Proposition 2.2.1
Suppose \(\mathbf C\) is a minimal cone in \(\mathbb{R}^{n+1}\), smooth away from the origin. Then we have the following Simons-type inequality on \(\mathbf C\setminus\{0\}\):
\[
\frac{1}{2}\Delta |A|^{2} \geq -|A|^4 + 2 \frac{|A|^{2}}{r^{2}}+|\nabla |A||^{2}.
\]
Proof. Choose \(\{ e_i \}\) to be a local orthonormal frame on \(\mathbf C\setminus\{0\}\) such that \(e_n\) is the radial direction \(\partial_r\). Recall the Simons identity for minimal hypersurfaces in \(\mathbb{R}^{n+1}\):
\[
\frac{1}{2}\Delta |A|^{2} = -|A|^4 + |\nabla A|^2.
\]
It suffices to show
\[
|\nabla A|^2 \geq 2 \frac{|A|^2}{r^2}+|\nabla |A||^2.
\]
Since \(\mathbf C\) is a cone, we have \(A_{in}=0\) and \(A_{ij,n}=-\frac{1}{r}A_{ij}\) for \(i,j\lt{}n\). Indeed,
\[
A(rp)=\frac{A(p)}{r}.
\]
Taking derivative in the radial direction gives
\[
A_{ij,n} (rp)= - \frac{A_{ij}(p)}{r^{2}}= -\frac{1}{r}A_{ij}(rp).
\]
So we have
\[
\sum_{i,j,k}^{}A_{ij,k}^{2}=\sum_{\alpha,\beta=1}^{n-1}3 A_{\alpha\beta,n}^{2} + \sum_{\alpha,\beta,\gamma=1}^{n-1}A_{\alpha\beta,\gamma}^{2} = 2 \frac{|A|^2}{r^2} + \sum_{\alpha,\beta=1}^{n-1}A_{\alpha\beta,n}^{2}+ \sum_{\alpha,\beta,\gamma=1}^{n-1}A_{\alpha\beta,\gamma}^{2}.
\]
At a fixed point, choose \(\{e_\alpha\}_{\alpha=1}^{n-1}\) so that
\[
A_{\alpha\beta}=\lambda_\alpha\delta_{\alpha\beta}.
\]
Then
\[
|\nabla |A||^2
=\frac{1}{|A|^2}\sum_{k=1}^{n}\Big(\sum_{\alpha=1}^{n-1}\lambda_\alpha A_{\alpha\alpha,k}\Big)^2
=\frac{1}{|A|^2}\sum_{\beta=1}^{n-1}\Big(\sum_{\alpha=1}^{n-1}\lambda_\alpha A_{\alpha\alpha,\beta}\Big)^2
+\frac{1}{|A|^2}\Big(\sum_{\alpha=1}^{n-1}\lambda_\alpha A_{\alpha\alpha,n}\Big)^2.
\]
By Cauchy–Schwarz and \(A_{\alpha\alpha,n}=-\frac1rA_{\alpha\alpha}\),
\[
|\nabla |A||^2
\leq \sum_{\alpha,\beta=1}^{n-1}A_{\alpha\alpha,\beta}^2+\frac{|A|^2}{r^2}
\leq \sum_{\alpha,\beta,\gamma=1}^{n-1}A_{\alpha\beta,\gamma}^2+\sum_{\alpha,\beta=1}^{n-1}A_{\alpha\beta,n}^2.
\]
Therefore,
\[
|\nabla A|^2
=2\frac{|A|^2}{r^2}
+\sum_{\alpha,\beta=1}^{n-1}A_{\alpha\beta,n}^2
+\sum_{\alpha,\beta,\gamma=1}^{n-1}A_{\alpha\beta,\gamma}^2
\geq 2\frac{|A|^2}{r^2}+|\nabla |A||^2.
\]
This completes the proof. ◻
A similar computation yields
\[
\frac{1}{2}\Delta|A|^{2}\geq p |\nabla |A||^{2} + (3-p)\frac{|A|^{2}}{r^{2}} - |A|^4,
\]
for any \(p\leq 1+\frac{2}{n-1}\).
Theorem 2.2.2
Let \(\mathbf C^n \subset \mathbb{R}^{n+1}\) be a minimal nonflat cone, smooth away from the origin. If \(n \leq 6\), \(\mathbf C\) is unstable. If \(n \geq 7\), there exists a stable minimal nonflat cone.
Proof. Suppose \(\mathbf C\) is stable. Testing the stability inequality with \(|A|\varphi\), we get
\[
\begin{aligned}
\int_{ } |A|^{4}\varphi^{2} \leq{}& \int_{ } |\nabla (|A|\varphi)|^2 = \int_{ } |\nabla |A||^2 \varphi^2 + |A|^2 |\nabla \varphi|^2 + \frac{1}{2} \langle\nabla |A|^{2}, \nabla \varphi^{2}\rangle\\
\leq{}& \int_{ } |\nabla |A||^2 \varphi^2 + |A|^{2}|\nabla \varphi|^2 - \frac{1}{2}\varphi^{2}\Delta |A|^{2} \\
\leq{}& \int_{ } |A|^{2}|\nabla \varphi|^2 + \varphi^{2}|A|^4 - 2 \frac{|A|^{2}}{r^{2}}\varphi^{2}.
\end{aligned}
\]
Hence, we obtain
\[
2 \int_{ } \frac{|A|^{2}}{r^{2}}\varphi^{2} \leq \int_{ } |A|^{2}|\nabla \varphi|^2.
\]
Now choose
\[
\varphi= \max\{ 1,r \}^{1-\frac{n}{2}-2\varepsilon} r^{1+\varepsilon}.
\]
We need to verify that this function is admissible. We compute
\[
\begin{aligned}
\int_{ } |A|^{2}|\nabla \varphi|^2={}&(1+\varepsilon)^{2}\int_{ \{ r\lt{}1 \}}|A|^{2} r^{2\varepsilon}+(2-\frac{n}{2}-\varepsilon)^{2}\int_{ \{ r\geq 1 \}} |A|^{2} r^{-n+2-\varepsilon}\\
={}&(1+\varepsilon)^{2}\int_{ 0}^1dr \int_{ \Sigma} |A_\Sigma|^{2} r^{2\varepsilon-2+n-1}d\Sigma\\
&+(2-\frac{n}{2}-\varepsilon)^{2}\int_{ 1}^\infty dr \int_{ \Sigma} |A_\Sigma|^{2} r^{-n-2\varepsilon+n-1}d\Sigma\\
={}&(\int_{ 0}^1 r^{2\varepsilon+n-3}dr + (2-\frac{n}{2}-\varepsilon)^{2}\int_{ 1}^\infty r^{-2\varepsilon-1}dr)\int_{ \Sigma} |A_\Sigma|^{2} d\Sigma\lt{}+\infty
\end{aligned}
\]
Here, \(\Sigma=\mathbf C\cap S^n\) is the link of the cone, which is a smooth closed minimal hypersurface in \(S^n\). Hence, \(\varphi\) is admissible. On the other hand, we have
\[
\begin{aligned}
2\int_{ } \frac{|A|^{2}}{r^{2}}\varphi^{2}={}& 2\int_{ \{ r\lt{}1 \}} |A|^{2} r^{2\varepsilon} + 2\int_{ \{ r\geq 1 \}} |A|^{2} r^{-n+2-4\varepsilon}\\
\end{aligned}
\]
When \(n\leq 6\), we can choose \(\varepsilon\gt{}0\) small enough such that \(2\gt{}(1+\varepsilon)^{2}\) and \(2\gt{}(2-\frac{n}{2}-\varepsilon)^{2}\). Hence \(|A|^2\equiv 0\), so \(\mathbf C\) is flat, a contradiction. Therefore, \(\mathbf C\) is unstable.
It remains to construct a stable minimal nonflat cone in \(\mathbb{R}^8\).
Define the Simons-type cone by
\[
\mathbf C_{p,q}
=\left\{(x,y)\in \mathbb{R}^{p+1}\times \mathbb{R}^{q+1}:\ q|x|^2=p|y|^2\right\}.
\]
Lemma 2.2.3
\(\mathbf C_{p,q}\) is a minimal cone in \(\mathbb{R}^{p+q+2}\), smooth away from the origin, and it is stable if and only if \(p+q\geq 6\).
One can verify directly that \(\mathbf C_{p,q}\) is minimal by computing its mean curvature. In particular, its principal curvatures are
\[
\kappa_1=\cdots=\kappa_p=\sqrt{\frac{q}{p}}, \qquad \kappa_{p+1}=\cdots=\kappa_{p+q}=-\sqrt{\frac{p}{q}},\quad \text{and}\quad \kappa_{p+q+1}=0.
\]
Its second fundamental form satisfies \(|A|^{2}=\frac{n-1}{r^{2}}\). Choose \(X= \frac{\varphi^{2}}{r^{2}}x\) and insert it into the first variation formula. Then
\[
0=\int_{ } \mathrm{div}^{\mathbf C_{p,q}}(X)=\int_{ } n\frac{\varphi^{2}}{r^{2}}-2\frac{\varphi^{2}}{r^{4}}|x|^2+2 \frac{\varphi \nabla^{\mathbf C_{p,q}} \varphi \cdot x}{r^{2}}=\int_{ } (n-2)\frac{\varphi^{2}}{r^{2}}+2 \frac{\varphi \nabla^{\mathbf C_{p,q}} \varphi \cdot x}{r^{2}}.
\]
Hence
\[
\int_{ } (n-2) \frac{\varphi^{2}}{r^{2}}\leq 2 \sqrt{\int_{ } \frac{\varphi^{2}}{r^{2}} \int_{ } |\nabla^{\mathbf C_{p,q}} \varphi|^2} \implies \int_{ } \frac{(n-2)^{2}}{4} \frac{\varphi^{2}}{r^{2}} \leq \int_{ } |\nabla^{\mathbf C_{p,q}} \varphi|^2.
\]
Since \(|A|^2=\frac{n-1}{r^2}\), this can be rewritten as
\[
\int_{ } \frac{(n-2)^{2}}{4(n-1)}|A|^{2} \varphi^{2} \leq \int_{ } |\nabla^{\mathbf C_{p,q}} \varphi|^2.
\]
In particular, if \(n\geq 7\), then \(\frac{(n-2)^2}{4(n-1)}\geq 1\), so \(\mathbf C_{p,q}\) is stable. ◻
In fact, one has the stronger result:
Theorem 2.2.4
Each \(\mathbf C_{p,q}\) is area-minimizing if \(p+q\geq 6\) except for the case \(p,q= (1,5)\) or \((5,1)\).
The case \(\mathbf C_{p,p}\) with \(p\geq 3\) was proved by Bombieri–De Giorgi–Giusti [BDGG69]. Lawson [Law72] later proved the result for \(p+q\gt{}6\), and Simões [Sim74] handled the remaining cases \(p,q=(2,4)\) or \((4,2)\).
The original proof is quite involved and relies on calibrations. One seeks a vector field \(\xi\) such that \(\operatorname{div}^{\mathbf C} \xi=0\) and \(\xi=\nu_{\mathbf C}\) on \(\mathbf C\setminus\{0\}\), which implies that \(\mathbf C\) is area-minimizing. Finding such a calibration for \(\mathbf C_{p,q}\) is difficult and requires solving an ODE.
Here we present a different proof due to De Philippis–Paolini [DPP09], which uses a sub-calibration argument together with an explicit construction of a sub-calibration for \(\mathbf C_{p,p}\).
