Regularity in the Immersed Setting

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. ◻

Regularity in the Immersed Setting

Gaoming Wang

Assistant Professor