We have to be slightly careful with our flow (6.9) in the Shapere-Vafa example as the metric is not quite Ricci-flat. The proof of the evolution equation for the phase function $\theta$ (2.5) and volume (2.6) show that the equations must be modified to

	$\displaystyle{d\over dt}\theta\!$	$\displaystyle=$	$\displaystyle\!-\Delta\,\theta+{\langle d\theta,d|\Omega|\rangle\over|\Omega|},$		(7.4)
	$\displaystyle{d\over dt}(|\Omega|\vol_{L})\!$	$\displaystyle=$	$\displaystyle\!-|d\theta|^{2}(|\Omega|\vol_{L}).$		(7.5)

when $|\Omega|=|\Omega/\!\vol\!|$ is not $\equiv 1$. Therefore the maximum principle still holds for $\theta$, and the condition (7.1) is again preserved by the flow. Similarly if we measure volume with respect to $|\Omega|\vol_{L}$ then this is decreasing and (7.2) is preserved by the flow. We can now prove the appropriate version of our conjecture in this example. From the proof it will also be clear that the original conjecture could be proved in this case in the $O(n)$-invariant Ricci flat metric if we knew it, we would just not be able to be as explicit about the flow equations.

We break the proof up into existence of the flow (best dealt with at the level of $\gamma^{n}\subset X^{n}$), controlling the angle variation (using $\gamma\subset\mathbb{C}\,$) to ensure no $180^{o}$ kinks appear in $\gamma$, and using this to show the flow exists for all time (for which we use $\gamma^{1}\subset X^{1}$ and $\theta$ on $\gamma^{n}$). We follow [An], in parts heavily modified to take care of the endpoints of $\gamma$. Finally we will show that the flow converges to a SLag.

While the curvature of the curve $\gamma^{1}\subset X^{1}$ is bounded, so is the norm of the flow vector (the last term in (6.9) is always bounded, and the second term can be bounded by the curvature at an intermediate point by Taylor’s theorem). So at any finite time $T$ the flow converges to a limit curve $\gamma_{T}^{1}$ pointwise. Parametrising the curves by their arclength on $X^{1}$, their first and second derivatives as maps to $X^{1}$ are therefore bounded, which by Arzelà-Ascoli implies that for a subsequence of $t$ we have convergence in $C^{1}$ to a $C^{1}$ curve with bounded (weak) curvature. By the uniqueness of the limit, then, $\gamma_{T}^{1}\subset X^{1}$ has bounded curvature.

Bounds on (the derivative of) the phase of $\gamma^{1}$ give corresponding bounds on (the derivative of) the phase of $\gamma^{n}$ (via (6.9) for $n$ and $n=1$). So the phase function $\theta$ of $\gamma^{n}$ is also $C^{0}$ convergent to the phase of $\gamma_{T}^{n}$, and satisfies the parabolic equation (7.4). Putting this into local coordinates and differentiating with respect to arclength $s$, we get a uniformly parabolic equation with bounded coefficients and a bounded solution $\theta_{s}$ on $t\in[0,T]$. By ([LSU] Section III Theorem 10.1), then, $\theta_{s}$ is in fact $\alpha$-Hölder continuous for some $\alpha>0$, and $\gamma^{1}_{T}$ is $H^{2+\alpha}$. By the existence of the flow for $H^{2+\alpha}$ initial conditions, then, the flow exists for some time $t>T$. $\square$

Unfortunately, in this region, the bounds we want for $\theta(\gamma^{\prime})$ do not follow directly from (6.4) and the bounds we have for $\theta$, and in fact only follow from comparison with known solutions. Draw the SLags of phase $S,\,I$ emanating from a zero of $p$, i.e. the curves in $\mathbb{C}\,$ solving $\theta(\gamma^{\prime})+(n/2-1)\theta(p(\gamma))=S$ or $I$. (Since this is an ODE, there is no problem in finding solutions and extending them to either infinity or another zero of $p$; see [SV].) In the tangent space to the zero of $p$ this gives a cone of angle $(S-I)/n<2\pi/3n$ which $\gamma$ lies inside and cannot cross either at $t=0$ or any later time in the flow. So in a sufficiently small neighbourhood of an endpoint of $\gamma$, we may bound $\theta(\gamma)$ inside a cone of angle less than $2\pi/3n$, and also take $\theta(p(\gamma))$ to be within any given $\epsilon$ of $\theta(\gamma)+C$ (since the zero of $p$ is nondegenerate; here $C$ is the phase of $\dot{p}$ at the zero of $p$). Thus $\theta(p(\gamma))$ can be bounded inside a similar cone, so that (6.4) bounds the variation of $\theta(\gamma^{\prime})$ by $2\pi/3+(n/2-1)2\pi/3n<\pi$.

The bounds on $\theta(\gamma^{\prime})$ imply that the curve does not spiral round its endpoints but moves away from them with nonzero derivative inside the above cone until it is outside the small neighbourhood employed above. So the remaining case to consider is if the curve can pass arbitrarily close to one of its own endpoints at some bounded-below arclength from its endpoints, i.e. if the cone of SLags above starting from a zero of $p$ passes either side of that same zero at some nonzero arclength. But then there would be a Slag fibred over a curve starting and ending at the same root of $p$, with our Lagrangian $\gamma^{n}$ the connect sum of this SLag and some other $\gamma_{1}^{n}$. But $\gamma^{n}$ cannot flow arbitrarily close to such a connect sum as its phase variation would approach at least the difference between the phase of the SLag and $\phi(\gamma^{1}_{n})$, contradicting (7.1). $\square$

By Lemma 7.7 the flow exists for all time unless, as we suppose now, the curvature of $\gamma^{1}$ becomes unbounded in finite time. Then to get a contradiction we start by scaling as in [An]. Pick $s_{i},\,t_{i},\ i=1,2,\ldots$ such that $t_{i}\to\infty$ and the curvature $\kappa_{i}$ of $\gamma^{1}(s_{i},t_{i})=y_{i}$ is maximal over the curvatures of $\gamma^{1}(s,t)$ for all $s$, and all $t\leq t_{i}$ (here we parameterise by arclength $s$ on $X^{1}$, centred at a zero $z$ of $p$, i.e. $\gamma^{1}\arrowvert_{s=0}=z$ lies over an endpoint of $\gamma$).

How we handle the blow up depends on whether it happens at the branch points of $\gamma^{1}$ (i.e. where $\gamma^{1}$ branches over the endpoints of $\gamma$), by which we mean $|s_{i}|=O(|\kappa_{i}^{-1}|)$, or in the interior $|s_{i}|\gg|\kappa_{i}^{-1}|$, due to the different nature of (6.9) at the branch points. We first deal with the interior where the flow is a perturbation of mean curvature flow and so can be handled by [An]:

Then we rescale as in [An];

$$s\mapsto\kappa_{i}s,\hskip 10.00002ptg\mapsto\kappa_{i}g,\hskip 10.00002ptt\mapsto\kappa_{i}^{2}(t-t_{i}),$$

(7.10)

where $g$ is the metric on $X^{1}$. This rescaled flow for $\gamma^{1}_{i}$ has the same form as (6.9),

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

but with $|p|$ and $|\dot{p}|$ replaced by their pullbacks to the new Riemannian surface (here it is important that $\dot{p}$ is still computed in the old coordinates, then pulled back). Therefore their gradients are scaled by $\kappa_{i}^{-1}$. $\mathbf{n}$ denotes the unit normal to $\gamma^{1}$ in the new metric on $X^{1}$. The curvature gets scaled by $\kappa_{i}^{-1}$ and so has a maximum, over $t\leq 0$, of 1 at $y_{i}$ (at time $t=0$). We want to show that the two perturbation terms on the right hand side of the above flow tend to zero as $i\to\infty$.

In the rescaled variables, work in a geodesic disc of radius $r_{i}/2$ (defined above; this tends to infinity as $i\to\infty$, importantly) about $y_{i}$. As $s_{i}\to 0$, for $i$ sufficiently large this is within an arbitrarily small neighbourhood of $z$ in the original metric, in which $\gamma^{\prime}$ varies within an angle $<\pi$ cone as in the proof of Lemma 7.8, i.e. $(\gamma^{1})^{\prime}$ varies within an angle $<\pi/2$ cone on the double cover $X^{1}$. So arclength $s$ on $\gamma^{1}$ and radial distance $r$ in $X^{1}$ are equivalent metrics on $\gamma^{1}$ in this disc; $r_{s}:=\partial r/\partial s$ and $s_{r}=\partial s/\partial r$ are both bounded.

As $y_{i}\in\gamma_{i}^{1}$ is of arclength $r_{i}$ from $z$ (the zero of $p$) at $s=0$, we deduce that all points of our disc are of distance $cr_{i}/2$ from the zero of $p$ (for some constant $c>0$ fixed for all $i\gg 1$) in the new metric. Thus, for $i$ large enough, we have

$$|p_{i}|^{1/2}\geq C\kappa_{i}^{-1}(cr_{i}/2),$$

where $C$ is a constant just less than the norm of the derivative of $p^{1/2}$ at the zero $z$ in the original metric on $X^{1}$ ($p^{1/2}$ pulls back to a well defined function on $X^{1}$ with a simple zero at $z$). We can therefore bound

$$|\dot{\gamma}_{i}^{1}-\MCV|\leq(n-1){\kappa_{i}^{-1}\sup|d(p^{1/2})|\over C\kappa_{i}^{-1}(cr_{i}/2)}+{1\over 2}\kappa_{i}^{-1}\sup|d\log(|p|+|\dot{p}|^{2}/4)|,$$

where both sups are taken over small neighbourhoods of $z$ in the original metric on $X^{1}$. As $i\to\infty$, $\kappa_{i},\,r_{i}\to\infty$, so the above bound tends to zero, while the radius of the disc we are working on $r_{i}/2\to\infty$. It follows that in the limit we get exactly mean curvature flow of an infinite disc $\mathbb{R}^{n}$; see ([An] Section 9) for how to pass to the limit to conclude that for this blow up to occur a $180^{o}$ kink must appear in the curve $\gamma^{1}$ (by which we mean the limsup in Lemma 7.8 is $\geq\pi$). But this contradicts Lemma 7.8.

(We do not repeat Angenent’s argument here as we will give a slightly harder, $n$-dimensional, version of it around the endpoints of $\gamma$ in Lemma 7.11 below. The point is just that in the rescaling we can get rid of the last two terms of our flow to reduce to the results of [An].) $\square$

The remaining case we must dismiss is that of $\kappa_{i}$ blowing up at points $y_{i}=\gamma^{1}(s_{i},t_{i})$ with $|s_{i}|<C/|\kappa_{i}|$ for some fixed $C$. Here we must work harder than in [An].

Taking $i$ sufficiently large that the metric on the radius $\kappa_{i}^{1/2}$ disc about $z$ in $X^{1}$ is sufficiently close to being flat, define geodesic polar coordinates on $X^{1}$, $r^{1}:=|\gamma^{1}|,\ \theta^{1}:=\theta(\gamma^{1})$ (which is $\theta(\gamma)/2$ to within a constant). Then we can assume that $\theta^{1}_{s}$ is arbitrarily $C^{1}$ close to

$${1\over r}\sin(\theta(\gamma^{1}_{s})-\theta^{1}),$$

(7.12)

which is the exact formula for a flat metric and polar coordinates. This bounds $|r\theta^{1}_{s}|$. Since by construction the curvature of $\gamma^{1}$ is not more than one, i.e. $|(\theta(\gamma^{1}_{s}))_{s}|\leq 1$, we can bound $|\theta(\gamma^{1}_{s})|\leq s$.

Note that (7.12) in flat space gives us the differential equation

$$f_{s}=\kappa_{i}-{\sin f\over r}$$

for $f=\theta(\gamma^{1}_{s})-\theta^{1}$, with $f(0)=0$ and $|\kappa_{i}|\leq 1$. This implies that $|f(s)|\leq|s|$ (consider a point where the graph of $f$ crosses that of $\pm s$, where $|f_{s}|\geq 1$, for a contradiction), so for $i$ sufficiently large that our polar coordinates are sufficiently close to flat coordinates we can deduce a bound on $|\theta(\gamma^{1}_{s})-\theta^{1}|/s$. Thus $\theta^{1}/s$ is also bounded, and $\theta^{1}/r$ by the uniform comparison bounds of $r$ and $s$ given by the cone argument in Lemma 7.8.

Instead of considering the equation (6.9) for $\gamma^{1}$, we analyse the equation (7.4) for $\theta$. After rescaling it becomes

$$\dot{\theta}=\Delta\theta+{\langle d\theta,d(|\Omega|_{i})\rangle\over|\Omega|_{i}},$$

(7.13)

where $|\Omega|_{i}$ is the pullback of $|\Omega|:=|\Omega/\!\vol\!|$ to $\gamma^{1}$ with its new metric. Note also that pulling functions up from $\gamma$ and taking their exterior derivative $d$ on either $\gamma^{1}$ or on $\gamma^{n}$ gives the same result via the obvious inclusion $\gamma^{1}\subset\gamma^{n}$ commuting with the projections to $\gamma$. Again we want to control this evolution equation as $i\to\infty$.

$|d\theta|=|\theta_{s}|=|\partial_{s}[\theta(\gamma^{1}_{s})/2+(n/2-1)\theta(p(\gamma^{1}))]$, and this is bounded by the estimates above, for $i$ sufficiently large that $\theta(p(\gamma^{1}))$ is $C^{1}$ close to $\theta^{1}/2$ in the disc in which we are working. So we can bound the last term in (7.13) by a constant times

$$\kappa_{i}^{-1}{\sup|d\Omega|\over\inf|\Omega|},$$

where the sup and inf are taken in the original metric over a small neighbourhood of $(0,\ldots,0,z)\in X^{n}$. This tends to zero as $i\to\infty$.

Computing the Laplacian on the space $\gamma^{n}$, with a radial coordinate $s$ and rotational symmetry about the origin $s=0$, makes (7.13)

$$\theta_{t}=\theta_{ss}+(n-1){R^{i}_{s}\over R^{i}}\theta_{s}+O(\kappa_{i}^{-1}),$$

(7.14)

where $R^{i}=R^{i}(s)$ is the radius of the sphere $S^{n-1}$ at $s$ in the new metric.

To compute $R^{i}$, we use our $\gamma^{1}\subset X^{1}$ arclength coordinate $s$, the radial coordinate $r$ on $X^{1}$, and a radial coordinate $\rho$ on $\mathbb{C}\,$. For $i$ sufficiently large, for $s\leq\kappa_{i}^{1/2}$ in the new metric, we can approximate $p$ linearly about $z$ and so assume that $R^{i}$ is as close as we like to $\kappa_{i}\sqrt{|\dot{p}(z)|\rho}$ in $C^{2}$. Therefore $R^{i}_{r}$ is approximated by

$$R^{i}_{r}={R^{i}_{\rho}\over r_{\rho}}\approx{1\over\sqrt{1+{4\rho\over\kappa_{i}^{2}|\dot{p}(z)|}}},$$

which is bounded and tends to $1$ in the interval $s\in[0,\kappa_{i}^{1/2})$. Similarly $R^{i}_{rr}$ can be taken to be arbitrarily small for $i$ sufficiently large.

Since $r_{s}$ is bounded, this gives bounds on $R^{i}_{s}$, implying that, on passing to a subsequence if necessary, the functions $R^{i}(s)$ are convergent as $i\to\infty$ by the Arzelà-Ascoli theorem.

Note also that for all $i$, $R^{i}_{s}(0)=1$. But to preserve this in the limit, we must similarly bound $R^{i}_{ss}$. Differentiating $r^{2}_{s}+r^{2}(\theta^{1}_{s})^{2}=1$ and $R^{i}_{s}=R^{i}_{r}r_{s}$ gives

$$R^{i}_{ss}=R^{i}_{rr}r_{s}^{2}-R^{i}_{r}(r(\theta^{1}_{s})^{2}+r^{2}\theta^{1}_{s}\theta^{1}_{ss}/r_{s}).$$

(7.15)

We have bounded $R^{i}_{rr},\ r_{s},\ r_{s}^{-1},\ r\theta^{1}_{s}$ and $\theta^{1}_{s}$; all this leaves is the last term in (7.15).

We have approximated $\theta^{1}_{s}$ in $C^{1}$ by ${1\over r}\sin(\theta(\gamma^{1}_{s})-\theta^{1})$; differentiating approximates $r^{2}\theta^{1}_{s}\theta^{1}_{ss}/r_{s}$ as closely as we like (as $i\to\infty$) to

$${r\theta^{1}_{s}\over r_{s}}(\kappa_{i}-\theta^{1}_{s})\cos(\theta(\gamma^{1}_{s})-\theta^{1})-\theta^{1}_{s}\sin(\theta(\gamma^{1}_{s})-\theta^{1}),$$

which we have bounded already. In conclusion, after passing to a subsequence if necessary, $R^{i}$ is $C^{1}$ convergent to some $R$ with $R_{s}(0)=1$, and the phase function $\theta^{\infty}$ of the limit curve $\gamma^{1}_{\infty}$ (which exists by Arzelà-Ascoli since $|\gamma^{1}_{s}|=1$ and $|\gamma^{1}_{t}|$ is bounded by the bound on its curvature $\kappa$) satisfies the limit of (7.14):

$$\theta^{\infty}_{t}=\theta^{\infty}_{ss}+(n-1){R_{s}\over R}\theta^{\infty}_{s}.$$

(7.16)

But this is just the heat equation for $\theta$ on an $n$-dimensional space with $O(n)$ symmetry, radial coordinate $s$, and radius $R(s)$ of the $S^{n-1}$ fibre over $s$. By construction of the time rescaling (7.10) it exists for all time $t\leq 0$, and the solution $\theta^{\infty}$ is bounded. Therefore, by Moser’s Harnack inequality [Mo], $\theta^{\infty}$ is in fact constant.

But for all $i$, max $\theta_{s}=1$ by construction, and passing to a subsequence if necessary the point where the maximum is obtained is convergent. To show then that max $\theta^{\infty}_{s}=1$, to get our contradiction, we need only know that $\theta_{s}$ is, say, uniformly (in $i$) Hölder continuous. This is again a consequence of ([LSU] Section III Theorem 10.1) as at the end of the proof of Lemma 7.7. By the boundedness of $\theta_{s}$ and the parabolic equation it satisfies (different for each $i$), $\theta_{s}$ is in fact $\alpha$-Hölder continuous for some $\alpha>0$, and its $H^{\alpha}$ norm can be bounded by the bounds on the coefficients of the parabolic equations. But these are bounded uniformly in $i$, as a glance at (7.13) confirms: the correction term tends to zero, and the Laplacian term (and its derivative with respect to $s$) is controlled by $C^{2}$ bounds on the metric which we provided above by bounding $R^{i}(s),\,R^{i}_{s}$ and $R^{i}_{ss}$. $\square$

Finally we show that this infinite time flow converges using standard techniques (see for instance [C] for a harder result). Notice that the same scaling proof (7.11) that the curvature of $\gamma^{1}$ does not blow up in finite time shows the same for our now infinite time flow. So using, the $O(n)$ symmetry, the curvature of the metric on $\gamma^{n}$ stays uniformly bounded, and we have a $C^{1}$ bound on $\theta$. Therefore in the equation (7.4) for $\theta$ on $\gamma^{n}$, which we rewrite as

$$\dot{\theta}=-\Delta^{\Omega}\theta:=|\Omega|^{-1}*d(|\Omega|*d\theta),$$

(7.17)

the coefficients have at least uniform $C^{1}$ bounds; $\theta$ then acquires a uniform $C^{3}$ bound by parabolic theory (see [LSU] III Theorem 12.1, for instance). Again by $O(n)$ symmetry we now get uniform $C^{2}$ bounds on $\gamma^{n}$’s curvature. And so it goes on, allowing us to extract a subsequence of times for which the flow converges in $C^{\infty}$ to a Slag.

To see that the flow converges without having to pass to a subsequence we need only show convergence of $\theta$ in $L^{2}$; this way no other subsequence of the flow can converge to a different limit in $C^{\infty}$. In fact we use an $L^{2}$-norm weighted by $|\Omega|$, and compute using (7.17) and (7.5):

$${d\over dt}\int_{\gamma^{n}}(\theta-\bar{\theta})^{2}|\Omega|\vol=\int_{\gamma^{n}}\left\{2(\theta-\bar{\theta})(-\Delta^{\Omega}\theta-{d\over dt}\bar{\theta})-(\theta-\bar{\theta})^{2}|d\theta|^{2}\right\}|\Omega|\vol,$$

where $\bar{\theta}=\int\theta|\Omega|\vol/\int|\Omega|\vol$ is constant on $\gamma^{n}$ (but not in time). This is then bounded above by

$$-2\int(\theta-\bar{\theta})\Delta^{\Omega}(\theta-\bar{\theta})|\Omega|\vol.$$

It is easily checked that $\Delta^{\Omega}=d_{\,}^{*^{\Omega}}d$, where $d_{\,}^{*^{\Omega}}$ is the adjoint of $d$ with respect to the $L^{2}$-metric $\int_{\gamma^{n}}\langle\ \cdot\ ,\ \cdot\ \rangle|\Omega|\vol$ we are using. So its kernel is just the constants, and by the uniform $C^{\infty}$ bounds on the metric of $\gamma^{n}$ and $|\Omega|$ we can get a uniform lower bound $\lambda>0$ for its first nonzero eigenvalue. This then gives a bound $\int(\mu\Delta\mu)|\Omega|\vol\geq\lambda\int\mu^{2}|\Omega|\vol$ for functions $\mu$ of integral zero. Setting $\mu=\theta-\bar{\theta}$ gives

$${d\over dt}\int(\theta-\bar{\theta})^{2}|\Omega|\vol\leq-2\lambda\int(\theta-\bar{\theta})^{2}|\Omega|\vol,$$

which then tends to zero as required.

We end by noting that one can get cone-type bounds similar to those of Lemma 7.8 on the tangent direction $\theta(\gamma^{1}_{s})$ even as a curve $\gamma$ representing an unstable Lagrangian $\gamma^{n}$ approaches and breaks across a zero of $p$. Suppose that the initial phase variation of $\gamma^{n}$ is, without loss of generality, in some $(-\delta,\delta)$. Draw the cone with boundary the SLags emanating from the zero of $p$ with phase $-\delta,\ \pi+\delta$ (the straight lines emanating from the zeros of $p$ drawn in Figures 2 and 4 display the $\delta=0$ cone for dimensions 2 and 3 respectively). Then in a sufficiently small neighbourhood of the zero, the variation in $\theta(p)$ can be taken to be less than $2\pi/n+2\delta+\epsilon$ for any $\epsilon>0$. Since we are free to make $\delta$ slightly smaller without violating the initial bounds on phase, we can ensure that there is a neighbourhood of the zero of $p$, and a cone with vertex at $p$ whose walls $\gamma$ cannot cross, such that for the part of $\gamma$ lying in this neighbourhood, the variation

$$\sup\theta(p(\gamma))-\inf\theta(p(\gamma))<2\pi/n+2\delta.$$

Comparing with (6.4) gives

$$\sup\theta(\gamma^{\prime})-\inf\theta(\gamma^{\prime})<(n/2-1)(2\pi/n+2\delta)+2\delta=(1-2/n)\pi+n\delta$$

so that again no $180^{o}$ kinks can occur while this is less than $\pi$, i.e. for $\delta\leq 2\pi/n^{2}$. So again the analysis should be tractable in this case (more general spiralling around a zero would make matters worse). However, we have not carried out the analysis necessary to show that at the moment $\gamma$ reaches the zero of $p$ it is sufficiently smooth that the two resulting curves it splits into give $C^{2}$ Lagrangians (whose flow we could restart).

Of course by just studying the simple examples above we cannot hope to know how bad the singularities are that arise in finite time in the general case. Also, as mentioned in [Th], we should perhaps restrict to those Lagrangians whose Floer cohomology is well defined [FO3]. This includes all homology spheres, however.

We should also point out the obvious fact that most of the evidence for our conjecture, other than perhaps the mirror symmetry and study of Joyce’s examples in [Th], has been essentially one-dimensional (either for $T^{2}$ in [Th], or by symmetry reduction in the examples above). This is unrepresentative, essentially because the angles at which Lagrangians intersect (the $\alpha_{i}$s of (3.3)) are all the same in this situation, and so are determined by the phase (their sum). So interesting phenomena, where degrees in Floer cohomology change (e.g. a Hom becomes an Extⁱ on the mirror while the phase remains fixed) are largely lost due to them being controlled entirely by the phase.