We first give a well-known geometric calculation which we learnt from unpublished lectures of Rick Schoen on his work with Jon Wolfson, but which dates back at least as far as [HaL], [Oh1] and others.

Picking an orthonormal basis of $T_{p}L$ and parallel transporting it along rays in $L$ to a frame field $(e_{i})$, $(e_{i},Je_{i})$ forms a local basis for $TX$ around $p$. Letting $(f_{j},g_{j}=-f_{j}\circ J)$ be the dual basis of 1-forms, it is clear that at $p$,

$$\Omega=e^{-i\theta}\bigwedge_{j}(f_{j}+ig_{j}),$$

with $\theta$ the phase function of $L$. Since $\Omega$ is parallel, $\nabla_{X}\Omega=0$ yields

	$\displaystyle iX(\theta)\bigwedge_{j}(f_{j}+ig_{j})$	$\displaystyle=$	$\displaystyle\sum_{k}(f_{1}+ig_{1})\wedge\ldots\wedge\nabla_{X}(f_{k}+ig_{k})\wedge\ldots\wedge(f_{n}+ig_{n})$		(2.2)
		$\displaystyle=$	$\displaystyle\sum_{k}\left[\nabla_{X}(f_{k}+ig_{k})\left({1\over 2}(e_{k}-iJe_{k})\right)\right]\bigwedge_{j}(f_{j}+ig_{j}).$

Taking covariant derivatives on the Calabi-Yau (i.e. not on $L$), we have

$$-\langle\MCV,JX\rangle=-\langle\sum_{i}\nabla_{e_{i}}e_{i},JX\rangle=\sum_{i}\langle\nabla_{e_{i}}Je_{i},X\rangle,$$

since $J$ is both skew adjoint and parallel. But as $Je_{i}$ and $X$ are orthogonal, this is

$$-\sum_{i}\langle Je_{i},\nabla_{e_{i}}X\rangle=\sum_{i}\langle e_{i},J\nabla_{X}e_{i}\rangle,$$

as we may choose $X$ to have zero bracket with the $e_{i}$s.

So comparing with (2.2) we are left with showing that

$$\sum_{k}\left[\nabla_{X}(f_{k}+ig_{k})\left({1\over 2}(e_{k}-iJe_{k})\right)\right]=i\sum_{i}\langle e_{i},J\nabla_{X}e_{i}\rangle,$$

i.e. that $(\nabla_{X}(f_{k}+ig_{k}))(e_{k}-iJe_{k})=2i\langle e_{k},J\nabla_{X}e_{k}\rangle$.

But $(f_{k}+ig_{k})(e_{k}-iJe_{k})=2$ is a constant, so the left hand side is $-(f_{k}+ig_{k})(\nabla_{X}(e_{k}-iJe_{k}))=-f_{k}(\nabla_{X}(-iJe_{k}))-ig_{k}(\nabla_{X}e_{k})$; the other terms vanish as $\nabla_{X}e_{i}$ was chosen to be perpendicular to $L$. Using $\nabla_{X}J=0$ and recalling that $g_{k}=-f_{k}\circ J$, we obtain $2i\langle e_{k},J\nabla_{X}e_{k}\rangle$. $\square$

Another simple but important result is how the phase $\theta$ and Riemannian volume form $\vol_{L}$ vary under a hamiltonian deformation $J\,\widetilde{\!dh\,}$ of $L$. This can be found in [Oh1], [Sm], for instance; we give short geometric proofs for completeness.

We next show that, given a suitable metric on the Lie algebra $C^{\infty}(L,\mathbb{R})/\mathbb{R}$, the gradient flow of the norm square $-\frac{1}{2}|m|^{2}$ of the moment map $m=\,\mathrm{Im}\,\Omega\arrowvert_{L}$ of [Th] is mean curvature flow. The following standard calculation, applicable in any such Kähler reduction picture, shows that the gradient flow of $-|m|^{2}$ is given by $2JX_{m^{*}}$, where $J$ is the complex structure, $m^{*}$ is the element of the Lie algebra $C^{\infty}(L)$ corresponding to the moment map $m=\,\mathrm{Im}\,\Omega\arrowvert_{L}$ in the dual of the Lie algebra under the metric on $C^{\infty}(L)$, and $X_{m^{*}}$ is its induced action on the space {Lagrangians with flat $U(1)$ connections on them}.

$$X(-|m|^{2})=-2\langle m,Xm\rangle=-2X(m(m^{*}))=-2\omega(X,X_{m^{*}})=2\langle X,JX_{m^{*}}\rangle.$$

By the definition of the group action in [Th], this deformation $JX_{m^{*}}$ is just the hamiltonian deformation of the Lagrangian $L$ with hamiltonian function $m^{*}$ on $L$.

Choosing the volume form Re $\Omega\arrowvert_{L}$ on $L$ to define an $L^{2}$ metric on $C^{\infty}(L)$ gives $m^{*}=\tan\theta$, since $m=\,\mathrm{Im}\,\Omega\arrowvert_{L}=\sin\theta\,$vol$\,=\tan\theta\,$Re $\Omega\arrowvert_{L}$. Similarly using the induced Riemannian volume form vol gives $m^{*}=\sin\theta$, while using

$${\sin\theta\over\theta}\,\mathrm{vol}$$

as volume form on $L$ yields $m^{*}=\theta$. Any of these are suitable for small phase $\theta:\,L\to\mathbb{R}$, and give similar flows down which the moment map decreases. The last one, however, is precisely mean curvature flow, by Lemma 2.1.

This and the previous lemma show that under mean curvature flow, the phase $\theta$ satisfies a (time dependent) heat equation while the Riemannian volume form decreases (as usual):

	$\displaystyle\dot{\theta}\!$	$\displaystyle=$	$\displaystyle\!-\Delta\,\theta,$		(2.5)
	$\displaystyle{d\over dt}\vol_{L}\!$	$\displaystyle=$	$\displaystyle\!-|d\theta|^{2}\vol_{L}.$		(2.6)

We therefore obtain a maximum principle for $\theta$, whose range must always decrease, but it is important to note that the Laplacian $\Delta$ is time dependent as the metric on $L$ used to define it varies.

From these follow a series of identities and estimates, many of which we use later, but none are strong enough to give long term existence of the mean curvature flow, and with good reason. Mean curvature flow is a complicated and much-studied subject (understood only in codimension 1, dimension 1 [Gr], and, in special cases, in two dimensions [W1], [W2]), with known examples of finite time blow-up. While we might expect it to behave better for Lagrangians (locally functions of one variable instead of $n$), examples in Section 6 show similar phenomena. But in our examples there will be a way round these problems, and we will be able to make a conjecture about the flow which may help in its study.