2.2 Special Lagrangian m -folds in ℂ m [03MU]

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2.2 Special Lagrangian $m$ -folds in ${\mathbin{\mathbb{C}}}^{m}$

Definition 2.3.

Let ${\mathbin{\mathbb{C}}}^{m}$ have coordinates $(z_{1},\dots,z_{m})$ and complex structure $J$ , and define a Kähler metric $g$ , Kähler form $\omega$ and $(m,0)$ -form $\Omega$ on ${\mathbin{\mathbb{C}}}^{m}$ by

\begin{split}g=|{\rm d}z_{1}|^{2}+\cdots+|{\rm d}z_{m}|^{2},\quad\omega&=\textstyle\frac{i}{2}({\rm d}z_{1}\wedge{\rm d}\bar{z}_{1}+\cdots+{\rm d}z_{m}\wedge{\rm d}\bar{z}_{m}),\\ \text{and}\quad\Omega&={\rm d}z_{1}\wedge\cdots\wedge{\rm d}z_{m}.\end{split}

(2.2)

Then $({\mathbin{\mathbb{C}}}^{m},J,g,\Omega)$ is the simplest example of a Calabi–Yau $m$ -fold.

Define a real 1-form $\lambda$ on ${\mathbin{\mathbb{C}}}^{m}$ called the Liouville form by

\lambda=-{\textstyle\frac{1}{2}}\mathop{\rm Im}(z_{1}{\rm d}\bar{z}_{1}+\cdots+z_{m}{\rm d}\bar{z}_{m}).

Then ${\rm d}\lambda=\omega$ . Thus, if $L$ is a Lagrangian in ${\mathbin{\mathbb{C}}}^{m}$ then ${\rm d}(\lambda|_{L})=0$ . We call $L$ an exact Lagrangian if $\lambda|_{L}={\rm d}f$ for some smooth $f:L\rightarrow{\mathbin{\mathbb{R}}}$ .

A (singular) Lagrangian $C$ in ${\mathbin{\mathbb{C}}}^{m}$ is called a cone if $C=tC$ for all $t>0$ , where $tC=\{t\,{\bf z}:{\bf z}\in C\}$ . Let $C$ be a closed Lagrangian cone in ${\mathbin{\mathbb{C}}}^{m}$ with an isolated singularity at 0. Then $\Sigma=C\cap{\cal S}^{2m-1}$ is a compact, nonsingular Legendrian $(m\!-\!1)$ -submanifold of ${\cal S}^{2m-1}$ , not necessarily connected. Let $g_{\smash{\scriptscriptstyle\Sigma}}$ be the metric on $\Sigma$ induced by the metric $g$ on ${\mathbin{\mathbb{C}}}^{m}$ in (2.2), and $r$ the radius function on ${\mathbin{\mathbb{C}}}^{m}$ . Define $\iota:\Sigma\times(0,\infty)\rightarrow{\mathbin{\mathbb{C}}}^{m}$ by $\iota(\sigma,r)=r\sigma$ . Then the image of $\iota$ is $C\setminus\{0\}$ , and $\iota^{*}(g)=r^{2}g_{\smash{\scriptscriptstyle\Sigma}}+{\rm d}r^{2}$ is the cone metric on $C\setminus\{0\}$ .

Let $L$ be a closed, nonsingular Lagrangian $m$ -fold in ${\mathbin{\mathbb{C}}}^{m}$ , e.g. $L$ could be special Lagrangian, or a Lagrangian LMCF expander. We call $L$ asymptotically conical (AC) with rate $\rho<2$ and cone $C$ if there exists a compact subset $K\subset L$ and a diffeomorphism $\varphi:\Sigma\times(T,\infty)\rightarrow L\setminus K$ for some $T>0$ , such that

\big|\nabla^{k}(\varphi-\iota)\big|=O(r^{\rho-1-k})\quad\text{as $r\rightarrow\infty$, for all $k=0,1,2,\ldots.$}

Here $\nabla,|\,.\,|$ are computed using the cone metric $\iota^{*}(g)$ . Note that if $\rho<\sigma<2$ and $L$ is AC with rate $\rho$ , then $L$ is also AC with rate $\sigma$ .

Asymptotically conical special Lagrangians are an important class of SL $m$ -folds in ${\mathbin{\mathbb{C}}}^{m}$ . McLean’s Theorem, Theorem 2.2, was generalized to AC SL $m$ -folds by Marshall [51] and Pacini [60]. Here is a special case of their results:

Theorem 2.4.

Let $L$ be an asymptotically conical SL $m$ -fold in ${\mathbin{\mathbb{C}}}^{m}$ for $m\geqslant 3$ with cone $C$ and rate $\rho\in(2-m,0),$ and write ${\mathbin{\cal M}}_{\scriptscriptstyle L}^{\rho}$ for the moduli space of deformations of $L$ as an AC SL $m$ -fold in ${\mathbin{\mathbb{C}}}^{m}$ with cone $C$ and rate $\rho$ . Then ${\mathbin{\cal M}}_{\scriptscriptstyle L}^{\rho}$ is a smooth manifold of dimension $b^{1}_{\rm cs}(L)=b^{m-1}(L)$ .

The next family of AC SL $m$ -folds in ${\mathbin{\mathbb{C}}}^{m}$ was first found by Lawlor [45], and rewritten by Harvey [26, p. 139–140]. They are often called Lawlor necks.

Example 2.5.

Let $m>2$ and $a_{1},\ldots,a_{m}>0$ , and define polynomials $p,P$ by

p(x)=(1+a_{1}x^{2})\cdots(1+a_{m}x^{2})-1\quad\text{and}\quad P(x)=\frac{p(x)}{x^{2}}.

(2.3)

Define real numbers $\phi_{1},\ldots,\phi_{m}$ and $A$ by

\phi_{k}=a_{k}\int_{-\infty}^{\infty}\frac{{\rm d}x}{(1+a_{k}x^{2})\sqrt{P(x)}}\quad\text{and}\quad A=\int_{-\infty}^{\infty}\frac{{\rm d}x}{2\sqrt{P(x)}}\,.

Clearly $\phi_{k},A>0$ . But writing $\phi_{1}+\cdots+\phi_{m}$ as one integral gives

\phi_{1}+\cdots+\phi_{m}=\int_{0}^{\infty}\frac{p^{\prime}(x){\rm d}x}{(p(x)+1)\sqrt{p(x)}}=2\int_{0}^{\infty}\frac{{\rm d}w}{w^{2}+1}=\pi,

making the substitution $w=\sqrt{p(x)}$ . So $\phi_{k}\in(0,\pi)$ and $\phi_{1}+\cdots+\phi_{m}=\pi$ . This yields a 1-1 correspondence between $m$ -tuples $(a_{1},\ldots,a_{m})$ with $a_{k}>0$ , and $(m\!+\!1)$ -tuples $(\phi_{1},\ldots,\phi_{m},A)$ with $\phi_{k}\in(0,\pi)$ , $\phi_{1}+\cdots+\phi_{m}=\pi$ and $A>0$ .

For $k=1,\ldots,m$ , define a function $z_{k}:{\mathbin{\mathbb{R}}}\rightarrow{\mathbin{\mathbb{C}}}$ by

z_{k}(y)={\rm e}^{i\psi_{k}(y)}\sqrt{a_{k}^{-1}+y^{2}},\quad\text{where}\quad\psi_{k}(y)=a_{k}\int_{-\infty}^{y}\frac{{\rm d}x}{(1+a_{k}x^{2})\sqrt{P(x)}}\,.

Now write ${\boldsymbol{\phi}}=(\phi_{1},\ldots,\phi_{m})$ , and define a submanifold $L_{{\boldsymbol{\phi}},A}$ in ${\mathbin{\mathbb{C}}}^{m}$ by

L_{{\boldsymbol{\phi}},A}=\bigl\{(z_{1}(y)x_{1},\ldots,z_{m}(y)x_{m}):y\in{\mathbin{\mathbb{R}}},\;x_{k}\in{\mathbin{\mathbb{R}}},\;x_{1}^{2}+\cdots+x_{m}^{2}=1\bigr\}.

Then $L_{{\boldsymbol{\phi}},A}$ is closed, embedded, and diffeomorphic to ${\cal S}^{m-1}\times{\mathbin{\mathbb{R}}}$ , and Harvey [26, Th. 7.78] shows that $L_{{\boldsymbol{\phi}},A}$ is special Lagrangian. Also $L_{{\boldsymbol{\phi}},A}$ is asymptotically conical, with rate $\rho=2-m$ and cone $C$ the union $\Pi_{0}\cup\Pi_{\boldsymbol{\phi}}$ of two special Lagrangian $m$ -planes $\Pi_{0},\Pi_{\boldsymbol{\phi}}$ in ${\mathbin{\mathbb{C}}}^{m}$ given by

\Pi_{0}=\bigl\{(x_{1},\ldots,x_{m}):x_{j}\in{\mathbin{\mathbb{R}}}\bigr\},\;\>\Pi_{\boldsymbol{\phi}}=\bigl\{({\rm e}^{i\phi_{1}}x_{1},\ldots,{\rm e}^{i\phi_{m}}x_{m}):x_{j}\in{\mathbin{\mathbb{R}}}\bigr\}.

Apply Theorem 2.4 with $L=L_{{\boldsymbol{\phi}},A}$ and $\rho\in(2-m,0)$ . As $L\cong{\cal S}^{m-1}\times{\mathbin{\mathbb{R}}}$ we have $b^{1}_{\rm cs}(L)=1$ , so Theorem 2.4 shows that $\mathop{\rm dim}\nolimits{\mathbin{\cal M}}_{\scriptscriptstyle L}^{\rho}=1$ . This is consistent with the fact that when $\boldsymbol{\phi}$ is fixed, $L_{{\boldsymbol{\phi}},A}$ depends on one real parameter $A>0$ . Here $\boldsymbol{\phi}$ is fixed in ${\mathbin{\cal M}}_{\scriptscriptstyle L}^{\rho}$ as the cone $C=\Pi_{0}\cup\Pi_{\boldsymbol{\phi}}$ of $L$ depends on $\boldsymbol{\phi}$ , and all $\hat{L}\in{\mathbin{\cal M}}_{\scriptscriptstyle L}^{\rho}$ have the same cone $C$ , by definition.

Imagi, Oliveira dos Santos and the author [31, Th. 1.1] prove a uniqueness theorem for Lawlor necks. The proof involves Lagrangian Floer cohomology and Fukaya categories, and was motivated by the ideas of this paper.

Theorem 2.6.

Suppose $L$ is a closed, embedded, exact, asymptotically conical special Lagrangian in ${\mathbin{\mathbb{C}}}^{m}$ for $m\geqslant 3,$ asymptotic at rate $\rho<0$ to a union $\Pi_{1}\cup\Pi_{2}$ of two transversely intersecting special Lagrangian planes $\Pi_{1},\Pi_{2}$ in ${\mathbin{\mathbb{C}}}^{m}$ . Then $L$ is equivalent under an $\mathop{\rm SU}(m)$ rotation to one of the ‘Lawlor necks’ $L_{\boldsymbol{\phi},A}$ found by Lawlor [45], and described in Example 2.5.

Here is an example based on Harvey and Lawson [27, §III.3.A]:

Example 2.7.

Define a special Lagrangian $T^{2}$ -cone $C$ in ${\mathbin{\mathbb{C}}}^{3}$ by

C=\bigl\{(z_{1},z_{2},z_{3})\in{\mathbin{\mathbb{C}}}^{3}:|z_{1}|=|z_{2}|=|z_{3}|,\;\>z_{1}z_{2}z_{3}\in[0,\infty)\bigr\}.

(2.4)

This will be important in §3.6 as it is a ‘stable’ special Lagrangian singularity in the sense of [33, Def. 3.6]. There are three families of explicit asymptotically conical SL 3-folds $L^{A}_{1},L^{A}_{2},L^{A}_{3}$ for $A>0$ in ${\mathbin{\mathbb{C}}}^{3},$ each diffeomorphic to ${\mathbin{\cal S}}^{1}\times{\mathbin{\mathbb{R}}}^{2}$ and asymptotic at rate $\rho=0$ to the cone $C$ , where

L^{A}_{1}=\bigl\{(z_{1},z_{2},z_{3})\in{\mathbin{\mathbb{C}}}^{3}:|z_{1}|^{2}-A=|z_{2}|^{2}=|z_{3}|^{2},\;\>z_{1}z_{2}z_{3}\in[0,\infty)\bigr\},

(2.5)

and $L^{A}_{2},L^{A}_{3}$ are obtained from $L^{A}_{1}$ by cyclic permutation of $z_{1},z_{2},z_{3}$ .

Example 2.8.

In [37, 38, 39] we study SL 3-folds in ${\mathbin{\mathbb{C}}}^{3}$ invariant under the ${\rm U}(1)$ -action

{\rm e}^{i\theta}:(z_{1},z_{2},z_{3})\longmapsto({\rm e}^{i\theta}z_{1},{\rm e}^{-i\theta}z_{2},z_{3})\quad\text{for ${\rm e}^{i\theta}\in{\rm U}(1)$.}

The three papers are surveyed in [40]. A ${\rm U}(1)$ -invariant SL 3-fold $N$ may locally be written in the form

\begin{split}N=\bigl\{(z_{1},z_{2},z_{3})\in{\mathbin{\mathbb{C}}}^{3}:\,&z_{1}z_{2}=v(x,y)+iy,\quad z_{3}=x+iu(x,y),\\ &|z_{1}|^{2}-|z_{2}|^{2}=2a,\quad(x,y)\in S\bigr\},\end{split}

(2.6)

where $S$ is a domain in ${\mathbin{\mathbb{R}}}^{2}$ , $a\in{\mathbin{\mathbb{R}}}$ and $u,v:S\rightarrow{\mathbin{\mathbb{R}}}$ satisfy (in a weak sense if $a=0$ ) the nonlinear Cauchy–Riemann equations

\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}\quad\text{and}\quad\frac{\partial v}{\partial x}=-2\bigl(v^{2}+y^{2}+a^{2}\bigr)^{1/2}\frac{\partial u}{\partial y}.

(2.7)

If $S$ is simply-connected, as $\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}$ there exists a potential $f$ for $u,v$ with $\frac{\partial f}{\partial y}=u$ , $\frac{\partial f}{\partial x}=v$ , satisfying

\Bigl(\Bigl(\frac{\partial f}{\partial x}\Bigr)^{2}+y^{2}+a^{2}\Bigr)^{-1/2}\frac{\partial^{2}f}{\partial x^{2}}+2\,\frac{\partial^{2}f}{\partial y^{2}}=0.

(2.8)

In [37, 38], for suitable strictly convex domains $S\subset{\mathbin{\mathbb{R}}}^{2}$ and boundary data $\phi:\partial S\rightarrow{\mathbin{\mathbb{R}}}$ , we prove the existence of a unique $f:S\rightarrow{\mathbin{\mathbb{R}}}$ satisfying (2.8) and $f|_{\partial S}=\phi$ , and then $u=\frac{\partial f}{\partial y}$ , $v=\frac{\partial f}{\partial x}$ satisfy (2.7) (possibly in a weak sense if $a=0$ ), and $N$ in (2.6) is special Lagrangian.

When $v=y=a=0$ , equations (2.7)–(2.8) become singular, and the SL 3-fold $N$ in (2.6) has a singularity at $(0,0,z_{3})=\bigl(0,0,x+iu(x,0)\bigr)$ in ${\mathbin{\mathbb{C}}}^{3}$ . In the simplest cases $N$ is locally modelled on the cone $C$ in (2.4) near $(0,0,z_{3})$ , but there are also infinitely many other topological types of singularities not locally modelled on cones. Note that the existence and uniqueness results for $N$ are entirely independent of the singularities appearing in the interior of $N$ .

The following will be important in §3.6. Using the results of [37, 38, 39, 40], by choosing a suitable family $\phi^{t}:t\in(-\epsilon,\epsilon)$ of boundary conditions for the potential $f^{t}$ , we can construct a family $N^{t}:t\in(-\epsilon,\epsilon)$ of exact ${\rm U}(1)$ -invariant SL 3-folds in ${\mathbin{\mathbb{C}}}^{3}$ of the form (2.6) with $a=0$ , with the following properties:

(i)

$N^{t}$ depends continuously on $t\in(-\epsilon,\epsilon)$ in a suitable sense, for instance as special Lagrangian integral currents in Geometric Measure Theory.
(ii)

$N^{t}$ is nonsingular for $t<0$ .
(iii)

$N^{0}$ has one singular point at $(0,0,0)\in{\mathbin{\mathbb{C}}}^{3}$ , which has tangent cone $\Pi_{1}\cup\Pi_{2}$ , where $\Pi_{1},\Pi_{2}$ are ${\rm U}(1)$ -invariant special Lagrangian planes in ${\mathbin{\mathbb{C}}}^{3}$ intersecting non-transversely with $\Pi_{1}\cap\Pi_{2}={\mathbin{\mathbb{R}}}$ .
(iv)

$N^{t}$ for $t>0$ has two singular points at $(0,0,\pm z(t))$ , where $z(t)$ depends smoothly on $t$ and $z(t)\rightarrow 0$ as $t\rightarrow 0$ . Each singular point is locally modelled on the special Lagrangian $T^{2}$ -cone $C$ in (2.4).

Thus, isolated singular points of SL $3$ -folds modelled on the $T^{2}$ -cone $C$ in (2.4) can appear or disappear in pairs under continuous deformation.