|\) is the union of three or more embedded \(C^{1,\alpha}\) hypersurfaces-with-boundary meeting only along their common boundary.

We write the class of such varifolds as \(\mathcal{V}_\alpha(\Lambda)\).

Theorem 4.5.2

Let \(V_i\in \mathcal{V}_\alpha(\Lambda)\) such that \(V_i \to q|B_2^n(0) \times \{ 0 \}\) in the varifold sense. Then, for \(i\) large enough, we have

\[ V_{i}\left.\right|_{B_1^n \times \mathbb{R}} = \sum_{k=1}^{q} |\text{graph}u_{i,k}|, \]

with \(u_{i,1}\leq \cdots \leq u_{i,q}\) and \(u_{i,k}\) are smooth functions such that

\[ \|u_{i,k}\|_{C^{1,\alpha}(B_1^n)}^{2} \leq C \int_{B_2 } x_{n+1}^{2} d\|V_{i}\| \]

for some constant \(C=C(n,\alpha,\Lambda)\gt{}0\) independent of \(i\).

Theorem 4.5.3

Suppose \(\boldsymbol{C}\) is a classical cone. There is no sequence of varifolds \(V_i\in \mathcal{V}_\alpha(\Lambda)\) such that \(V_i \to \boldsymbol{C}|_{B_2(0)}\) in the varifold sense.

Recall that a classical cone

\[ \boldsymbol{C}:= \sum_{i=1}^{N} q_i |H_i| \]

where \(H_i\) are \(n\)-dimensional half-hyperplanes in \(\mathbb{R}^{n+1}\) such that they contains origin and share the same boundary and \(q_i\) are positive integers.

Sketch of the proof

The two theorems are proved simultaneously by induction on the density of planes and cones. Assume that the sheeting theorem holds for \(q\leq q_0\) and that the minimal distance theorem holds for \(\Theta(\boldsymbol{C},0)\leq q_0\).
Show that minimal distance theorem holds for \(\Theta(\boldsymbol{C},0)\leq q_0+1\). It is enough to treat the densities \(\Theta(\boldsymbol{C},0)=q_0+\frac{1}{2}\) and \(q_0+1\).
Show that the sheeting theorem holds for \(q=q_0+1\).
Dimension reduction argument plus the classification of stable cones implies that the singular set of \(V_\infty\) has Hausdorff dimension at most \(n-7\).

Proof of the minimal distance theorem

Assume the sheeting theorem holds for \(q\leq q_0\) and the minimal distance theorem holds for \(\Theta(\boldsymbol{C},0)\leq q_0\). Together with dimension reduction, this implies the desired regularity whenever \(\Theta(\|V\|,X)\leq q_0\) for all \(X\in B_2(0)\).
Suppose \(V\) is sufficiently close to a classical cone \(\boldsymbol{C}\) in the varifold sense and \(\Theta(\|V\|,0)\geq \Theta(\|\boldsymbol{C}\|,0)\). Then, outside neighborhood of the singular set of \(\boldsymbol{C}\), using the sheeting theorem, we can write \(V\) as a union of smooth graphs over the half-hyperplanes in \(\boldsymbol{C}\).
Establish the \(L^{2}\)-estimate for these graphs and key \(L^{2}\) improvement in a smaller scale, following Simon’s cylindrical-singularity argument [Sim93].
By iteration, we can obtain the set \(\{ \Theta(\|V\|,X)\geq \Theta(\|\boldsymbol{C}\|,0)\}\) is \(C^{1,\alpha}\)-regular for some \(\alpha\gt{}0\), which implies \(V\) has classical singularity at \(0\), which is a contradiction.

Proof of the sheeting theorem

Same induction assumption as above.
Write \(V_i\) as a union of Lipschitz graphs \(u_{i,k}\) over the plane \(\{ x_{n+1}=0 \}\) outside a small bad set.
Consider the blow-up class \(v\), which is the limit of \(E(V_i)^{-1}u_{i,k}\) where \(E(V_i)\) is the \(L^{2}\)-excess of \(V_i\).
(\(\star\)) Prove that \(v=(v_k)\) is harmonic.

First, we need a Lipschitz approximation for stationary varifolds.

Theorem 4.5.4

Fix \(q\in\mathbb{Z}_{\geq 1}\) and \(\sigma\in(0,1)\). Then there exists \(\varepsilon_0=\varepsilon_0(n,q,\sigma)\in(0,1)\) such that the following holds. Let \(V\) be a stationary integral \(n\)-varifold in \(B_2^{n+1}(0)\) such that:

\(\displaystyle \frac{1}{\omega_n 2^n}\,\|V\|\bigl(B_2^{n+1}(0)\bigr) \lt{} q+\frac12\) and \(\displaystyle q-\frac12 \leq \frac{1}{\omega_n}\,\|V\|\bigl(\mathbb{R}\times B_1^n(0)\bigr) \lt{} q+\frac12\);
\(\displaystyle \hat{E}^2(V) := \int_{\mathbb{R}\times B_1^n(0)} |x^1|^2\,\mathrm{d}\|V\|(X) \lt{} \varepsilon_0\).

Then there exists \(\mathcal{H}^n\)-measurable \(\Sigma\subset B_\sigma^n(0)\) such that:

\(\displaystyle \mathcal{H}^n(\Sigma)+\|V\|(\mathbb{R}\times\Sigma) \leq C\,\hat{E}_V^2\);
there exist Lipschitz functions \(u_1,\ldots,u_q:B_\sigma^n(0)\to\mathbb{R}\) with \(\operatorname{Lip}(u_j)\leq\frac12\), \(u_1\leq\cdots\leq u_q\), \(\displaystyle \sup_{B_\sigma^n(0)} |u_j| \leq C\,\hat{E}_V^{\frac{1}{n+1}}\), and

\[ V \mathbin{\llcorner} \bigl(\mathbb{R}\times (B_\sigma^n(0)\setminus\Sigma)\bigr) = \sum_{j=1}^q \bigl|\text{graph}(u_j|_{B_\sigma^n(0)\setminus\Sigma})\bigr|. \]

Here \(C=C(n,q,\sigma)\in(0,\infty)\).

Moreover, we can establish the following estimate

\[ \int_{B_\sigma(0) } \sum_k |u_k|^{2}+|Du_k|^{2} \leq C E^{2}(V). \]

Definition 4.5.5

We consider \(V_i\) as above such that \(\hat{E}^{2}(V_i) \to 0^+\) and \(\sigma_i \to 1\) as \(i\to \infty\). The above estimate implies that \(\hat{u}_k:=E^{-1}u_k\) satisfies

\[ \sum_{k}^{}\|u_k\|_{W^{1,2}(B_{\sigma_i}(0))}^{2} \leq C. \]

Up to a subsequence, we can find a weak limit function \(v\). The blow-up class \(\mathcal{B}_q\) is the collection of all such limits.

Theorem 4.5.6

The blow-up class \(\mathcal{B}_q\) has the following properties:

\(\mathcal{B}_q\subset W^{1,2}_{\mathrm{loc}}(B_1^n(0);\mathbb{R}^q)\cap L^2(B_1^n(0);\mathbb{R}^q)\).
If \(v\in\mathcal{B}_q\), then \(v^1\leq v^2\leq\cdots\leq v^q\).
If \(v\in\mathcal{B}_q\), then \(\Delta v_a=0\) in \(B_1^n(0)\), where \(v_a=q^{-1}\sum_{j=1}^q v^j\).
For each \(v\in\mathcal{B}_q\) and each \(z\in B_1^n(0)\), either (\(\mathcal{B}\)4 I**)** or (\(\mathcal{B}\)4 II**)** below holds:

(\(\mathcal{B}\)4 I**)** (Hardt–Simon inequality [HS79]). For each \(\rho\in\bigl(0,\tfrac{3}{8}(1-|z|)\bigr]\),

\[ \sum_{j=1}^q \int_{B_{\rho/2}^n(z)} R_z^{\,2-n}\left(\frac{\partial}{\partial R_z}\Bigl(\frac{v^j-v_a(z)}{R_z}\Bigr)\right)^{\!2}\,\mathrm{d}x \leq C\rho^{-n-2}\int_{B_\rho^n(z)}\bigl|v-\ell_{v,z}\bigr|^2\,\mathrm{d}x, \]

where \(R_z(x)=|x-z|\), \(\ell_{v,z}(x)=v_a(z)+Dv_a(z)\cdot(x-z)\), and \(v-\ell_{v,z}=\bigl(v^1-\ell_{v,z},\ldots,v^q-\ell_{v,z}\bigr)\).

**(**<span class="math inline">&#92;&#40;&#92;&#109;&#97;&#116;&#104;&#99;&#97;&#108;&#123;&#66;&#125;&#92;&#41;</span><!-- -->4 *II***)** There exists <span class="math inline">&#92;&#40;&#92;&#115;&#105;&#103;&#109;&#97;&#61;&#92;&#115;&#105;&#103;&#109;&#97;&#40;&#122;&#41;&#92;&#105;&#110;&#40;&#48;&#44;&#49;&#45;&#124;&#122;&#124;&#93;&#92;&#41;</span> such that <span class="math inline">&#92;&#40;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#118;&#61;&#48;&#92;&#41;</span> in <span class="math inline">&#92;&#40;&#66;&#95;&#92;&#115;&#105;&#103;&#109;&#97;&#94;&#110;&#40;&#122;&#41;&#92;&#41;</span>.

If \(v\in\mathcal{B}_q\), then:

(\(\mathcal{B}\)5 I**)** \(\displaystyle \tilde{v}_{z,\sigma}(\cdot)\equiv \|v(z+\sigma(\cdot))\|_{L^2(B_1^n(0))}^{-1}\,v(z+\sigma(\cdot))\in\mathcal{B}_q\) for each \(z\in B_1^n(0)\) and \(\sigma\in\bigl(0,\tfrac{3}{8}(1-|z|)\bigr]\) whenever \(v\not\equiv 0\) in \(B_\sigma^n(z)\);

(\(\mathcal{B}\)5 II**)** \(v\circ\gamma\in\mathcal{B}_q\) for each orthogonal rotation \(\gamma\) of \(\mathbb{R}^n\);

(\(\mathcal{B}\)5 III**)** \(\|v-\ell_v\|_{L^2(B_1^n(0))}^{-1}(v-\ell_v)\in\mathcal{B}_q\) whenever \(v-\ell_v\not\equiv 0\) in \(B_1^n(0)\), where \(\ell_v(x)=v_a(0)+Dv_a(0)\cdot x\) for \(x\in\mathbb{R}^n\) and \(v-\ell_v=\bigl(v^1-\ell_v,\ldots,v^q-\ell_v\bigr)\).
If \(\{v_k\}_{k=1}^\infty\subset\mathcal{B}_q\), then there exists a subsequence \(\{k'\}\) of \(\{k\}\) and a function \(v\in\mathcal{B}_q\) such that \(v_{k'}\to v\) locally in \(L^2(B_1^n(0))\) and locally weakly in \(W^{1,2}(B_1^n(0))\).
If \(v\in\mathcal{B}_q\) is such that for each \(j=1,\ldots,q\), there exist linear maps \(L_1^j,L_2^j:\mathbb{R}^n\to\mathbb{R}\) with \(v^j(x^2,y)=L_1^j(x^2,y)\) if \(x^2\gt{}0\), \(v^j(x^2,y)=L_2^j(x^2,y)\) if \(x^2\leq 0\), and \(L_1^j(0,y)=L_2^k(0,y)\) for all \(1\leq j,k\leq q\) and \(y\in\mathbb{R}^{n-1}\), where \((x^2,y)\) are coordinates on \(\mathbb{R}^n\), then \(v^1=v^2=\cdots=v^q=L\) for some linear map \(L:\mathbb{R}^n\to\mathbb{R}\).

Proof.

This a consequence of the definition of the blow-up class.
This is based on \(u_k\leq u_{k+1}\).
This is from the stationary property of \(V_i\).
We can consider a new sequence \(V_i'=(\eta_{z,\sigma})_{\#}V_i\), where its blow-up limit is \(\tilde{v}_{z,\sigma}\). Similarly, we can consider \(V_i'=\gamma_{\#}V_i\), where its blow-up limit is \(v \circ \gamma\). Finally, the last property is from the rotation involving direction \(e_{n+1}\).
This follows from the compactness of the blow-up sequence.
This is one of the key properties of the blow-up class, and the place that stable condition is used. The idea is, either we have good density points accumulating at a given point, which implies the Hardt–Simon inequality. Or we have a density gap, which by our induction assumption, implies that \(v\) is harmonic.
This is the most technical part of the proof. It rules out the possibility that a blow-up class contains the singular model of a classical cone.

◻

Theorem 4.5.7

For any \(v\in\mathcal{B}_q\), we have \(v\) is smooth and harmonic.

Proof. This proof is similar to the classical dimensional reduction argument.

We define the “singular set” of \(v\) as the set of points \(x\in B_1^n(0)\) such that \(\mathcal{B}4 \mathrm{I}\) holds.

First, we can study homogeneous degree-one functions in the blow-up class, namely tangent functions of a given function.

Claim. Each homogeneous 1 function in the blow-up class is a linear function.

Next, we can show that the “singular set” is empty by the dimensional reduction argument. ◻

After proving the blow-up class are all smooth and harmonic, we can show the sheeting theorem holds by using the standard regularity argument.

Proof of the property \(\mathcal{B}7\)

Suppose \(v\in\mathcal{B}_q\) is such that for each \(j=1,\ldots,q\), we can write

\[ v^j(x_1,y)=\lambda^j x_1 \text{ if } x_1\gt{}0, \text{ and } v^j(x_1,y)=\mu^j x_1 \text{ if } x_1\leq 0, \]

for some \(\lambda^j,\mu^j\in\mathbb{R}\).

We define the cone \(\boldsymbol{C}_k\) as the union of the graphs of the functions \(\hat{E}_i \lambda^j x_1\) for \(x_1\gt{}0\) and \(\hat{E}_i \mu^j x_1\) for \(x_1\leq 0\).

We can define the fine excess \(E(V_i)\) as follows:

\[ E(V_i):= \int_{ \mathbb{R} \times B_1^n(0)} d^{2}(X,\boldsymbol{C}_i) d\|V_i\|(X). \]

Clearly, we have \(E(V_i) \leq \hat{E}(V_i)\). But we expect that \(E(V_i)\) is much smaller than \(\hat{E}(V_i)\).

Indeed, we can establish the following under suitable assumptions.

\[ \lim \hat{E}^{-1}(V_i)E(V_i) = 0. \]

By the sheeting theorem, we can write \(V_i\) as a union of smooth graphs over the cones \(\boldsymbol{C}_i\) for \(|x_1|\geq \sigma\).

Now, we define \(h_i\) and \(w_i\) the vector valued function such that

\[ V_i = \sum_{j=1}^{q} |h_i^j+\hat{E}_i \lambda^jx_1| + \sum_{j=1}^{q} |w_i^j+\hat{E}_i \mu^jx_1|, \]

and we expect that \(h_i^j\) and \(w_i^j\) much smaller than \(\hat{E}_i\).

Establish the key \(L^{2}\)-estimate for \(h_i^j,w_i^j\)

Such estimate containst the following

\[ \int_{ B_{\frac{1}{2}}} \frac{|X^\bot|^{2}}{|X|^{n+2}}+\sum_{2\leq j\leq n}^{}|e_j^\bot|^{2}+\frac{d^{2}(X,\mathrm{spt}\,\|C_j\|)}{|X|^{n+2-\mu}} \leq C \hat{E}_i^2. \]

We can also establish the uniform bound for \(E^{-1}(V_i)h_i^j\) and \(E^{-1}(V_i)w_i^j\) and obtain the limit function \(h\) and \(w\). Such limit is called the fine blow-up limit.

Theorem 4.5.8

For the fine blow-up limit \(h\) and \(w\), we know \(h,w\) are at least \(C^{2}\) up to the boundary.

Proof. This relies on the key \(L^{2}\) esimate using the fine excess.

First, we need to show the \(C^{0,\alpha}\) estimate for \(h,w\). We need to use the \(L^{2}\)-estimate to finish the proof.

Now, we need to show the \(C^{2,\alpha}\) estimate for \(h,w\). This is based on the stationary property of \(V_i\) and the definition of the blow-ups, and the properties of harmonic functions.

Note that each \(h,w\) are all harmonic in its interior, which is a consequence of the stationary property of \(V_i\) and definition of the blow-ups.

The key is actually the smoothness up to the boundary. ◻

Improvement of the fine excess at a smaller scale

Using the properties of the fine blow-up limit, we can improve the fine excess at a smaller scale as follows.

Under suitable assumptions, with \(\hat{E}(V_i)\) small enough, \(\hat{E}^{-1}(V_i)E(V_i)\) is small enough, we can find a new cone \(\boldsymbol{C}_i'\), such that if we apply a rotation, a scaling to the original varifold \(V_i\), we can obtain a new varifold \(V_i'\), such that the fine excess of \(V_i'\) with respect to \(\boldsymbol{C}_i'\) is much smaller than the fine excess of \(V_i\) with respect to \(\boldsymbol{C}_i\). One can think of this as follows. Under suitable assumptions, if one zooms in the original varifold \(V_i\), it will look closer to a new cone \(\boldsymbol{C}_i'\).

Iterative process

If such steps can be done infinitely many times for a given varifold \(V_i\), we can obtain a limit cone \(\boldsymbol{C}_{i,\infty}\). Note that the difference between \(\boldsymbol{C}_{i,k}\) and \(\boldsymbol{C}_{i,k+1}\) is much smaller than the difference between \(\boldsymbol{C}_{i,k}\) and the plane \(\{ x_{n+1}=0 \}\), because the fine excess is much smaller than the original excess. Hence \(\boldsymbol{C}_{i,\infty}\) is still a classical cone. These steps also imply that we can find varifolds \(V_{i,k}\) which converge to \(\boldsymbol{C}_{i,\infty}\) in the varifold sense.

This is a contradiction to the minimal distance theorem.