Here we consider an example suggested to us by both Paul Seidel and Cumrun Vafa, used in [SV] and [KS]. Consider the affine algebraic variety $X^{n}$ given by

$$\sum_{i=1}^{n}x_{i}^{2}=p(t)$$

in $\mathbb{C}\,^{n}\times\mathbb{C}\,$, where $p$ is some polynomial in $t\in\mathbb{C}\,$ with only simple zeros. Denote by $\pi:\,X^{n}\to\mathbb{C}\,$ the projection to the $t$ coordinate. Here we use the Kähler structure restricted from $\mathbb{C}\,^{n+1}$, and the nowhere-zero holomorphic volume form given by taking the Poincaré residue ([GH] p 147) of the standard form $dx_{1\ldots n}:=dx_{1}\ldots dx_{2\,}dt$ on $\mathbb{C}\,^{n+1}$; this can be written as

$$(-1)^{n+i+1}{dx_{1\ldots\hat{\imath}\ldots n\,}dt\arrowvert_{X^{n}}\over 2x_{i}}={dx_{1}\ldots dx_{n}\arrowvert_{X^{n}}\over\dot{p}(t)}$$

(6.1)

for any $i$ (so where $x_{i}=0\ \forall i$ we can use the second expression). Here $\hat{\imath}$ means that we omit the $dx_{i}$ term from the wedge product. This is then not parallel, and the metric we have chosen is not the Ricci-flat one. Nonetheless it is a good explicit testing ground for the conjecture; we can still define $\theta$ as the phase of $\Omega\arrowvert_{L}$ and SLags as having constant phase, of course we then use flow by the $J\,\widetilde{\!d\theta\,}$ vector, rather than mean curvature flow in this metric. While the two flows are similar and would be the same in the Ricci-flat metric, only the former has SLags as its stationary points (for the latter we get minimal submanifolds, which in this metric are not quite SLag). As Edward Goldstein pointed out to us, the $J\,\widetilde{\!d\theta\,}$ flow is the gradient flow of the weighted volume $\int_{L}|\Omega|\vol$ instead of $\int_{L}\vol$; everything proceeds analogously to before on weighting all vols by $|\Omega|$, as we shall see.

Each smooth fibre over $t\in\mathbb{C}\,$ is an affine quadric with a natural Lagrangian $S^{n-1}$ ‘real’ slice, namely the intersection of the fibre with the slice

$$x_{i}\in\sqrt{p(t)}\,\mathbb{R}\hskip 10.00002pt\forall i.$$

It is invariant under the obvious $O(n)$ action on $X^{n}$, and is the vanishing cycle of every singular fibre (i.e. the fibres over the roots of $p$). Therefore any path $\gamma:\,I\to\mathbb{C}\,$ ($I\ni u$ being some interval in $\mathbb{R}$) from one zero of $p$ to another lifts to give a canonical $O(n)$-invariant Lagrangian $n$-sphere $\gamma^{n}$, $S^{n-1}$-fibred over $\gamma$ except at the endpoints where it closes up. Also, any vector $\gamma^{\prime}\partial_{t}$ in the base $\mathbb{C}\,\ni t$ lifts canonically to a vector

$$\gamma^{\prime}\left(\partial_{t}+{\dot{p}\over 2p}\sum x_{i}\partial_{x_{i}}\right)$$

(6.2)

tangent to the infinitesimal Lagrangian $\gamma^{n}$ lying above $\gamma^{\prime}$. Here ^′ denotes $d/du$. Note that $\gamma^{1}$ is a closed curve double covering $\gamma$, branched over $\gamma$’s endpoints. We will use this curve $\gamma^{1}$ later to study $\gamma^{n}$.

The phase function $\theta$ on $\gamma^{n}$ is also $O(n)$-invariant and so a function of $t\in\mathbb{C}\,$ which we may calculate at $x_{1}=\sqrt{p(t)},\,x_{i}=0\ \forall i\geq 2$. Choosing a basis of tangent vectors to $\gamma^{n}$ at this point,

$$\gamma^{\prime}(\partial_{t}+{\dot{p}\over 2p^{1/2}}\partial_{x_{1}}),\,\sqrt{p}\,\partial_{x_{2}},\ldots,\sqrt{p}\,\partial_{x_{n}},$$

(6.3)

wedging them together and evaluating against the $(n,0)$-form (6.1) gives

$$\gamma^{\prime}{\dot{p}\over 2p^{1/2}}{(\sqrt{p})^{n-1}\over\dot{p}}={1\over 2}\gamma^{\prime}p^{n/2-1}.$$

Therefore the phase function on $\gamma^{n}$ is given by

$$\theta:=\theta(\gamma^{n})=\theta(\gamma^{\prime})+\big({n\over 2}-1\big)\theta(p(\gamma)),$$

(6.4)

where $\theta(\gamma^{\prime})$ is the usual angle of the path $\gamma$, and $\theta(p)$ is the phase of the complex number $p$ evaluated at $t=\gamma$. So

$$d\theta=\left({d\theta(\gamma^{\prime})\over du}+\big({n\over 2}-1\big){d\theta(p)\over du}\right)du,$$

where $du$ is the pullback to $\gamma^{n}$ under the projection $\pi$ of the corresponding 1-form on $\gamma(I)\subset\mathbb{C}\,$.

Using the metric and the orthogonal basis (6.3) we see that

$$\,\widetilde{\!du\,}={\gamma^{\prime}(\partial_{t}+{\dot{p}\over 2p^{1/2}}\partial_{x_{1}})\over|\gamma^{\prime}(\partial_{t}+{\dot{p}\over 2p^{1/2}}\partial_{x_{1}})|^{2}}$$

at the point $x_{1}=\sqrt{p(t)},\,x_{i}=0\ \forall i\geq 2$.

Therefore, by $O(n)$-invariance and the holomorphicity of the projection $\pi$, $J\,\widetilde{\!d\theta\,}$ is the canonical lift (6.2) of

$${d(\theta(\gamma^{\prime})+\big({n\over 2}-1\big)\theta(p))\over du}\,{\pi_{*}\!\left[J\gamma^{\prime}(\partial_{t}+{\dot{p}\over 2p^{1/2}}\partial_{x_{1}})\right]\over|\gamma^{\prime}|^{2}(1+|\dot{p}|^{2}/4|p|)}={{1\over|\gamma^{\prime}|}{d\over du}(\theta(\gamma^{\prime})+\big({n\over 2}-1\big)\theta(p))\over 1+|\dot{p}|^{2}/4|p|}\,i{\gamma^{\prime}\over|\gamma^{\prime}|}\partial_{t}.$$

Denoting by $\mathbf{t}=\partial_{u}/|\gamma^{\prime}|=\gamma^{\prime}\partial_{t}/|\gamma^{\prime}|$ and $\mathbf{n}=i\mathbf{t}$ the unit tangent and normal vectors to $\gamma$ at a point $\gamma(u)$, the above is

$${\mathbf{t}[\theta(\gamma^{\prime})+\big({n\over 2}-1\big)\theta(p)]\over 1+|\dot{p}|^{2}/4|p|}\,\mathbf{n}.$$

By the Cauchy-Riemann equations for the holomorphic function $\log p=\log|p|+i\theta(p)$, $\mathbf{t}\theta(p)=-\mathbf{n}\log|p|$, so that our flow is the lift to $X^{n}$ of the flow of $\gamma$ with vector

$$V^{n}={1\over 1+|\dot{p}|^{2}/4|p|}(\MCV+\,(1-n/2)\mathbf{n}(\log|p|)\,\mathbf{n}),$$

(6.5)

where $\MCV$ is the usual mean curvature vector of $\gamma$ in the flat metric on $\mathbb{C}\,$.

So we can reduce studying our flow to studying the flow of a curve $\gamma$ with fixed endpoints (at zeros of $p$), under the above vector field. We would like to relate this to mean curvature flow of $\gamma\subset\mathbb{C}\,$ in a different metric, and also to both our flow and the mean curvature flow for the double $\gamma^{1}$ of $\gamma$ in the double cover $X^{1}$ of $\mathbb{C}\,$ branched over the zeros of $p$. The advantage of this is that we now have a flow for a closed curve instead of a boundary value problem (but since the flow has $O(1)=\mathbb{Z}/2$ symmetry it is equivalent to a flow of the original curve $\gamma$ with fixed endpoints). We need the following lemma.

Since the unit normal to $\gamma$ in the new metric is $g^{-1/2}\mathbf{n}$, the new mean curvature vector is

$${g\langle\gamma^{\prime\prime}+\Gamma_{\gamma^{\prime}}\gamma^{\prime},g^{-1/2}\mathbf{n}\rangle\over g|\gamma^{\prime}|^{2}}\,g^{-1/2}\mathbf{n}={1\over g}\left(\MCV-{1\over 2}\mathbf{n}(\log g)\,\mathbf{n}\right),$$

as claimed. $\square$

Using this we can get a number of geometrically interesting flows which are equivalent to our original flow in $X^{n}$. Namely, using the result (6.5), the above Lemma, and the fact that locally (away from branch points) $X^{1}$ is conformally equivalent to $\mathbb{C}\,$ with its metric scaled by $g=1+|\dot{p}|^{2}/4|p|$ (by holomorphicity and (6.2)), we can deduce the following.

Denote by $V^{n}$ the flow vector $\pi_{*}J\,\widetilde{\!d\theta\,}$ of the curve $\gamma\subset\mathbb{C}\,$ under our flow in $X^{n}$. Denote by $\MCV^{1}_{g}$ the flow vector of $\gamma\subset\mathbb{C}\,$ under mean curvature flow of $\gamma^{1}$ in $X^{1}$, with $X^{1}$’s natural metric scaled by a $\mathbb{Z}/2$-invariant function $g$ (and omit the $g$ in the notation if $g\equiv 1$). And denote by $\MCV_{g}$ the mean curvature vector of $\gamma$ in $\mathbb{C}\,$ with metric $g\langle\,.\,,\,.\,\rangle$.

Letting $\mathbf{n}$ be the unit normal to $\gamma$ calculated in the standard metric on $\mathbb{C}\,$, and letting $f={|p|^{n-1}\over|p|+|\dot{p}|^{2}/4}$ , we have the following relations between the various flows:

$$V^{n}=f.\MCV^{1}_{f}=f.\MCV_{|p|^{n-2}},$$

(6.7)

and

$$V^{n}=\MCV^{1}-\left[{1\over 2}|p|^{2-n}\mathbf{n}(f)\right]\mathbf{n}=V^{1}-{1\over 2}(n-1){\mathbf{n}(|p|)\over|p|+|\dot{p}|^{2}/4}\,\mathbf{n}.$$

(6.8)

The problem with the first two is that on $\mathbb{C}\,\supset\gamma$ the flow is not parabolic, it has degeneracies at the end points. As $X^{1}$ is so closely modelled on $X^{n}$ (and is in fact canonically embedded in it), however, we might expect better on $X^{1}$. This is more or less true; the result is that writing (6.8) in terms of the unit normal $\mathbf{n}^{1}$ on $X^{1}$, we get

The last term is bounded (as near a zero of $p$, $\dot{p}\neq 0$ by nondegeneracy of $p$’s zeros) and so unimportant, we shall see, and the flow resulting from the first term is well understood. The second term is more curious; it is of the order of $\mathbf{t}^{1}\theta(p)\approx\mathbf{t}^{1}\theta((\gamma^{1})^{\prime})/2$ (where ${}^{\prime}=\mathbf{t}^{1}$ denotes differentiation with respect to arclength on $X^{1}$) whenever we are close to a point where $\gamma$ emanates from a zero of $p$ (so that $p\approx Ct$ and $\theta(p)\approx\theta(\gamma)\approx\theta(\gamma^{\prime})$). (The last approximation is of course not true if $\gamma$ simply passes close to a zero of $p$; then the equation blows up quickly as a glance at (6.5) shows, $\gamma$ flowing to this zero and breaking across it as discussed below; in the stable case we will be able to rule out this behaviour and need only consider $\gamma$ ending at the zero.)

But this is half the curvature of $\gamma^{1}$, so we get an approximation to the first term again, and something like mean curvature flow for $\gamma^{1}\subset X^{1}$. In fact in a small neighbourhood of (the double cover of) a zero of $p$, in coordinates $(x,y)$ in which $\gamma^{1}$ is a graph $y(x)$, the evolution PDE is of the general shape

$$y_{t}=y_{xx}+{y_{x}\over x},$$

where by the $\mathbb{Z}/2$-symmetry $y_{x}\arrowvert_{x=0}=0$, so the second term is approximately $y_{xx}$. So for some analysis we use this flow for $\gamma^{1}$, while for the rest we pass back to $n$-dimensions, and work with the phase function $\theta$ instead, giving a more standard (but $n$-dimensional) parabolic equation.

We first assert how the flow behaves, before proving it in the stable case in the next section. Note that any deformation of $\gamma$ is a hamiltonian deformation of $\gamma^{n}$ (and SLag $\gamma^{n}$s have no moduli) since the $\gamma^{n}$s are spheres. We picture what happens in Figures 2 and 4 in the 2 and 3 dimensional cases respectively. The dots represents zeros of $p$ in both cases, and the epsilons and zeros are phases.

In two dimensions the curves $\gamma$ whose Lagrangians $\gamma^{2}$ have constant phase are the straight lines, as can be seen from (6.4). Curves such as those marked 1 and 3 in Figure 2 flow towards a straight line (of some non-zero angle) corresponding to a SLag, whereas curve 2 flows up until it ‘hangs’ on a zero of $p$ (in finite time), where, on restarting the flow for 2 different curves, the separate flows form a kink and in the limit converge to destabilising SLags of different phases. These unions of SLags of different phases are still stationary for the volume functional (satisfying the second order variational equations, just not the first order SLag equations), and in fact are minimising in odd dimensions (the angle criterion [N], [L] makes minimality of the singular union locally equivalent to the above destabilising phase condition; in even dimensions reversing the order of the Lagrangians reverses the inequality and the configuration is not minimal, just stationary).

Again we see how the phase or angle criterion comes to bear; curves 1 and 2 are in the same homology class, but the two different phase signs give very different results. As noted before in Figure 1, this is related to the necessity of the phase to vary non-monotonically to form the unstable connect sum; in Figure 3 we plot the phases of the two connect sums, and with dotted lines their limits under the heat flow (7.4) (this is the correct modification of (2.5) in the non Ricci-flat case).

Drawing $\gamma$ the other way round the zero of $p$ gives something in the same homology class (the Dehn twist around the root of $p$ does not alter the homology class of the $S^{1}$ fibre over $\gamma$), which is the opposite connect sum discussed in [Th] – once the phase inequality becomes unstable for one connect sum it becomes stable for the other.

The two connect sums are related by monodromy, as in [Th]. Take a one parameter family of polynomials $p$ which rotates two zeros $z_{1},\,z_{2}$ of $p$ around each other. Then under the resulting monodromy a curve joining $z_{1}$ to a third zero $z_{3}$ is taken from being ‘above’ $z_{2}$ to being below it, thus turning one connect sum into the other.

The three dimensional picture is similar. In Figure 4 we plot the lines corresponding to SLags of phase zero, and connect sums $L_{1}\#L_{2}$ for $\phi(L_{1})=0$ and $\phi(L_{2})=\pm\epsilon$ (curves 1 and 2). Again we see the same behaviour with the phases behaving as in the graphs in Figure 3 and the flow getting hung on a zero of $p$ in the unstable case, splitting the Lagrangian.

Reversing the order of the connect sum in this case involves taking the $S^{2}$ fibre once around the zero of $p$; this effects a Dehn twist, reversing its orientation. Thus although curve 3 appears to give a Lagrangian in the same homology class, it is not; the phase of $L_{1}$ once we have been round the root of $p$ has shifted by $\pi$ and we get the connect sum $L_{2}\#(L_{1}[-1])$ discussed in [Th]. As is also discussed there, this can be seen to be unstable.

Things are not quite as simple as we have portrayed them if the initial curve has very large phase variation. It is quite possible for a curve corresponding to a stable Lagrangian, which is nonetheless very far from being a SLag, to pass close to a zero of $p$ without the large negative curvature away from the zero that the SLags exhibit. It can then flow into the zero, the Lagrangian being split into unions of Lagrangians of which it was a connect sum, despite their phases being such that they do not destabilise it. This limit is in the closure of the hamiltonian deformation orbit of the original Lagrangian, but does not contradict stability.

(A similar often-ignored subtlety occurs with stable bundles: when moduli of semistable bundles are created by using GIT on part of a Quot scheme, the orbits of stable bundles are not closed in Quot – the closures contain gradeds coming from any extension of sheaves forming the bundle, stable or not – and are not stable points for the group action on Quot; the GIT quotient of Quot would usually be empty. It is only in the subscheme of Quot representing semistable sheaves that orbits of stable bundles are in fact stable, and the moduli space is the GIT quotient of this part of Quot only.)

The point about stable Lagrangians is that this can be avoided by choosing the Lagrangian to have sufficiently small variation in phase $\theta$ (or sufficiently small volume) that it cannot be split into destabilising Lagrangians of different phase (or higher total volume); this we discuss now.