3.3 On finite time singularities of Lagrangian MCF [03NX]

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3.3 On finite time singularities of Lagrangian MCF

Finite time singularities of Lagrangian MCF were discussed in §2.3. For graded Lagrangian MCF, Theorem 2.11 says that any finite time singularity must be of type II, and Theorem 2.12 that any finite time singularity must admit a ‘type II blow up’ modelled on a nontrivial eternal solution of Lagrangian MCF in ${\mathbin{\mathbb{C}}}^{m}$ . As in the end of §2.3, two natural classes of eternal solutions are provided by SL $m$ -folds in ${\mathbin{\mathbb{C}}}^{m}$ , and Lagrangian MCF translators.

Motivated by this, the next ‘principle’ gives heuristic pictures of how the author expects two different classes of finite time singularities to work.

Principle 3.9.

Let $(M,J,g,\Omega)$ be a compact Calabi–Yau $m$ -fold and $\{L^{t}:t\in[0,T)\}$ a family of compact, immersed, graded Lagrangians in $M$ satisfying Lagrangian MCF, with a finite time singularity at $t=T,$ and a singular point at $x\in M$ . Here are broad descriptions of two classes of such singularities:

(a)

Let $U$ be a small open neighbourhood of $x$ in $M,$ which we identify with a small open neighbourhood of $0$ in ${\mathbin{\mathbb{C}}}^{m}=T_{x}M,$ and $\epsilon>0$ be small. Then $L^{t}\cap U$ approximates a closed, exact SL $m$ -fold in ${\mathbin{\mathbb{C}}}^{m}$ for $t\in(T-\epsilon,T)$ .

Since SL $m$ -folds are stationary points of LMCF, to ‘first order’ $L^{t}\cap U$ is constant in $t,$ but to ‘second order’ $L^{t}\cap U$ wanders slowly in the moduli space of closed, exact SL $m$ -folds in ${\mathbin{\mathbb{C}}}^{m},$ until at time $t=T$ it hits a singular SL $m$ -fold. This ‘wandering’ is driven by ‘outside influences’ from the whole of $L^{t},$ not just from $L^{t}\cap U$ .

For example, if $N$ is an exact asymptotically conical SL $m$ -fold in ${\mathbin{\mathbb{C}}}^{m},$ we could have $L^{t}\cap U\approx f(t)\cdot N$ for $t\in(T-\epsilon,T),$ where $f:(T-\epsilon,T)\rightarrow(0,\infty)$ is smooth with $f(t)\rightarrow 0$ as $t\rightarrow T$ .
(b)

Let $U,\epsilon$ be as in (a). Then $L^{t}\cap U$ approximates a closed, exact LMCF translator in ${\mathbin{\mathbb{C}}}^{m}=T_{x}M$ for $t\in(T-\epsilon,T)$ . To ‘first order’ $L^{t}\cap U$ moves by translation in ${\mathbin{\mathbb{C}}}^{m}=T_{x}M,$ since it approximates a translating soliton. But to second order it also wanders slowly in the moduli space of closed, exact LMCF translators in ${\mathbin{\mathbb{C}}}^{m},$ driven by ‘outside influences’ from the whole of $L^{t},$ until at time $t=T$ it hits a singular soliton.

For example, if $N$ is an exact LMCF translator in ${\mathbin{\mathbb{C}}}^{m}$ with translating vector $v\in{\mathbin{\mathbb{C}}}^{m},$ we could have $L^{t}\cap U\approx f(t)\cdot N+g(t)\cdot v$ for $t\in(T-\epsilon,T),$ where $f,g:(T-\epsilon,T)\rightarrow(0,\infty)$ are smooth with $f(t)\rightarrow 0$ as $t\rightarrow T$ .

Remark 3.10.

(i) We will describe examples of behaviours (a),(b) in §3.5 and §3.8. Section 3.7 discusses a class of singularities not of type (a) or (b).

Note that in (a),(b) we do not simply mean that the singularity has a type II blow up $\{\tilde{L}^{s}:s\in{\mathbin{\mathbb{R}}}\}$ in Theorem 2.12 with $\tilde{L}^{s}$ special Lagrangian or an LMCF translator. In general type II blow ups describe only a small part of the singularity, and may give little idea of the global geometry and topology near the singular point. The point of (a),(b) is that in these cases we have a more complete picture of the singularity than a general type II blow up gives.

(ii) As in §2.3, Lagrangian MCF shrinkers do not occur in the graded case. The other major class of Lagrangian MCF solitons, Lagrangian MCF expanders (as in §2.3) are not relevant to the formation of singularities of the flow (that is, to describing the flow immediately before the singular time $t=T_{i}$ ). However, we can use Lagrangian MCF expanders to model the flow immediately after a surgery at a singular time $t=T_{i}$ , and we do this in §3.4.

If we believe Principle 3.9, stretching credulity a little further gives:

Principle 3.11.

Any type of (sufficiently well-behaved) singularity of SL $m$ -folds, which can appear as a limit of nonsingular, locally exact SL $m$ -folds, may provide a local model for finite time singularities of Lagrangian MCF.

Similarly, any (sufficiently well-behaved) singular Lagrangian in ${\mathbin{\mathbb{C}}}^{m}$ which can appear as a limit of nonsingular, exact Lagrangian MCF translators in ${\mathbin{\mathbb{C}}}^{m},$ may provide a local model for finite time singularities of Lagrangian MCF.

This suggests a class of research problems:

Problem 3.12.

(a) Choose from the literature your favourite family of explicit, nonsingular, exact SL $m$ -folds $N_{s}$ in ${\mathbin{\mathbb{C}}}^{m}$ which converge to an explicit singular SL $m$ -fold $N_{0}$ as $s\rightarrow 0$ . For example, let $N$ be an exact AC SL $m$ -fold in ${\mathbin{\mathbb{C}}}^{m}$ with cone $C,$ and take $N_{s}=s\cdot N$ for $s>0$ and $N_{0}=C$ .

Construct examples $\{L^{t}:t\in[0,T]\}$ of Lagrangian MCF in ${\mathbin{\mathbb{C}}}^{m}$ or in a Calabi–Yau $m$ -fold $(M,J,g,\Omega)$ with finite time singularities at $t=T$ for which $L^{T}$ has a singularity at $x\in{\mathbin{\mathbb{C}}}^{m}$ modelled on $N_{0},$ and $L^{t}$ near $x$ for $t\in(T-\epsilon,T)$ approximates $N_{s(t)},$ where $s(t)\rightarrow 0$ as $t\rightarrow T,$ as in Principle 3.9(a).

(b) If you can do (a), determine whether Lagrangian MCF starting from a small generic Hamiltonian perturbation of $L^{0}$ also develops finite time singularities of the same type. In this case, we call this type a generic singularity of Lagrangian MCF. If it is not generic, compute the expected codimension amongst Hamiltonian perturbations of $L^{0}$ in which singularities of this type occur.