2.3 Lagrangian mean curvature flow [03N1]

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2.3 Lagrangian mean curvature flow

Next we discuss (Lagrangian) mean curvature flow. A book on mean curvature flow (MCF) for hypersurfaces in ${\mathbin{\mathbb{R}}}^{n}$ is Mantegazza [50]. Two useful surveys on Lagrangian MCF are Smoczyk [67] and Neves [56].

Let $(M,g)$ be a Riemannian manifold, and $N$ a compact manifold with $\mathop{\rm dim}\nolimits N<\mathop{\rm dim}\nolimits M$ , and consider embeddings or immersions $\iota:N\hookrightarrow M$ , so that $\iota(N)$ is a submanifold of $M$ . Mean curvature flow (MCF) is the study of smooth 1-parameter families $\iota_{t}$ , $t\in[0,T)$ of such $\iota_{t}:N\hookrightarrow M$ satisfying

\frac{{\rm d}\iota_{t}}{{\rm d}t}=H_{\iota_{t}},

where $H_{\iota_{t}}\in C^{\infty}(\iota_{t}^{*}(TM))$ is the mean curvature of the submanifold $\iota_{t}:N\hookrightarrow M$ . We usually write $N^{t}$ rather than $\iota_{t}:N\hookrightarrow M$ , suppressing the immersion, so that $\{N^{t}:t\in[0,T)\}$ is a family of submanifolds satisfying MCF.

Mean curvature flow is the gradient flow of the volume functional for compact submanifolds $N$ in $M$ . It has a unique short-time solution starting from any compact submanifold $N$ .

Now let $(M,J,g,\Omega)$ be a Calabi–Yau $m$ -fold, and $L$ a compact Lagrangian submanifold in $M$ . Then the mean curvature of $L$ is $H=J\nabla\Theta_{L}$ , where $\Theta_{L}:L\rightarrow{\rm U}(1)$ is the phase function from Definition 2.1. Thus $H$ is an infinitesimal deformation of $L$ as a Lagrangian. Smoczyk [66] shows that MCF starting from $L$ preserves the Lagrangian condition, yielding a 1-parameter family of Lagrangians $\{L^{t}:t\in[0,\epsilon)\}$ with $L^{0}=L$ , which are all in the same Hamiltonian isotopy class if $L$ is Maslov zero. This is Lagrangian mean curvature flow (LMCF). Special Lagrangians are stationary points of Lagrangian MCF.

We will be especially interested in Lagrangian MCF for graded Lagrangians. Suppose $\{L^{t}:t\in[0,T)\}$ is a family of compact, graded Lagrangians satisfying Lagrangian MCF. Then $L^{t}$ are all Hamiltonian isotopic, that is, graded Lagrangian MCF stays within a fixed Hamiltonian isotopy class. Also, if the phase function $\theta_{L^{0}}$ takes values in an interval $[a,b]$ or $(a,b)$ , then so does $\theta_{L^{t}}$ for $t\in[0,T)$ . Thus, Lagrangian MCF preserves the almost calibrated condition.

It is an important problem to understand the singularities which arise in Lagrangian mean curvature flow. Singularities in Lagrangian MCF are often locally modelled on soliton solutions, Lagrangians in ${\mathbin{\mathbb{C}}}^{m}$ which move by rescaling or translation under Lagrangian MCF.

Definition 2.9.

A closed Lagrangian $L$ in ${\mathbin{\mathbb{C}}}^{m}$ is called an LMCF expander if $H=\alpha F^{\perp}$ in $C^{\infty}(T{\mathbin{\mathbb{C}}}^{m}|_{L})$ , where $H$ is the mean curvature of $L$ and $F^{\perp}$ is the orthogonal projection of the position vector $F$ (that is, the inclusion $F:L\hookrightarrow{\mathbin{\mathbb{C}}}^{m}$ ) to the normal bundle $TL^{\perp}\subset T{\mathbin{\mathbb{C}}}^{m}|_{L}$ , and $\alpha>0$ is constant.

This implies that (after reparametrizing by diffeomorphisms of $L$ ) the family of Lagrangians $L^{t}:=\sqrt{2\alpha t}\,L$ for $t\in(0,\infty)$ satisfy Lagrangian mean curvature flow. That is, Lagrangian MCF expands $L$ by dilations.

Similarly, we call $L$ an LMCF shrinker if $H=\alpha F^{\perp}$ for $\alpha<0$ , and then $L^{t}:=\sqrt{2\alpha t}\,L$ for $t\in(-\infty,0)$ satisfy LMCF, so LMCF shrinks $L$ by dilations.

We call $L$ an LMCF translator if $H=v^{\perp}$ , where $v\in{\mathbin{\mathbb{C}}}^{m}$ is the translating vector of $L$ , and $v^{\perp}$ the orthogonal projection of $v$ to $TL^{\perp}$ . Then $L^{t}:=L+tv$ for $t\in{\mathbin{\mathbb{R}}}$ satisfy LMCF, so Lagrangian MCF translates $L$ in ${\mathbin{\mathbb{C}}}^{m}$ .

Finite time singularities of MCF have a fundamental division into ‘type I’ and ‘type II’ singularities:

Definition 2.10.

Let $(M,g)$ be a compact Riemannian manifold (e.g. a Calabi–Yau $m$ -fold) and $\{L^{t}:t\in[0,T)\}$ a family of compact immersed submanifolds in $M$ (e.g. Lagrangians) satisfying mean curvature flow. We say that the family has a finite time singularity at $t=T$ if the flow cannot be smoothly continued to $[0,T+\epsilon)$ for any $\epsilon>0$ . As in Wang [71, Lem. 5.1] this implies that $\mathop{\rm lim\,sup}_{t\rightarrow T}\|A^{t}\|_{C^{0}}\rightarrow\infty$ , where $A^{t}$ is the second fundamental form of $L^{t}$ .

We call such a finite time singularity of type I if $\|A^{t}\|_{C^{0}}^{2}\leqslant\penalty C/(T-t)$ for some $C>0$ and all $t\in[0,T)$ . Otherwise we call the singularity of type II.

We call $x\in M$ a singular point of the flow if $\mathop{\rm lim\,sup}_{t\rightarrow T}\|A^{t}|_{U\cap L^{t}}\|_{C^{0}}=\infty$ for all open neighbourhoods $U$ of $x$ in $M$ .

Huisken [29] showed that type I singularities developing a singularity at $x\in M$ are locally modelled in a strong sense on MCF shrinkers in ${\mathbin{\mathbb{R}}}^{n}=T_{x}M$ , through a process known as ‘type I blow up’, as in Smoczyk [67, Prop. 3.17] or Mantegazza [50, §3].

However, we are interested in MCF of graded Lagrangians in Calabi–Yau $m$ -folds, and it turns out that type I singularities do not occur in graded Lagrangian MCF, as was proved by Wang [71, Rem. 5.1] and Chen and Li [13, Cor. 6.7] in the almost calibrated case (i.e. Lagrangians $L^{t}$ with phase variation less than $\pi$ ) and by Neves [55, Th. A] in the graded (or Maslov zero) case.

Theorem 2.11.

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. Then the flow cannot develop a type I singularity.

A parallel result of Neves [56, Cor. 3.5] says that there exist no nontrivial, immersed, graded Lagrangian MCF shrinkers in ${\mathbin{\mathbb{C}}}^{m}$ (satisfying a few extra conditions such as closed in ${\mathbin{\mathbb{C}}}^{m}$ and of bounded Lagrangian angle), so there are no possible local models for type I blow ups of graded Lagrangian MCF. Examples of Lagrangian MCF shrinkers in ${\mathbin{\mathbb{C}}}^{m}$ can be found in Abresch and Langer [1] for $m=1$ and in Anciaux [3] and Joyce, Lee and Tsui [43, Th. F] in higher dimensions, but none of them are graded.

So, for graded Lagrangian MCF, all finite time singularities are of type II. It is a well known ‘folklore’ theorem that type II singularities of MCF admit ‘type II blow ups’, eternal smooth solutions of MCF in ${\mathbin{\mathbb{R}}}^{n}$ modelling the formation of the singularity in the small region where the second fundamental form $A^{t}$ is largest as $t\rightarrow T$ . The idea of type II blow ups is due to Hamilton, and explanations can be found in Smoczyk [67, §3.4] and Mantegazza [50, §4.1], and for Lagrangian MCF in Han and Li [24, §2]. We state it for graded LMCF:

Theorem 2.12.

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 $t=T$ . Then at some singular point $x\in M$ of the flow there exists a type II blow up.

That is, identifying $M$ near $x$ with $T_{x}M\cong{\mathbin{\mathbb{C}}}^{m}$ near $0,$ there exist sequences $(t_{i})_{i=1}^{\infty}$ in $[0,T),$ $(x_{i})_{i=1}^{\infty}$ in $M$ and $(\lambda_{i})_{i=1}^{\infty}$ in $(0,\infty),$ such that $t_{i}\rightarrow T,$ $x_{i}\rightarrow x,$ $\lambda_{i}\rightarrow\infty$ and $\lambda_{i}^{2}(T-t_{i})\rightarrow 0$ as $i\rightarrow\infty,$ and for each $s\in{\mathbin{\mathbb{R}}}$ the limit

\tilde{L}^{s}=\lim_{i\rightarrow\infty}\lambda_{i}\cdot(L^{t_{i}+\lambda_{i}^{-2}s}-x_{i})

exists as a nonempty, noncompact, smooth, closed, immersed, exact, graded Lagrangian in ${\mathbin{\mathbb{C}}}^{m}$ whose mean curvature $\tilde{A}^{s}$ is nonzero (so that $\tilde{L}^{s}$ is not a union of Lagrangian planes in ${\mathbin{\mathbb{C}}}^{m}$ ). All derivatives of $\tilde{A}^{s},$ and the phase function $\theta_{\smash{\tilde{L}^{s}}},$ are uniformly bounded independently of $s\in{\mathbin{\mathbb{R}}}$ . Also $\tilde{L}^{s}$ depends smoothly on $s\in{\mathbin{\mathbb{R}}},$ and $\{\tilde{L}^{s}:s\in{\mathbin{\mathbb{R}}}\}$ satisfies Lagrangian MCF in ${\mathbin{\mathbb{C}}}^{m}$ .

A solution $\{\tilde{L}^{s}:s\in{\mathbin{\mathbb{R}}}\}$ of MCF for all $s\in{\mathbin{\mathbb{R}}}$ is called an eternal solution. Two obvious classes of eternal solutions of Lagrangian MCF in ${\mathbin{\mathbb{C}}}^{m}$ are

(a)

$\tilde{L}^{s}=L$ is independent of $s\in{\mathbin{\mathbb{R}}}$ , and is an SL $m$ -fold in ${\mathbin{\mathbb{C}}}^{m}$ .
(b)

$\tilde{L}^{s}=L+sv$ for $s\in{\mathbin{\mathbb{R}}}$ , where $L$ is a Lagrangian MCF translator in ${\mathbin{\mathbb{C}}}^{m}$ with translating vector $v\in{\mathbin{\mathbb{C}}}^{m}$ .

Many examples of special Lagrangian $m$ -folds in ${\mathbin{\mathbb{C}}}^{m}$ are known suitable for use in (a), but for (b) there are few, as we explain in §2.4.