2.1 Special Lagrangian submanifolds in ℂ m [03KD]

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2.1 Special Lagrangian submanifolds in $\mathbin{\mathbb{C}}^{m}$

We begin by defining calibrations and calibrated submanifolds, following Harvey and Lawson [9].

Definition 2.1 Let $(M,g)$ be a Riemannian manifold. An oriented tangent $k$ -plane $V$ on $M$ is a vector subspace $V$ of some tangent space $T_{x}M$ to $M$ with $\mathop{\rm dim}V=k$ , equipped with an orientation. If $V$ is an oriented tangent $k$ -plane on $M$ then $g|_{V}$ is a Euclidean metric on $V$ , so combining $g|_{V}$ with the orientation on $V$ gives a natural volume form $\mathop{\rm vol}_{V}$ on $V$ , which is a $k$ -form on $V$ .

Now let $\varphi$ be a closed $k$ -form on $M$ . We say that $\varphi$ is a calibration on $M$ if for every oriented $k$ -plane $V$ on $M$ we have $\varphi|_{V}\leqslant\mathop{\rm vol}_{V}$ . Here $\varphi|_{V}=\alpha\cdot\mathop{\rm vol}_{V}$ for some $\alpha\in\mathbin{\mathbb{R}}$ , and $\varphi|_{V}\leqslant\mathop{\rm vol}_{V}$ if $\alpha\leqslant 1$ . Let $N$ be an oriented submanifold of $M$ with dimension $k$ . Then each tangent space $T_{x}N$ for $x\in N$ is an oriented tangent $k$ -plane. We say that $N$ is a calibrated submanifold if $\varphi|_{T_{x}N}=\mathop{\rm vol}_{T_{x}N}$ for all $x\in N$ .

It is easy to show that calibrated submanifolds are automatically minimal submanifolds [9, Th. II.4.2]. Here is the definition of special Lagrangian submanifolds in $\mathbin{\mathbb{C}}^{m}$ , taken from [9, §III].

Definition 2.2 Let $\mathbin{\mathbb{C}}^{m}$ have complex coordinates $(z_{1},\dots,z_{m})$ , and define a metric $g$ , a real 2-form $\omega$ and a complex $m$ -form $\Omega$ on $\mathbin{\mathbb{C}}^{m}$ by

\begin{split}g=|{\rm d}z_{1}|^{2}+\cdots+|{\rm d}z_{m}|^{2},\quad\omega&=\frac{i}{2}({\rm d}z_{1}\wedge{\rm d}\bar{z}_{1}+\cdots+{\rm d}z_{m}\wedge{\rm d}\bar{z}_{m}),\\ \text{and}\quad\Omega&={\rm d}z_{1}\wedge\cdots\wedge{\rm d}z_{m}.\end{split}

(1)

Then $\mathop{\rm Re}\Omega$ and $\mathop{\rm Im}\Omega$ are real $m$ -forms on $\mathbin{\mathbb{C}}^{m}$ . Let $L$ be an oriented real submanifold of $\mathbin{\mathbb{C}}^{m}$ of real dimension $m$ . We say that $L$ is a special Lagrangian submanifold of $\mathbin{\mathbb{C}}^{m},$ or SL $m$ -fold for short, if $L$ is calibrated with respect to $\mathop{\rm Re}\Omega$ , in the sense of Definition 2.1.

As in [10, 11] there is a more general definition of special Lagrangian $m$ -fold involving a phase ${\rm e}^{i\theta}$ , but we will not use it here. Harvey and Lawson [9, Cor. III.1.11] give the following alternative characterization of special Lagrangian submanifolds.

Proposition 2.3

Let $L$ be a real $m$ -dimensional submanifold of $\mathbin{\mathbb{C}}^{m}$ . Then $L$ admits an orientation making it into an SL submanifold of $\mathbin{\mathbb{C}}^{m}$ if and only if $\omega|_{L}\equiv 0$ and $\mathop{\rm Im}\Omega|_{L}\equiv 0$ .

An $m$ -dimensional submanifold $L$ in $\mathbin{\mathbb{C}}^{m}$ is called Lagrangian if $\omega|_{L}\equiv 0$ . Thus special Lagrangian submanifolds are Lagrangian submanifolds satisfying the extra condition that $\mathop{\rm Im}\Omega|_{L}\equiv 0$ , which is how they get their name.

Next we give a result characterizing SL 3-planes $\mathbin{\mathbb{R}}^{3}$ in $\mathbin{\mathbb{C}}^{3}$ , which will be useful in §7. Define an anti-bilinear cross product $\times:\mathbin{\mathbb{C}}^{3}\times\mathbin{\mathbb{C}}^{3}\rightarrow\mathbin{\mathbb{C}}^{3}$ by

(r_{1},r_{2},r_{3})\times(s_{1},s_{2},s_{3})=(\bar{r}_{2}\bar{s}_{3}-\bar{r}_{3}\bar{s}_{2},\bar{r}_{3}\bar{s}_{1}-\bar{r}_{1}\bar{s}_{3},\bar{r}_{1}\bar{s}_{2}-\bar{r}_{2}\bar{s}_{1}).

(2)

It is equivariant under the $\mathop{\rm SU}(3)$ -action on $\mathbin{\mathbb{C}}^{3}$ . Using this notation, we prove

Proposition 2.4

Let ${\bf r},{\bf s}\in\mathbin{\mathbb{C}}^{3}$ be linearly independent over $\mathbin{\mathbb{R}}$ , with $\omega({\bf r},{\bf s})=0$ . Then ${\bf r},{\bf s}$ and ${\bf r}\times{\bf s}$ are linearly independent over $\mathbin{\mathbb{R}}$ , and $\langle{\bf r},{\bf s},{\bf r}\times{\bf s}\rangle_{\scriptscriptstyle\mathbb{R}}$ is the unique special Lagrangian $3$ -plane in $\mathbin{\mathbb{C}}^{3}$ containing $\langle{\bf r},{\bf s}\rangle_{\scriptscriptstyle\mathbb{R}}$ .

Proof. Explicit calculation using (2) shows that

	$\displaystyle g({\bf r},{\bf r}\times{\bf s})=g({\bf s},{\bf r}\times{\bf s})=0,$		(3)
	$\displaystyle\omega({\bf r},{\bf r}\times{\bf s})=\omega({\bf s},{\bf r}\times{\bf s})=0,$		(4)
	$\displaystyle\|{\bf r}\times{\bf s}\|^{2}=\|\bf r\|^{2}\|\bf s\|^{2}-g({\bf r},{\bf s})^{2}-\omega({\bf r},{\bf s})^{2},$		(5)
	$\displaystyle\text{and}\quad(\mathop{\rm Im}\Omega)({\bf r},{\bf s},{\bf r}\times{\bf s})=0,$		(6)

for all ${\bf r},{\bf s}\in\mathbin{\mathbb{C}}^{3}$ . When ${\bf r},{\bf s}$ are linearly independent and $\omega({\bf r},{\bf s})=0$ , equation (3) shows that ${\bf r}\times{\bf s}$ is orthogonal to ${\bf r},{\bf s}$ , and (5) that $|{\bf r}\times{\bf s}|\neq 0$ . Therefore ${\bf r},{\bf s}$ and ${\bf r}\times{\bf s}$ are linearly independent.

Also we have $\omega({\bf r},{\bf s})=\omega({\bf r},{\bf r}\times{\bf s})=\omega({\bf s},{\bf r}\times{\bf s})=0$ by (4), so that $\langle{\bf r},{\bf s},{\bf r}\times{\bf s}\rangle_{\scriptscriptstyle\mathbb{R}}$ is a Lagrangian 3-plane. Then (6) shows that $\langle{\bf r},{\bf s},{\bf r}\times{\bf s}\rangle_{\scriptscriptstyle\mathbb{R}}$ is a special Lagrangian 3-plane, by Proposition 2.7. It is easy to see that this is the only SL 3-plane in $\mathbin{\mathbb{C}}^{3}$ containing $\langle{\bf r},{\bf s}\rangle_{\scriptscriptstyle\mathbb{R}}$ . $\square$

2.1 Special Lagrangian submanifolds in ℂ m [03KD]

2.1 Special Lagrangian submanifolds in ℂm\mathbin{\mathbb{C}}^{m}

Proposition 2.3

Proposition 2.4

2.1 Special Lagrangian submanifolds in $\mathbin{\mathbb{C}}^{m}$