3.5 ‘Neck pinches’ using Lawlor necks [03P6]

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3.5 ‘Neck pinches’ using Lawlor necks

The programme of §3.2 requires a flow $\{L^{t}:t\in[0,\infty)\}$ starting from a single Lagrangian $L^{0}=L$ , but converging as $t\rightarrow\infty$ to a union $L_{1}\cup\cdots\cup L_{n}$ of several (possibly intersecting) special Lagrangians of different phases, where we regard $L_{1}\cup\cdots\cup L_{n}$ as a single immersed Lagrangian. Thus, we need a local model for how one Lagrangian $L$ can break up into a union $L_{1}\cup L_{2}$ of two Lagrangians under the flow, at some singular time $t=T_{i}$ , in the notation of §3.2.

We call this local model a ‘neck pinch’, as it involves the Lawlor necks $L_{\boldsymbol{\phi},A}$ of Example 2.5 as $A\rightarrow 0$ , so that the ‘neck’ pinches to a point. It is an example of Principles 3.9(b) and 3.11, where the special Lagrangian local models are the Lawlor necks $L_{\boldsymbol{\phi},A}$ . The possibility of such pinching behaviour is clear from Thomas and Yau [70], and Neves [55, §4] proves that it occurs in an example, where both [70, 55] work with $\mathop{\rm SO}\nolimits(m)$ -equivariant Lagrangians, so that Lagrangian MCF is reduced to understanding evolution of real curves.

Conjecture 3.16.

The following behaviour, which we call a ‘neck pinch’, can occur in Lagrangian MCF with surgeries in Calabi–Yau $m$ -folds for $m\geqslant 2,$ as in §3.2. Furthermore, ‘neck pinches’ are a generic singularity. That is, if Lagrangian MCF beginning from $L^{0}$ develops a neck pinch, then Lagrangian MCF beginning from any sufficiently small Hamiltonian perturbation $\tilde{L}^{0}$ of $L^{0}$ also develops a neck pinch.

Let $(M,J,g,\Omega)$ be a Calabi–Yau $m$ -fold, and extend $D^{b}{\mathbin{\mathscr{F}}}(M)$ to include immersed Lagrangians, as in [2]. Suppose $\{(L^{t},E^{t}):t\in(T-\epsilon,T+\epsilon)\}$ for $\epsilon>0$ small is a family of immersed Lagrangian branes in $M$ with $HF^{*}$ unobstructed, and $\{b^{t}:t\in(T-\epsilon,T+\epsilon)\}$ a corresponding family of bounding cochains, satisfying the following conditions:

(i)

The $(L^{t},E^{t},b^{t})$ for $t\in(T-\epsilon,T+\epsilon)$ are all isomorphic in $D^{b}{\mathbin{\mathscr{F}}}(M)$ .
(ii)

When $t<T,$ $L^{t},E^{t}$ depend smoothly on $t\in(T-\epsilon,T),$ and $\{(L^{t},E^{t}):t\in(T-\epsilon,T)\}$ satisfies Lagrangian MCF, with a finite time singularity at $t=T,$ with one singular point $p\in M$ .

Similarly, when $t\geqslant T,$ $L^{t},E^{t}$ depend smoothly on $t\in[T,T+\epsilon),$ and $\{(L^{t},E^{t}):t\in[T,T+\epsilon)\}$ satisfies Lagrangian MCF. The topology of $L^{t}$ for $t\in(T-\epsilon,T+\epsilon)$ changes discontinuously at $t=T$ . Nonetheless, the family $\{L^{t}:t\in(T-\epsilon,T+\epsilon)\}$ is continuous at $t=T$ in a suitable sense, e.g. as graded Lagrangian integral currents in Geometric Measure Theory.
(iii)

Identifying $M$ near $p$ with $T_{p}M\cong{\mathbin{\mathbb{C}}}^{m}$ near $0,$ for each $t\in(T-\epsilon,T),$ $L^{t}$ approximates a ‘Lawlor neck’ $L_{\boldsymbol{\phi}(t),A(t)}$ from Example 2.5, after a translation and a ${\rm U}(m)$ rotation in ${\mathbin{\mathbb{C}}}^{m}$ . Here $A(t)>0$ is small and $A(t)\rightarrow 0$ as $t\rightarrow T,$ so that $L_{\boldsymbol{\phi}(t),A(t)}$ converges to a union $\Pi_{0}\cup\Pi_{\boldsymbol{\phi}(T)}$ of transversely intersecting special Lagrangian planes in ${\mathbin{\mathbb{C}}}^{m}$ as $t\rightarrow T$ .
(iv)

For $t\in[T,T+\epsilon),$ there is a self-intersection point $p^{t}$ of $L^{t}$ where two local sheets $L^{t}_{\pm}$ of $L^{t}$ intersect transversely with $\mu_{L^{t}_{+},L^{t}_{-}}(p^{t})=1$ . Here $p^{t},L^{t}_{\pm}$ depend smoothly on $t\in[T,T+\epsilon),$ with $p^{T}=p$ .
(v)

We have $\theta_{L^{T}_{+}}(p^{T})=\theta_{L^{t}_{-}}(p^{T}),$ and $\theta_{L^{t}_{+}}(p^{t})<\theta_{L^{t}_{-}}(p^{t})$ for $t\in(T,T+\epsilon)$ .
(vi)

The ${\mathbin{\mathbb{F}}}$ -local systems $E^{t}$ for $t\in[T,T+\epsilon)$ are constructed from the ${\mathbin{\mathbb{F}}}$ -local systems $E^{t^{\prime}}$ for $t^{\prime}\in(T-\epsilon,T)$ by deleting the ‘neck’ in $L^{t^{\prime}}$ and extending $E^{t^{\prime}}$ over $p^{t}$ in $L^{t}_{\pm}$ in the unique possible way (at least for $m\geqslant 3$ ).
(vii)

When $t\in[T,T+\epsilon),$ the bounding cochain $b^{t}$ for $L^{t}$ includes an element $b^{t}_{p^{t}}\in\mathop{\rm Hom}\nolimits_{\mathbin{\mathbb{F}}}\bigl(E_{+}^{t}|_{p^{t}},E_{-}^{t}|_{p^{t}}\bigr)\otimes_{\mathbin{\mathbb{F}}}\Lambda_{\rm nov}^{\geqslant 0}$ as in §2.6. This is of the form

$b^{t}_{p^{t}}=a_{0}P^{\lambda(t)}+\text{higher order terms,}$

where $a_{0}\in\mathop{\rm Hom}\nolimits_{\mathbin{\mathbb{F}}}\bigl(E_{+}^{t}|_{p^{t}},E_{-}^{t}|_{p^{t}}\bigr)$ is the natural isomorphism induced from $E^{t^{\prime}}$ for $t^{\prime}\in(T-\epsilon,T)$ using (vi), and $\lambda(t)=\int_{T}^{t}\bigl(\theta_{L^{s}_{-}}(p^{s})-\theta_{L^{s}_{+}}(p^{s})\bigr){\rm d}s,$ so that $\lambda(T)=0$ and $\lambda(t)>0$ for $t\in(T,T+\epsilon)$ by (v).

Remark 3.17.

(a) The ‘neck pinching’ behaviour of Conjecture 3.16 is inverse to the ‘opening a neck’ behaviour of §3.4. So, for example, we can imagine a flow $\{L^{t}:t\in[0,\infty)\}$ satisfying the programme of §3.2, with two singular times $0<T_{1}<T_{2}$ , which starts with a single $L^{t}$ for $0\leqslant\penalty t<T_{1}$ , undergoes a ‘neck pinch’ at $t=T_{1}$ and becomes a union $L^{t}=L^{t}_{1}\cup L^{t}_{2}$ of Lagrangians $L^{t}_{1},L^{t}_{2}$ intersecting at one point $p^{t}$ for $T_{1}<t<T_{2}$ , and then at $t=T_{2}$ ‘opens the neck’ at $p^{t}$ and turns back into a single Lagrangian $L^{t}$ for $t>T_{2}$ .

Note that these inverse singular behaviours involve different (though related) geometric local models, Lawlor necks $L_{\boldsymbol{\phi},A}$ and Joyce–Lee–Tsui expanders $L_{\boldsymbol{\phi}}^{\alpha}$ . We do not just naïvely run the local picture for the flow in reverse. Note too that ‘neck pinching’ works only for $m\geqslant 2$ , whereas ‘opening necks’ works for $m\geqslant 1$ , so when $m=1$ , ‘opening necks’ has no inverse behaviour.

In a similar way, the author expects that many types of finite time singularity possible in the programme of §3.2 should have a corresponding inverse type, so that changes in the topology of $L^{t}$ , and other qualitative features, are reversible. An exception to this is that when $m=1$ , the flow can only decrease the number of self-intersection points, making the curve ‘less immersed’.

(b) Theorem 2.6 shows that Lawlor necks $L_{\boldsymbol{\phi},A}$ are the only possible geometric local models for such ‘neck pinches’.

(c) The inequality $\theta_{L^{t}_{+}}(p^{t})<\theta_{L^{t}_{-}}(p^{t})$ in (v) is the opposite of (3.8) in §3.4. Heuristically, we expect ‘small necks’ to shrink under Lagrangian MCF when $\theta_{L^{t}_{+}}(p^{t})<\theta_{L^{t}_{-}}(p^{t})$ , and to grow when $\theta_{L^{t}_{+}}(p^{t})>\theta_{L^{t}_{-}}(p^{t})$ .

(d) The case $m=2$ in Conjecture 3.16 is special. For $m\geqslant 3$ , the family ${\mathbin{\cal F}}$ of AC special Lagrangian ‘Lawlor necks’ $L$ in ${\mathbin{\mathbb{C}}}^{m}$ asymptotic to $\Pi_{0}\cup\Pi_{\boldsymbol{\phi}}$ is (isomorphic to) $(0,\infty)$ , and all such $L$ are exact. When $m=2$ , the family ${\mathbin{\cal F}}$ is ${\mathbin{\mathbb{R}}}^{2}\setminus\{0\}$ , and the subfamily ${\mathbin{\cal F}}_{\rm exact}$ of exact $L$ is ${\mathbin{\mathbb{R}}}\setminus\{0\}\subset{\mathbin{\mathbb{R}}}^{2}\setminus\{0\}$ , since then ${\mathbin{\cal F}}_{\rm exact}$ contains both the $L_{\boldsymbol{\phi},A}$ for $A>0$ and $\tilde{L}_{\boldsymbol{\phi},A}$ for $A<0$ in Example 2.5.

Also, when $m=2$ the local systems $E^{t^{\prime}}$ for $t^{\prime}\in(T-\epsilon,T)$ could have nontrivial holonomy around the ‘neck’. If so, the definition of $E^{t}$ for $t\in[T,T+\epsilon)$ in part (vi) no longer makes sense, since we cannot extend $E^{t^{\prime}}$ over $p^{t}$ in $L^{t}_{\pm}$ .

One conclusion is that for $m=2$ , though neck pinches should be generic under Hamiltonian perturbations, they may be nongeneric (and of index 1) under Lagrangian perturbations, since Lagrangian perturbations may allow the flow to wander in ${\mathbin{\cal F}}={\mathbin{\mathbb{R}}}^{2}\setminus\{0\}$ rather than ${\mathbin{\cal F}}_{\rm exact}={\mathbin{\mathbb{R}}}\setminus\{0\}$ , and will only hit the singularity $0\in{\mathbin{\mathbb{R}}}^{2}$ in real codimension 1 amongst initial Lagrangians.

We can also ask: if Lagrangian MCF $\{L^{t}:t\in(T-\epsilon,T)\}$ develops a singularity as $t\rightarrow T$ modelled on Lawlor necks $L_{\boldsymbol{\phi},A}$ for $A\in(0,\infty)\subset{\mathbin{\cal F}}_{\rm exact}={\mathbin{\mathbb{R}}}\setminus\{0\}$ , rather than continuing for $t>T$ using immersed SL 2-folds as in Conjecture 3.16, why not continue using Lawlor necks $\tilde{L}_{\boldsymbol{\phi},A}$ for $A\in(-\infty,0)\subset{\mathbin{\cal F}}_{\rm exact}={\mathbin{\mathbb{R}}}\setminus\{0\}$ , immediately opening the neck again, in a similar way to §3.4?

The author expects that this is the correct thing to do if $E^{t^{\prime}}$ for $t^{\prime}\in(T-\epsilon,T)$ has nontrivial holonomy around the ‘neck’. But in the trivial holonomy case, it would change the isomorphism class of $(L^{t},E^{t},b^{t})$ in $D^{b}{\mathbin{\mathscr{F}}}(M)$ , and so should be avoided according to the philosophy of §3.2.