Sub-minimal Sets and Sub-calibrations

Let \(\Omega\subset\mathbb{R}^n\) be an open set. For a set \(E\) with smooth boundary, the perimeter \(P(E,A)\) in a smooth bounded open set \(A\subset\Omega\) is the \((n-1)\)-dimensional Hausdorff measure of \(\partial E\cap A\).

Definition 2.3.1

A set \(E\) with smooth boundary is sub-minimal in \(\Omega\) if for every smooth bounded open set \(A\subset\Omega\) and every \(F\subset E\) with \(E\setminus F\subset\subset A\),

\[ P(E,A)\leq P(F,A). \]

Proposition 2.3.2

If \(E\) and \(E^c=\Omega\setminus E\) are both sub-minimal in \(\Omega\), then \(E\) is minimal in \(\Omega\).

Proof. Let \(A\subset\Omega\) be a smooth bounded open set, and let \(F\) satisfy \(E\triangle F\subset\subset A\). Define \(F'=E\cap F\subset E\) and \(F''=(E\cup F)^c\subset E^c\). Then \(E\setminus F'\subset\subset A\) and \(E^c\setminus F''\subset\subset A\). By sub-minimality:

\[ P(E,A)\leq P(F',A),\quad P(E^c,A)\leq P(F'',A). \]

Since \(P(E^c,A)=P(E,A)\) and \(P(F'',A)=P(E\cup F,A)\), we have

\[ 2P(E,A)\leq P(E\cap F,A)+P(E\cup F,A). \]

Using the identity

\[ P(E\cap F,A)+P(E\cup F,A)\leq P(E,A)+P(F,A), \]

we conclude \(P(E,A)\leq P(F,A)\), so \(E\) is minimal. ◻

Definition 2.3.3

Let \(E\) have smooth boundary. A \(C^1\) vector field \(\xi\) on \(\Omega\) is a sub-calibration of \(E\) if:

\(\xi(x)=\nu_E(x)\) (exterior unit normal) for all \(x\in\partial E\);
\(\operatorname{div}\xi(x)\leq0\) for all \(x\in E\);
\(\lvert\xi(x)\rvert\leq1\) for all \(x\in\Omega\).

Theorem 2.3.4

If \(E\) admits a sub-calibration \(\xi\), then \(E\) is sub-minimal.

Proof. Let \(A\) be a bounded open set, and let \(F\subset E\) with \(E\setminus F\subset\subset A\). Choose \(\eta_j\in C_c^1(A)\) with \(\eta_j=1\) on \(E\setminus F\), \(0\leq\eta_j\leq1\), and \(\bigcup_j\{x:\eta_j(x)=1\}=A\). Let \(\xi_j=\eta_j\xi\). Then

\[ \int_{E\cap A}\operatorname{div}\xi_j - \int_{F\cap A}\operatorname{div}\xi_j = \int_{E\setminus F}\operatorname{div}\xi \leq 0, \]

\[ \int_{E\cap A}\operatorname{div}\xi_j \leq \int_{F\cap A}\operatorname{div}\xi_j \leq P(F,A). \]

By the divergence theorem:

\[ \int_{E\cap A}\operatorname{div}\xi_j = \int_{\partial E\cap A}\langle\xi_j,\nu_E\rangle d\mathcal{H}^{n-1} = \int_{\partial E\cap A}\eta_j d\mathcal{H}^{n-1} \geq \mathcal{H}^{n-1}\bigl(\partial E\cap\{\eta_j=1\}\bigr). \]

Taking \(j\to\infty\) yields \(P(E,A)\leq P(F,A)\), so \(E\) is sub-minimal. ◻

Let \(n=2m\). The Simons cone is

\[ \mathbf C=\bigl\{(x_1,\dots,x_m,y_1,\dots,y_m)\in\mathbb{R}^{2m}: x_1^2+\dots+x_m^2=y_1^2+\dots+y_m^2\bigr\}. \]

Define

\[ \mathcal{C}=\bigl\{(x,y)\in\mathbb{R}^m\times\mathbb{R}^m: |x|\leq|y|\bigr\},\quad f(x,y)=\frac{1}{4}\bigl(|x|^4-|y|^4\bigr). \]

Define the vector field

\[ \xi=\frac{Df}{|Df|}. \]

Proposition 2.3.5

\(\xi\) is a sub-calibration for \(E\) in \(\mathbb{R}^{2m}\setminus\{0\}\), where \(E:=\{(x,y):|x|\leq |y|\}\), and \(-\xi\) is a sub-calibration for \(E^c\) in \(\mathbb{R}^{2m}\setminus\{0\}\).

Proof.

\(\lvert\xi\rvert=1\) everywhere, and \(\xi\) is the exterior normal to \(E\) on \(\partial E\). We compute

\[ Df=(|x|^2 x,\,-|y|^2 y), \qquad |Df|^2 = |x|^6 + |y|^6. \]

Hence

\[ \Delta f = (m+2)|x|^{2} - (m+2)|y|^{2},\quad D|Df|^{2}= 6\bigl(|x|^{4} x,\, |y|^{4} y\bigr). \]

Thus

\[ \begin{aligned} |Df|^3\mathrm{div}\xi ={}& |Df|^2 \Delta f - \frac{1}{2}\langle D|Df|^2, Df\rangle\\ ={}& \bigl((m+2)|x|^{2} - (m+2)|y|^{2}\bigr)(|x|^6 + |y|^6) - 3\bigl(|x|^{8} + |y|^{8}\bigr)\\ ={}& (m-1)|x|^8 - (m-1)|y|^8 + (m+2)|x|^{2}|y|^{6} - (m+2)|x|^{6}|y|^{2}\\ ={}& (m-1)(|x|^4-|y|^4)(|x|^4+|y|^4) + (m+2)|x|^{2}|y|^{2}(|y|^4 - |x|^4)\\ ={}& (|x|^4 - |y|^4)\bigl((m-1)(|x|^4 + |y|^4) - (m+2)|x|^2 |y|^2\bigr). \end{aligned} \]

Note that

\[ (m-1)a^{2} - (m+2)ab + (m-1)b^{2} \]

is nonnegative if and only if

\[ \mathrm{det}\,\begin{pmatrix} m-1 & -\frac{m+2}{2}\\ -\frac{m+2}{2} & m-1 \end{pmatrix} = (m-1)^2 - \frac{(m+2)^2}{4} = \frac{3m(m-4)}{4} \geq 0. \]

◻

Theorem 2.3.6

The Simons cone \(\mathbf C\) is area-minimizing in \(\mathbb{R}^8\).

Proof. By the preceding theorem, \(E\) and \(E^c\) are sub-minimal in \(\mathbb{R}^8\setminus\{0\}\). Since the origin has codimension \(\gt{}1\), the perimeter is unchanged, and hence \(E\) and \(E^c\) are sub-minimal in \(\mathbb{R}^8\). By the preceding proposition, \(E\) is minimal, so its boundary \(\mathbf C\) is area-minimizing. ◻

One may consider the gradient of the function

\[ \frac{1}{4}\left( q^{2}|x|^4- p^{2}|y|^4 \right) \]

in the case of the minimizing cone \(\mathbf C_{p,q}\).

Sub-minimal Sets and Sub-calibrations

Gaoming Wang

Assistant Professor