Surjectivity of a Gluing for Stable $T^2$-cones in Special Lagrangian Geometry
Yohsuke Imagi
1112.4309.tex
\documentclass{article}
\usepackage{amsmath,amsthm,amssymb,amscd,bm}
\usepackage{ascmac}

\usepackage[arrow,matrix]{xy}

\newtheorem*{theorem*}{Theorem}
\newtheorem{theorem}{Theorem}
\newtheorem*{lemma*}{Lemma}
\newtheorem{lemma}{Lemma}
\newtheorem*{corollary*}{Corollary}
\newtheorem{corollary}{Corollary}
\newtheorem*{proposition*}{Proposition}
\newtheorem{proposition}{Proposition}
\theoremstyle{definition}
\newtheorem*{definition*}{Definition}
\newtheorem{definition}{Definition}
\newtheorem*{remark*}{Remark}
\newtheorem{remark}{Remark}



\DeclareMathOperator{\grad}{grad}
\DeclareMathOperator{\hess}{hess}
\DeclareMathOperator{\graph}{Graph}
\DeclareMathOperator{\di}{div}
\DeclareMathOperator{\area}{area}
\DeclareMathOperator{\re}{Re}
\DeclareMathOperator{\im}{Im}
\DeclareMathOperator{\spt}{Spt}
\DeclareMathOperator{\Ker}{Ker}
\DeclareMathOperator{\cyl}{cyl}
\DeclareMathOperator{\crit}{Crit}

\title{Surjectivity of a Gluing
for Stable $T^2$-cones in Special Lagrangian Geometry}
\author{Yohsuke Imagi}
\begin{document}
\maketitle
\textbf{Abstract.}
We suppose that $M$ is a Calabi--Yau (or more generally almost Calabi--Yau)
manifold of dimension $3$ over $\mathbb{C}$.
We define a moduli space $\mathcal{V}$ of special
Lagrangian varifolds (or currents) in $M$, using
geometric measure theory.
We suppose that $X$ is a compact special Lagrangian $3$-fold
in $M$ with singularity at one point and modelled on
a stable $T^2$-cone of multiplicity $1$.
We regard $X$ as a point of $\mathcal{V}$,
and determine a neighbourhood of $X$ in $\mathcal{V}$.

%We suppose that $M$ is a Calabi--Yau (or more generally almost Calabi--Yau)
%manifold of dimension $3$ over $\mathbb{C}$.
%We take $a\in H_3(M;\mathbb{Z})$.
%We denote by $\mathcal{N}$ the space of all compact special
%Lagrangian submanifolds of $X$ with homology class $a$.
%We denote by $\mathcal{V}$ the space of all
%special Lagrangian integral currents in $M$
%with support compact and connected, boundary $0$ and homology class $a$.
%We denote by $\mathcal{X}$ the space of all $X\in\mathcal{V}$
%with singularity only at one point
%and modelled upon multiplicity $1$ stable $T^2$-cones
%in the sense of Joyce.
%We can embed $\mathcal{N}$ and $\mathcal{X}$ into $\mathcal{V}$.
%We denote by $\overline{\mathcal{N}}$ the closure of $\mathcal{N}$ in $\mathcal{V}$.
%We put $\partial\mathcal{N}=\overline{\mathcal{N}}\setminus\mathcal{N}$.
%We suppose $X\in\mathcal{X}\cap\partial\mathcal{N}$.
%We prove that $X$ satisfies a topological condition of Joyce so that
%we can smooth $X$ by his method of gluing.
%We can then take
%a real number $\tau>0$ and a neighbourhood $\mathcal{Y}$
%of $X$ in $\mathcal{X}$ with the following property:
%by the method of gluing
%we can construct an open embedding
%$G:(0,\tau)\times\mathcal{Y}\to\mathcal{N}$
%with $G(t,Y)$ tending to $Y$ as $t\to+0$
%for each $Y\in\mathcal{Y}$.
%Setting $G(0,Y)=Y$ we get a continuous map
%$G:[0,\tau)\times\mathcal{Y}\to\overline{\mathcal{N}}$.
%We prove that $G$ maps $[0,\tau)\times\mathcal{Y}$
%onto a neighbourhood of $X$ in $\mathcal{V}$.

\section{Introduction}\label{intro}
Gluing is a method of constructing
blowing-up solutions
to partial differential equations,
and we shall say that the gluing is surjective
if all blowing-up solutions may be re-constructed
by the gluing.
In Yang--Mills gauge theory, for example,
Taubes~\cite{Taubes} constructs some instantons
by the method of gluing,
and Donaldson~\cite{Donaldson} proves its surjectivity;
for the details we refer also to
Freed and Uhlenbeck~\cite{Freed Uhlenbeck}.
In special Lagrangian geometry,
by a theory of Joyce~\cite{Joyce1,Joyce2,Joyce3,Joyce4,Joyce5},
we can smooth some isolated conical singularities
by the method of gluing,
and we wish to prove its surjectivity.

We can formulate the surjectivity problems in terms of moduli spaces.
In the situation of Donaldson's result~\cite{Donaldson}
indeed one has the moduli space of Yang--Mills instantons modulo gauge equivalence;
using a result of Uhlenbeck~\cite{Uhlenbeck} one can compactify it
by adding some singular instantons;
one defines the boundary of the moduli space
as the set of all the singular instantons added for the compactification;
using the surjectivity result one can determine
a neighbourhood of the boundary in the compactified space.

We wish to do a similar thing
in special Lagrangian geometry.
We suppose that $M$ is a Calabi--Yau (or more generally almost Calabi--Yau) manifold
of dimension $m$ over $\mathbb{C}$.
We take $a\in H_m(M;\mathbb{Z})$.
We denote by $\mathcal{N}$ the set
of all compact special Lagrangian submanifolds of $M$
with homology class $a$.
We give $\mathcal{N}$ the $C^\infty$-topology.
Mclean~\cite[Theorem~3.6]{Mclean} proves that
$\mathcal{N}$ is a manifold of dimension finite.

For compactifying $\mathcal{N}$ we shall use varifolds and currents
in geometric measure theory.
As special Lagrangian varifolds are calibrated in the sense of Harvey
and Lawson~\cite{Harvey and Lawson}
they may be oriented and regarded
as currents.
By a fundamental theorem of Harvey and Lawson
special Lagrangian currents are homologically area-minimizing,
and so their area will not be cancelled;
for the details we refer e.g. to Federer~\cite[5.4.2]{Federer}
or Simon~\cite[Theorem~34.5]{Simon3}.
In effect we may use varifolds in place of currents
and vice versa.
We shall mainly use varifolds rather than currents.

We denote by $\mathcal{V}$ the set of all
special Lagrangian integral varifolds (or currents)
in $M$ with support compact, boundary $0$ and homology class $a$.
We give $\mathcal{V}$ the weak topology of varifolds.
From an integral compactness theorem of Allard
\cite[Theorem~6.4]{Allard}
we see that $\mathcal{V}$ is compact.
We may regard $\mathcal{N}$
as a subset of $\mathcal{V}$.
From a regularity theorem of Allard~\cite[Theorem~8.19]{Allard}
we see that $\mathcal{N}$ is an open subset of $\mathcal{V}$,
and the $C^\infty$-topology on $\mathcal{N}$ is the same as
the one induced from $\mathcal{V}$.
We denote by $\overline{\mathcal{N}}$ the closure of
$\mathcal{N}$ in $\mathcal{V}$.
We put $\partial\mathcal{N}=\overline{\mathcal{N}}\setminus\mathcal{N}$.

In the situation of Donaldson's result~\cite{Donaldson}
the boundary of the moduli space consists of instantons
with isolated singularities, to which one can apply
the gluing method of Taubes.
In our situation, however,
$\partial\mathcal{N}$ consists of special Lagrangian
varifolds whose singularities may be non-isolated or branched,
and it will be difficult therefore to
determine a neighbourhood of $\partial\mathcal{N}$ in $\mathcal{V}$.
What we shall do is to take a certain point $X$ of $\partial\mathcal{N}$
and determine a neighbourhood of $X$ in $\mathcal{V}$.

We have mainly two things to do:
the analysis of bubbling-off and
and the classification of local models.
In the proof of Donaldson's surjectivity theorem one
uses a technique of Uhlenbeck~\cite{Uhlenbeck}
for the analysis of bubbling-off,
and a result of Atiyah, Hitchin and Singer~\cite{AHS}
for the classification of local models.

We suppose that $X$ has singularity only at one point
and modelled on Jacobi-integrable smooth cones of multiplicity $1$.
We can then analyse the bubbling-off near $X$,
using a technique of Allard--Almgren~\cite{AA}, Simon~\cite{Simon,Simon2}
and the author~\cite{Imagi}.
On the other hand it seems difficult in general to
classify local models for smoothing isolated conical singularities.

For stable $T^2$-cones
in the sense of Joyce~\cite[Definition~3.6]{Joyce2}, however, we can
classify the local models.
We can describe the stable $T^2$-cones as follows.
We put
\begin{equation}\label{C}
C=\{(z_1,z_2,z_3)\in\mathbb{C}^3\setminus\{0\}
:|z_1|=|z_2|=|z_3|,z_1z_2z_3\in(0,\infty)\}.
\end{equation}
Harvey and Lawson~\cite[Chapter~III.3.A, Theorem~3.1]{Harvey and Lawson}
prove that $C$ is a special Lagrangian submanifold of $\mathbb{C}^3\setminus\{0\}$.
We put $\Sigma=C\cap S^5$.
It is easy to see that $\Sigma$ is diffeomorphic to $T^2$.
Joyce proves that $C$ is a stable cone.
Haskins~\cite{Haskins} proves that every stable $T^2$-cone may be expressed as $a(C)$
for some $a\in SU_3$.
%For the proof we shall use the stability and a symmetry of $C$.
%Our method seems to generalize to stable cones with a suitable symmetry,
%such as examples of Ohnita~\cite{Ohnita}.
%For instance, one can embed $SU_3/SO_3$ into $S^{11}$,
%and prove that the cone on $SU_3/SO_3$ is a special Lagrangian cone in $\mathbb{C}^6$,
%which is stable by a result of Ohnita~\cite{Ohnita}.
%Haskins and Pacini~\cite[Remark~2.3]{Haskins Pacini}
%point out that $SU_3/SO_3$ 
%has a non-zero Stiefel--Whitney number,
%and so does not bound any compact manifold of dimension $6$.
%This implies that one cannot smooth
%compact special Lagrangian $6$-folds with singularities modelled on
%the cone on $SU_3/SO_3$ of multiplicity $1$.
%If one has an analogue to Theorem~\ref{uniqueness} for
%the cone on $SU_3/SO_3$, then
%using our bubbling-off analysis one can detemine
%neighbourhoods in $\mathcal{V}$
%of compact special Lagrangian $6$-folds with singularities modelled on
%the cone on $SU_3/SO_3$ of multiplicity $1$.

We suppose now that $M$ is of dimension $3$ over $\mathbb{C}$.
We suppose also that $X$ has singularity only at one point
and modelled on $C$ with multiplicity $1$.
We prove that we have a local model $L$
compatible with $X$
in the situation of Joyce~\cite[Theorem~10.4]{Joyce5};
$L$ is a special Lagrangian submanifold of $\mathbb{C}^3$
asymptotic at infinity to $C$ with multiplicity $1$.
For each $t>0$ we can define $tL$ by
re-scaling $L$ by $t$.
We can find a real number
$\tau>0$ such that for each $t\in(0,\tau)$
we can glue $tL$ to $X$ by the method of gluing.
Hence we get a smooth embedding $G:(0,\tau)\to\mathcal{N}$
with $G(t)$ tending to $X$ as $t\to+0$.

We shall parametrize $G:(0,\tau)\to\mathcal{N}$ by a family of
special Lagrangian $3$-folds with isolated conical singularities
modelled on $C$.
We denote by $\mathcal{X}$ the set of all elements of $\mathcal{V}$
with supported connected, singularity only at one point and modelled on $C$
with multiplicity $1$.
Joyce~\cite[Definition~5.6]{Joyce2}
defines a topology on $\mathcal{X}$.
We prove that it is actually the same as
the one induced from $\mathcal{V}$.
By a result of Joyce~\cite[Corollary~6.11]{Joyce2}
we can take a neighbourhood $\mathcal{Y}$ of $X$ in $\mathcal{X}$
which has a structure of manifold.
Joyce proves~\cite[Proposition~10.3]{Joyce5}
also $\dim\mathcal{Y}=\dim\mathcal{N}-1$.
Making $\tau>0$ and $\mathcal{Y}$
smaller if necessary
we can construct a
one-to-one local diffeomorphism
$G:(0,\tau)\times\mathcal{Y}\to\mathcal{N}$
with $G(t,Y)$ tending to $Y$ as $t\mapsto+0$
for each $Y\in\mathcal{Y}$.

We can extend $G$ continuously to $[0,\tau)\times\mathcal{Y}$ by
setting $G(0,Y)=Y$ for each $Y\in\mathcal{Y}$.
We note that $G:[0,\tau)\times\mathcal{Y}
\to\overline{\mathcal{N}}$ is
a one-to-one open mapping.
Our surjectivity result is:
\begin{theorem}\label{main}
$G$ maps $[0,\tau)\times\mathcal{Y}$ onto
a neighbourhood of $X$ in $\mathcal{V}$.
\end{theorem}

%Although we suppose $X\in\partial\mathcal{N}$ in Theorem~\ref{main}
%yet our method applies also to the case $X\notin\partial\mathcal{N}$,
%and in that case we can find a neighbourhood $\mathcal{U}$ of $X$ in $\mathcal{V}$
%with $\mathcal{U}\subset\mathcal{X}$.




We begin in \S\ref{SL section}
with a review of special Lagrangian geometry
and geometric measure theory.
In \S\ref{bubble section} we analyse the bubbling-off of special Lagrangian varifolds
near Jacobi-integrable smooth cones of multiplicity $1$
(which need not be stable nor $T^2$-cones).
In \S\ref{uniqueness section} we classify the local models for smoothing $C$.
In \S\ref{completion section} we prove Theorem~\ref{main}.

\textbf{Acknowledgements.}
I would like to thank Kenji Fukaya,
Manabu Akaho, Mark Haskins,
Lorenzo Foscolo, Yoshihiro Tonegawa
and the referees
for many useful comments.
I was supported in part by a JSPS research fellowship 22-699
and a GCOE programme at Kyoto University.
I wrote a part of this paper during a visit to Imperial College London.



\section{Review of SL Geometry and GMT}
\label{SL section}
In this section we shall give a review of
special Lagrangian geometry and geometric measure theory.
First of all we shall define almost Calabi--Yau manifolds and
their special Lagrangian submanifolds.
%Harvey and Lawson
%discuss special Lagrangian geometry in Calabi--Yau manifolds,
%but we can define special Lagrangian geometry also in almost
%Calabi--Yau manifolds.
%Goldstein~\cite{Goldstein} and others:
%Goldstein constructs some examples of almost Calabi--Yau manifolds
%and their special Lagrangian submanifolds;
%Joyce~\cite{JoyceSYZ} discusses generic
%special Lagrangian submanifolds of almost Calabi--Yau manifolds.

Let $(M,\omega)$ be a symplectic manifold of dimension $2m$,
and let $J$ be a complex structure on $M$
such that if we put
$\hat{g}(v,w)=\omega(v,Jw)$
then $\hat{g}$ will be a K\"ahler metric on $(M,J)$.
Let $\Omega$ be a holomorphic $(m,0)$-form on $(M,J)$
with $\Omega|_x\neq0$ for every $x\in M$.
Then we shall call $(M,\omega,J,\Omega)$ an almost Calabi--Yau manifold,
and $(\omega,J,\Omega)$ an almost Calabi--Yau structure on $M$.
We can define a smooth function $\psi:M\to(0,\infty)$ such that
\begin{equation}\label{psi}
\frac{\psi^{2m}}{m!}\omega^{\wedge m}=
(-1)^s\left(\frac{i}{2}\right)^m
\Omega\wedge\overline{\Omega}
\text{ where }s=\frac{m(m-1)}{2}.
\end{equation}
We put $g=\psi^2\hat{g}$.
We shall call $g$ the almost Calabi--Yau metric on $(M,\omega,J,\Omega)$.
Here $g$ need not be a K\"ahler metric on $(M,J)$.
If we have $\psi(x)=1$ for every $x\in M$ then
$g$ will be a K\"ahler metric of Ricci curvature $0$.
In that case we shall call $(\omega,J,\Omega)$ a Calabi--Yau structure on $M$,
and $g$ the Calabi--Yau metric on $(M,\omega,J,\Omega)$.

We define a Calabi--Yau structure on $\mathbb{R}^{2m}=\mathbb{C}^m$ as follows.
Let $(z^1,\cdots,z^m)$ be the co-ordinates of $\mathbb{C}^m$.
Let
$\omega_0=\frac{i}{2}(dz^1\wedge d\overline{z^1}
+\cdots+dz^m\wedge d\overline{z^m})$,
$g_0=dz^1\otimes d\overline{z^1}+
\cdots+dz^m\otimes d\overline{z^m}$
and $\Omega_0=dz^1\wedge\cdots\wedge dz^m$.
Let $J_0$ be
the complex structure of $\mathbb{C}^m$.
Then $(\omega_0,J_0,\Omega_0)$
is a Calabi--Yau structure on $\mathbb{R}^{2m}$,
and $g$ is the almost Calabi--Yau metric on $(\mathbb{R}^{2m},\omega_0,J_0,\Omega_0)$.

Let $(M,\omega,J,\Omega)$ be an almost Calabi--Yau manifold,
and let $g$ be the almost Calabi--Yau metric on $(M,\omega,J,\Omega)$.
Then $\re\Omega$ will be a calibration on $(M,g)$
in the sense of Harvey and Lawson~\cite{Harvey and Lawson},
and so we can define $\re\Omega$-submanifolds of $(M,g)$,
which we shall call special Lagrangian submanifolds of $(M,\omega,J,\Omega)$.
Special Lagrangian submanifolds of $(M,\omega,J,\Omega)$
will be Lagrangian submanifolds of $(M,\omega)$.

We can also define $\re\Omega$-varifolds and currents in $(M,g)$,
which we shall call special Lagrangian varifolds and currents in
$(M,\omega,J,\Omega)$, respectively.
In the remainder of this section we shall consider
calibrated geometry not limited to special Lagrangian geometry.

We suppose that $M$ is a manifold.
For each $x\in M$
we denote by $G_p(T_xM)$
the Grassmann manifold of all vector subspaces of $T_xM$ of dimension $p$.
We put $G_p(TM)=\bigcup_{x\in M}G_p(T_xM)$.
By a varifold of dimension $p$ in $M$
we shall mean a Radon measure on $G_p(TM)$.

We suppose that $g$ is a Riemannian metric on $M$,
and $\phi$ is a calibration of degree $p$ on $M$.
For each $x\in M$ we put
$G_\phi(T_xM)=\{S\in G_p(T_xM):|\phi|_S|_g=1\}$.
We also put $G_\phi(TM)=\bigcup_{x\in M}G_\phi(T_xM)$.
By a $\phi$-varifold in $M$
we shall mean a Radon measure on $G_\phi(TM)$.

For each $x\in M$ and $S\in G_{\phi}(T_xM)$
we define $\overrightarrow{S}\in\bigwedge^p T_xM$ as follows.
We take an orthonormal basis $(e_1,\cdots,e_p)$ for $T_xM$ with respect to $g|_x$
such that $\langle\phi|_x,e_1\wedge\cdots\wedge e_p\rangle=1$.
We set
$\overrightarrow{S}=e_1\wedge\cdots\wedge e_p$.
It is easy to see
that $\overrightarrow{S}$ is independent of the choice of $(e_1,\cdots,e_p)$,
and so well-defined.

Let $V$ be a $\phi$-varifold in $M$.
Then we can define a $p$-current $\overrightarrow{V}$ in $M$ by setting
\[\overrightarrow{V}(\chi)=\int_{G_{\phi}(TM)}
\langle\chi|_x,\overrightarrow{S}\rangle dV(x,S)\]
for every $p$-form $\chi$ on $M$ with support compact.

Harvey and Lawson~\cite[Chapter~II.1, Definition~1.4]{Harvey and Lawson}
define positive $\phi$-currents in $M$,
which we shall explain next.
First of all we recall a definition of
Harvey and Lawson~\cite[Chapter~II.A, Definition~A.1]{Harvey and Lawson}:
by a $\phi$-non-negative $p$-form on $M$
we shall mean a $p$-form $\chi$ on $M$
with $\langle\chi|_x,\overrightarrow{S}\rangle\geq0$ for every
$x\in M$ and $S\in G_\phi(T_xM)$.
Harvey and Lawson~\cite[Chapter~II.A, Proposition~A.2]{Harvey and Lawson}
prove that a $p$-current $T$ in $M$
is a positive $\phi$-current
if (and only if) we have $T(\chi)\geq0$
for every $\phi$-non-negative $p$-form on $M$ with support compact.
We have:
\begin{theorem*}
Let $V$ be a $\phi$-varifold in $M$.
Then $\overrightarrow{V}$ is a positive $\phi$-current in $M$
in the sense of Harvey and Lawson.
\end{theorem*}
\begin{proof}
We have only to prove that
if $\chi$ is a $\phi$-non-negative $p$-form on $M$ with support compact
then we have $\overrightarrow{V}(\chi)\geq0$.
By the definition of $\phi$-non-negative $p$-forms we have
$\langle\chi|_x,\overrightarrow{S}\rangle\geq0$
for every $x\in M$ and $S\in G_\phi(T_xM)$.
Hence we get
$\overrightarrow{V}(\chi)=\int
\langle\chi|_x,\overrightarrow{S}\rangle dV(x,S)\geq0$,
completing the proof.
\end{proof}


We suppose now that $M$ is compact.
We take $a\in H_p(M;\mathbb{R})$.
We denote by $\mathcal{V}$
the set of all integral $\phi$-varifolds $V$ in $M$
with support compact,
$\partial\overrightarrow{V}=0$ and $[\overrightarrow{V}]=a$.
We give $\mathcal{V}$ the weak topology in the sense of Allard
\cite[Definition~2.6(2)]{Allard},
i.e. the topology of the Radon measures on $G_p(TM)$.
We have then:
\begin{theorem*}
$\mathcal{V}$ is compact.
\end{theorem*}
\begin{proof}
It is easy to see that $\mathcal{V}$ is a metrizable space.
It suffices therefore to prove that if $V_1,V_2,V_3,\cdots\in\mathcal{V}$
then there exists a subsequence of $(V_n)_{n=1}^\infty$
converging in $\mathcal{V}$.
Since $M$ is compact it is clear that $\phi$ is compactly-supported,
and for each $n=1,2,3,\cdots$ therefore we have
$\area V_n=\overrightarrow{V_n}(\phi)=a\cdot[\phi]$
where $[\phi]$ denotes the de Rham cohomology class of $\phi$.
This implies that $\area V_n$ is bounded with respect to $n$.
By an integral compactness theorem of Allard~\cite[Theorem~6.4]{Allard},
therefore, we can find a subsequence of $(V_n)_{n=1}^\infty$
converging as Radon measures on $G_p(TM)$.
We may identify $(V_n)_{n=1}^\infty$
with the subsequence.
We denote its limit by $V$.
It suffices then to prove $V\in\mathcal{V}$.
Since $V_n$ tends to $V$ as varifolds
we see that $\overrightarrow{V_n}$ tends to $\overrightarrow{V}$
as $p$-currents in $M$.
As $\partial\overrightarrow{V_n}=0$
for every $n=1,2,3,\cdots$ so we have $\partial\overrightarrow{V}=0$.
As $[\overrightarrow{V_n}]=a$
for every $n=1,2,3,\cdots$ so we have $[\overrightarrow{V}]=a$.
We have therefore $V\in\mathcal{V}$ as we want.
\end{proof}

Let $V$ be a varifold of dimension $p$ in $M$.
Then we shall denote by $\|V\|$ the Radon measure on $M$ defined by
setting $\|V\|(f)=V(\pi\circ f)$ for every $f\in C_c(M;\mathbb{R})$,
where $\pi$ denotes the projection of $G_p(TM)$ onto $M$,
and $C_c(M;\mathbb{R})$ denotes the set of all continuous functions on $M$
with support compact.

We denote by $\mathcal{R}$ the set of all Radon measures on $M$
which may be expressed as $\|V\|$ for some $V\in\mathcal{V}$.
We give $\mathcal{R}$ the topology of the Radon measures on $M$.
We have then:
\begin{theorem*}
The mapping $V\mapsto\|V\|$
is a homeomorphism of $\mathcal{V}$ onto $\mathcal{R}$.
\end{theorem*}
\begin{proof}
From the definition of $\mathcal{R}$ it is clear that
$V\mapsto\|V\|$ maps $\mathcal{V}$ onto $\mathcal{R}$.
Notice also that each $V\in\mathcal{V}$ is rectifiable by a result of
Allard~\cite[Theorem~5.5]{Allard}.
Then we see that $V\mapsto\|V\|$ is one-to-one.
From the definition of $\|V\|$, moreover, we see that
$V\mapsto\|V\|$ is continuous.
We have thus proved that $V\mapsto\|V\|$
is a continuous bijection of $\mathcal{V}$ onto $\mathcal{R}$.
Notice also that $\mathcal{V}$ is compact and $\mathcal{R}$ is Hausdorff.
Then we see that
$V\mapsto\|V\|$
is a homeomorphism of $\mathcal{V}$ onto $\mathcal{R}$.
\end{proof}

We denote by $\mathcal{T}$ the set of all positive $\phi$-currents in $M$
which may be expressed as $\overrightarrow{V}$ for some $V\in\mathcal{V}$.
We give $\mathcal{T}$ the topology of $p$-currents in $M$.
We have then:
\begin{theorem*}
The mapping $V\mapsto\overrightarrow{V}$
is a homeomorphism of $\mathcal{V}$ into $\mathcal{T}$.
\end{theorem*}
\begin{proof}
From the definition of $\mathcal{T}$ it is clear that
$V\mapsto\overrightarrow{V}$ maps $\mathcal{V}$ onto $\mathcal{T}$.
Notice also that $V\mapsto\|V\|$ is one-to-one by the previous proposition.
Then we see that
$V\mapsto\overrightarrow{V}$ is one-to-one.
It is also easy to see that $V\mapsto\overrightarrow{V}$ is continuous.
We have thus proved that $V\mapsto\overrightarrow{V}$
is a continuous bijection of $\mathcal{V}$ onto $\mathcal{T}$.
Notice also that $\mathcal{V}$ is compact and $\mathcal{T}$ is Hausdorff.
Then we see that
$V\mapsto\overrightarrow{V}$
is a homeomorphism of $\mathcal{V}$ onto $\mathcal{T}$.
\end{proof}

%(see also \cite[Theorem~23.1]{Simon3})
%One can therefore define the homology class $[\overrightarrow{V}]\in H_3(M;\mathbb{Z})$
%by a theory of Federer~\cite[Chapter~4.4]{Federer}.
%Let $\mathcal{V}$ be the set of all special Lagrangian varifolds $V$ of first variation $0$ 
%in $M.$
%Put $\mathcal{U}_{f;a,b}=\{V\in\mathcal{V}:a<V(f)<b\}$
%whenever $a<b$
%and $f$ is a continuous function with compact support in $G^\mathbb{R}_{\re\Omega}(TM).$
%The family $\{\mathcal{U}_{f;a,b}\}$ generates a topology on $\mathcal{V}.$
%One easily proves that it is locally compact and Hausdorff.
%
%Let $V$ be a varifold of dimension $3$ in $M.$
%Put
%\begin{align}
%\|V\|(A)=V\bigl(\bigcup_{x\in A}G_\mathbb{R}(3;T_xM)\bigr)
%\end{align}
%for any subset $A$ of $M.$
%It is a Radon measure on $M.$






\section{Analysis of Bubbling-off}
\label{bubble section}
In this section we shall analyse bubbling-off near
special Larangian Jacobi-integrable smooth cones of multiplicity $1$.
We can summarize our results as follows.

We take a special Lagrangian varifold $X$ with singularity only
at one point $x$ and asymptotic at $x$ to
a multiplicity $1$ special Lagrangian
Jacobi-integrable smooth cone $C$.
We consider special Lagrangian varifolds $V$ close to $X$.
We define an energy functional for $V$.
We prove that if the energy of $V$ is small
then $V$ has singularity only at one point $y$ and
asymptotic at $y$ to
a multiplicity $1$ smooth cone close to $C$,
and if the energy of $V$ is large then
$V$ bubbles off.

We begin in \S\ref{energy section}
with the definition of an energy functional
and a discussion about related results.
In \S\ref{integrability section}
we discuss an important property of
special Lagrangian Jacobi-integrable cones.
In \S\ref{bubble-off section} we analyse the bubbling-off.



%Uhlenbeck's removable singularities theorem~\cite[Theorem~4.1]{Uhlenbeck};

\subsection{Energy Functional}
\label{energy section}
We define an energy functional as follows.
For each varifold $V$ of dimension $m$ in $\mathbb{R}^n$ we put
\[\mathcal{E}(V)=
\int_{\mathbb{R}^n\times G_m(\mathbb{R}^n)}
\frac{|S^{\perp}y|^2}{|y|^{m+2}}dV(y,S)\in[0,\infty]\]
where $S^\perp$ denotes the orthogonal complement to $S$ in $\mathbb{R}^n$,
and
$S^\perp y$ denotes the projection of $y$ onto $S^\perp$.

We shall recall a monotonocity formula for stationary varifolds.
We put
$B_\rho=\{y\in\mathbb{R}^n:|y|<\rho\}$ for each $\rho>0$
and $A_{\sigma,\rho}=B_\rho\setminus\overline{B_\sigma}$
for each $\rho>\sigma>0$.
For each $Z\subset\mathbb{R}^n$ we put $\widetilde{Z}=Z\times G_m(\mathbb{R}^n)$.
If $V$ is a stationary varifold of dimension $m$ in $(B_1,g_0)$
and if $0<\sigma<\rho<1$
then we have
\begin{equation}\label{flat version}
\frac{\|V\|(B_\rho)}{\rho^m}
-\frac{\|V\|(B_\sigma)}{\sigma^m}
=\mathcal{E}(V\llcorner\widetilde{A}_{\sigma,\rho});
\end{equation}
for the proof we refer e.g. to Allard~\cite[Theorem~5.1(1)]{Allard}
or Simon~\cite[Equation~17.4]{Simon3}.
It is easy to extend \eqref{flat version} to Riemannian metrics in place of $g_0$:
\begin{proposition}\label{mono}
Let $g$ be a Riemannian metric on $B_1$
with $\epsilon=|g-g_0|_{C^1(B_1)}<\infty$.
Then there exists $k>0$ such that
if $V$ is a stationary varifold of dimension $m$ in $(B_1,g)$
and if $0<\sigma<\rho<1$
then we have
\[e^{k\epsilon\rho}\frac{\|V\|(B_\rho)}{\rho^m}
-e^{k\epsilon\sigma}\frac{\|V\|(B_\sigma)}{\sigma^m}
\geq\mathcal{E}(V\llcorner\widetilde{A}_{\sigma,\rho}).\]
\end{proposition}

We suppose now that $n=2m$ and $(\omega,J,\Omega)$
is an almost Calabi--Yau structure on $B_1$.
We denote by $\mathcal{V}_0$ the space of all special Lagrangian
integral varifolds $V$ with $\partial\overrightarrow{V}=0$
in $(B_1,\omega,J,\Omega)$.

We define $r:\mathbb{R}^{2m}\to[0,\infty)$
by setting $r(y)=|y|$ for each $y\in\mathbb{R}^{2m}$.
We put $g_{\cyl}=r^{-2}g_0$,
which we shall call the cylindrical metric on $\mathbb{R}^{2m}\setminus\{0\}$.

We regard $(0,\infty)$ as a multiplicative group
acting upon $\mathbb{R}^{2m}$ as re-scaling.
By a smooth cone in $\mathbb{R}^{2m}$
we shall mean a closed submanifold of
$\mathbb{R}^{2m}\setminus\{0\}$
invariant under the re-scaling by $(0,\infty)$.
We denote by $\mathcal{C}$ the set of all special
Lagrangian smooth cones in $(\mathbb{R}^{2m},\omega_0,J_0,\Omega_0)$.

We suppose $C\in\mathcal{C}$.
We denote by $NC$ the normal bundle to $C$ in $\mathbb{R}^{2m}$
with respect $g_0$.
We get the same bundle
even if we use $g_{\cyl}$ in place of $g_0$.

For each $u\in C^k(C;NC)$ and $\rho<\sigma$
we define $|u|^{k,\cyl}_{[\rho,\sigma]}$ as follows.
We put $t=-\log r$ and
$\partial_t=\frac{\partial}{\partial t}$.
We put $\Sigma=C\cap S^{2m-1}$.
By the definition of smooth cones
$\Sigma$ is a compact submanifold of $S^{2m-1}$.
We denote by $\nabla_\Sigma$ the Levi-Civita connexion over $\Sigma$
induced from $g_0$.
We put
\[|u|^{k,\cyl}_{[\rho,\sigma]}=
\sup_{C\cap A_{\sigma,\rho}}
\sum_{i,j\geq0,i+j\leq k}|\partial_t^i\nabla_\Sigma^ju|.\]

If $|u|^{1,\cyl}_{[\rho,\sigma]}$ is sufficient small,
then we can define
the exponential map $\exp u:C\cap A_{\sigma,\rho}\to A_{\sigma,\rho}$
with respect to the metric $g_{\cyl}$,
and the image of $\exp u$ will be a submanifold of $A_{\sigma,\rho}$,
which we shall denote by $\graph_{\cyl}u$.
We put $|{\graph_{\cyl}u}|=\mathcal{H}^m\llcorner\graph_{\cyl}u$
where $\mathcal{H}^m$ denotes the Hausdorff $m$-dimensional measure
with respect to the almost Calabi--Yau metric $g$.

We suppose that $(\omega,J,\Omega)$ is an almost Calabi--Yau structure on $B_1$,
and $g$ is the almost Calabi--Yau structure on $B_1$.
From the proof of the author~\cite[Theorem~2.2]{Imagi}
we get:
\begin{theorem}\label{AAS}
There exists $\epsilon>0$ depending only on $m$ and $C$
such that if we have
\[|\Omega-\Omega_0|_{C^0(B_1)}+
|g-g_0|_{C^1(B_1)}<\epsilon,\;
V\in\mathcal{V}_0,\;0<\rho<1,\;
\mathcal{E}(V\llcorner\widetilde{A}_{\rho,1})<\epsilon\]
\[\text{and }
v\in C^\infty(C\cap A_{1/2,1};NC),\;|v|^{2,\cyl}_{[1/2,1]}<\epsilon,\;
|{\graph_{\cyl}v}|=\|V\|\llcorner A_{1/2,1}\]
then we can extend $v$ to $C\cap A_{\rho,1}$
so that $\|V\|\llcorner A_{\rho,1}=|{\graph_{\cyl}v}|$
and $|v|^{2,\cyl}_{[\rho,1]}<k\epsilon^\alpha$
for some $k>0$ and $\alpha\in(0,1)$ depending only on $m$ and $C$.
\end{theorem}
\begin{remark*}
For the proof one has to use a result of \L{}ojasiewicz~\cite{Lojasiewicz},
following Simon~\cite{Simon}.
\end{remark*}
We shall give a corollary to the theorem above.
We give $\mathcal{C}$ the $C^\infty$-topology.
%by Allard's regularity theorem, actually,
%it is the same as the one induced from $\mathcal{V}_0$.
Let $\mathcal{C}'$ be a neighbourhood of $C$ in $\mathcal{C}$.
Then we have:
\begin{corollary}\label{energy small}
There exists $\epsilon>0$ depending only on $m$, $C$ and $\mathcal{C}'$
such that if we have
\[|\Omega-\Omega_0|_{C^0(B_1)}+
|g-g_0|_{C^1(B_1)}<\epsilon,\;
V\in\mathcal{V}_0,\;\mathcal{E}(V)<\epsilon\]
\[\text{and }
v\in C^\infty(C\cap A_{1/2,1};NC),\;|v|^{2,\cyl}_{[1/2,1]}<\epsilon,\;
|{\graph_{\cyl}v}|=\|V\|\llcorner A_{1/2,1}\]
then $V$ is singular only at $0$
and asymptotic at $0$ to some $C'\in\mathcal{C}'$ with multiplicity $1$.
\end{corollary}
\begin{remark*}
By a result of Simon~\cite[Theorem~5]{Simon}
$C'$ will be a unique tangent cone to $V$ at $0$.
\end{remark*}
\begin{proof}
Let $C'$ be a tangent cone to $V$ at $0$.
Then we can take $\delta_1>\delta_2>\delta_3>\cdots$
tending to $0$ with $\delta_n^{-1}V$ tending to $C'$
as $n\to\infty$.
We can also take $\eta>0$ such that if
$v\in C^\infty(\Sigma;NC)$ and $|v|_{C^1(\Sigma)}\leq\eta$
then we have $\graph_{\cyl}v\in\mathcal{C}'$.
By Theorem~\ref{AAS} we can find $v_n\in C^\infty(C\cap A_{\delta_n,1};NC)$
such that $\|V\|\llcorner A_{\delta_n,1}
=|{\graph_{\cyl}v_n}|$ with $|v_n|^{2,\cyl}_{[\delta_n,1]}\leq\eta$,
and so we can find a subsequence of
$(\|\delta_n^{-1}V\|\llcorner A_{1/2,1})_{n=1}^\infty$
converging to $|{\graph_{\cyl}v_\infty}|$ for some
$w\in C^\infty(C\cap A_{1/2,1};NC)$
with $|w|^{1,\cyl}_{[1/2,1]}\leq\eta$.
This implies that $C'$ is smooth of multiplicity $1$
and we have $C'\in\mathcal{C}'$ as we want.
\end{proof}


\subsection{Special Lagrangian Jacobi-integrable Cones}
\label{integrability section}
We shall discuss an important property of
special Lagrangian Jacobi-integrable cones.
For a moment we shall consider Jacobi-integrable cones
which need not be special Lagrangian.

We denote by $\mathcal{K}$ the set of all smooth cones in $\mathbb{R}^n$.
We define a functional $F:\mathcal{K}\to(0,\infty)$
by setting $F(K)=\area(K\cap B_1)$
for each $K\in\mathcal{K}$.
We denote by $\crit F$ the set of all critical points of $F$.

Let $K\in\mathcal{K}$.
Then we have $T_K\mathcal{K}=C^\infty(K;NK)$.
We can also define the hessian of $F$ at $K$
as a linear operator $\hess_KF:T_K\mathcal{K}\to T_K\mathcal{K}$.
We shall say that $K$ is Jacobi-integrable
if for every $v\in\Ker\hess_CF$ there exists
a one parameter family $(K_t)_{t\in\mathbb{R}}$ in $\crit F$
with
\[
K_0=K\text{ and }\frac{dK_t}{dt}\Big|_{t=0}=v.
\]
We suppose that $K$ is Jacobi-integrable.
Adams and Simon~\cite[Lemma~1]{AS} prove:
\begin{proposition}\label{CCC}
There exists a neighbourhood of $K$ in $\mathcal{K}$
on which $F$ is constant.
\end{proposition}
\noindent
For the proof one has to
use the real-analyticity of $F$.
Adams and Simon also prove that
$\crit F$ is a non-degenerate critical manifold
in the sense of Bott~\cite{Bott}.

We shall do a similar thing for Lagrangian cones.
We denote by $\widehat{\mathcal{C}}$ the set of all
Lagrangian smooth cones in $(\mathbb{R}^{2m},\omega_0)$.
We have $\widehat{\mathcal{C}}\subset\mathcal{K}$.
We put $\widehat{F}=F|_{\widehat{\mathcal{C}}}$.
We denote by $\crit\widehat{F}$ the set of all critical points of $\widehat{F}$.

We suppose $\widehat{C}\in\widehat{\mathcal{C}}$.
By a result of Weinstein~\cite[Corollary~6.2]{Weinstein}, then,
$T_{\widehat{C}}\widehat{\mathcal{C}}$ may be identified with
the set of all closed $1$-forms on $\widehat{C}$.
By a result of Oh~\cite[Theorem~3.5]{Oh}
the hessian of $\widehat{F}$ at $\widehat{C}$ will be
the Laplacian acting upon closed $1$-forms on $\widehat{C}$.
We shall denote the Laplacian by $\Delta$, and
say that $\widehat{C}$ is Lagrangian Jacobi-integrable
if for every closed $1$-form $\xi$ on $\widehat{C}$
with $\Delta\xi=0$ there exists
a one parameter family $(\widehat{C}_t)_{t\in\mathbb{R}}$ in $\crit\widehat{F}$
with
\[\widehat{C}_0=\widehat{C}\text{ and }
-\frac{d\widehat{C}_t}{dt}\Big|_{t=0}\lrcorner\omega_0=\xi.\]
In a way similar to the proof of Proposition~\ref{CCC} we can prove:
\begin{proposition}\label{CC}
If $\widehat{C}$ is Lagrangian Jacobi-integrable
then there exists a neighbourhood of $\widehat{C}$ in $\widehat{\mathcal{C}}$
on which $\widehat{F}$ is constant.
\end{proposition}

Joyce~\cite[Definition~6.7]{Joyce1} defines
special Lagrangian Jacobi-integrable cones,
which we shall recall next.
As in \S\ref{energy section}
we denote by $\mathcal{C}$ the set of
all special Lagrangian smooth cones in $(\mathbb{R}^{2m},
\omega_0,J_0,\Omega_0)$.
We take $C\in\mathcal{C}$.
We shall say that $C$ is special Lagrangian Jacobi-integrable
if for every closed $1$-form $\xi$ on $C$
with $d*\xi=0$ there exists
a one parameter family $(C_t)_{t\in\mathbb{R}}$ in $\mathcal{C}$ with
\[C_0=C\text{ and }
-\frac{dC_t}{dt}\Big|_{t=0}\lrcorner\omega_0=\xi.\]
We have:
\begin{lemma*}
If $C$ is special Lagrangian Jacobi-integrable
then $C$ is Lagrangian Jacobi-integrable.
\end{lemma*}
\begin{proof}
Let $\xi$ be a closed $1$-form on $C$ with $\Delta\xi=0$.
Then we have
$d*d*\xi=0$ and so
\begin{equation}\label{a}
*d*\xi=a
\end{equation}
for some $a\in\mathbb{R}$.
We define a map $f_t^a:\mathbb{C}^m\to\mathbb{C}^m$
by setting $f_t^a(z_1,z_2,\cdots,z_m)=(e^{iat}z_1,z_2,\cdots,z_m)$.
We put
$C_t^a=f_t^a(C)$.
We have then
\begin{equation}\label{Omega}
\Omega_0|_{C_t^a}=(f_t^{a*}\Omega_0)|_{C}=e^{iat}\Omega_0|_{C}=e^{iat}*1.
\end{equation}
We put
\[\xi^a=-\frac{dC^a_t}{dt}\Big|_{t=0}\lrcorner\omega_0.
\]
Since $C^a_t$ is Lagrangian we see that $\xi^a$ is a closed $1$-form on $C$.
We may also write $\xi^a=g_0(\tilde{\xi}^a,\bullet)$
for some $\tilde{\xi}^a$.
We have then
$\frac{dC^a_t}{dt}|_{t=0}=J_0\tilde{\xi}^a$
and so
\[\frac{d}{dt}\Omega_0|_{C_t^a}\Big|_{t=0}=d(J_0\tilde\xi^a
\lrcorner\Omega_0|_C)=d(i\tilde\xi^a\lrcorner\Omega_0|_C)
=d(i\tilde{\xi}^a\lrcorner*1)=id*\xi^a.\]
This combined with \eqref{Omega} implies that on $C$ we have
\[id*\xi^a=ia*1\]
and so $*d*\xi^a=a$. This combined with \eqref{a}
implies $*d*(\xi-\xi^a)=0$.
By the special Lagrangian Jacobi-integrability of $C$, therefore,
we can find
a one-parameter family
$(C_t)_{t\in\mathbb{R}}$ in $\mathcal{C}$ with
\[C_0=C\text{ and }-\frac{dC_t}{dt}\Big|_{t=0}\lrcorner\omega_0=\xi-\xi^a.\]
Putting $\widehat{C}_t=f^a_t(C_t)$ we get
\[\widehat{C}_t\in\widehat{\mathcal{C}},\;\widehat{C}_0=C\text{ and }-
\frac{d\widehat{C}_t}{dt}\Big|_{t=0}\lrcorner\omega_0=\xi,\]
completing the proof.
\end{proof}

By Proposition~\ref{CC} and the lemma above we have: 
\begin{proposition}\label{CC'}
Let $C$ be special Lagrangian Jacobi-integrable.
Then there exists a neighbourhood $\mathcal{C}'$ of $C$ in $\mathcal{C}$
on which the area functional is constant;
i.e.
for each $C'\in\mathcal{C}'$
we have $\area(C'\cap B_1)=\area(C\cap B_1)$.
\end{proposition}

\subsection{Bubbling-off}
\label{bubble-off section}
We suppose that $(\omega,J,\Omega)$ is an almost Calabi--Yau structure on $B_1$
with $J|_0=J_0$ and $\Omega|_0=\Omega_0$.
We denote by $g$ the almost Calabi--Yau metric on $(B_1,\omega,J,\Omega)$.
We write $g=\sum_{i,j=1}^{2m}g_{ij}dy^idy^j$
and suppose
\begin{equation*}
g_{ij}(0)=\delta_{ij}\text{ and }
\frac{\partial g_{ij}}{\partial y^k}(0)=0
\text{ for each }i,j,k=1,\cdots,2m.
\end{equation*}

If $W$ is a varifold in $\mathbb{R}^n$
and if $\delta>0$ then we can define $\delta^{-1}W$
by re-scaling $W$ by $\delta^{-1}$.
If $b\in\mathbb{R}^n$ then
we can also define $W-b$
by translating $W$ by $-b$.


We take $C\in\mathcal{C}$
and a neighbourhood $\mathcal{C}'$ of $C$ in $\mathcal{C}$.
$X\in\mathcal{V}_0$.
We denote by $\mathcal{X}'$
the space of all elements of $\mathcal{V}_0$
with support connected,
with singularity only at one point $y$
and asymptotic at $y$ to some element of $\mathcal{C}'$
with multiplicity $1$.
We take $X\in\mathcal{V}_0$ and suppose that
$0$ is the singular point of $X$
and $C$ is the multiplicity $1$ smooth tangent cone to $X$ at $0$.
We have then:
\begin{theorem}\label{X top}
Let $(X_n)_{n=1}^\infty$ be a sequence in $\mathcal{X}'$
converging to $X$.
For each $n=1,2,3,\cdots$
let $x_n$ be the singular point of $X_n$,
and let $C_n$ be the multiplicity $1$ smooth tangent cone to $X_n$ at $x_n$.
Then $x_n$ tends to $0$ and $C_n$ tends to $C$ as $n\to\infty$.
\end{theorem}
\begin{proof}
By Allard's regularity theorem $x_n$ tends to $0$ as $n\to\infty$.
By Proposition~\ref{mono} we have
\[\mathcal{E}(X_n-x_n)\leq e^{k\epsilon\rho}\frac{\|X_n-x_n\|(B_\rho)}{\rho^m}
-e^{k\epsilon\sigma}\frac{\|X_n-x_n\|(B_\sigma)}{\sigma^m}\]
where $0<\sigma<\rho<1$.
Letting $\sigma\to0$ we get
\[\mathcal{E}(X_n-x_n)\leq e^{k\epsilon\rho}\frac{\|X_n-x_n\|(B_\rho)}{\rho^m}
-\area(C_n\cap B_1).\]
This combined with Proposition~\ref{CC'} implies
\begin{equation}\label{EE}
\mathcal{E}(X_n-x_n)\leq e^{k\epsilon\rho}\frac{\|X_n-x_n\|(B_\rho)}{\rho^m}
-\area(C\cap B_1).
\end{equation}
Making $\rho$ smaller if necessary we may suppose
\[e^{k\epsilon\rho}\frac{\|X\|(B_\rho)}{\rho^m}-\area(C\cap B_1)<\epsilon.\]
For $n$ sufficiently large, therefore, we have
\[e^{k\epsilon\rho}\frac{\|X_n-x_n\|(B_\rho)}{\rho^m}-\area(C\cap B_1)<\epsilon.\]
This combined with \eqref{EE} implies
\[\mathcal{E}(X_n-x_n)<\epsilon.\]
By Corollary~\ref{energy small}, therefore, $C_n$ tends to $C$ as $n\to\infty$.
\end{proof}




We have also:
\begin{theorem}\label{bubble-off}
Let $(V_n)_{n=1}^\infty$ be a sequence in
$\mathcal{V}_0\setminus\mathcal{X}'$
converging to $X$.
Then there exists a sequence $(\delta_n)_{n=1}^\infty$
of positive real numbers converging to $0$,
a sequence $(y_n)_{n=1}^\infty$ in $B_1$
converging to $0$,
and a subsequence of $\bigl(\delta_n^{-1}(V_n-y_n)\bigr)_{n=1}^\infty$
converging to some varifold $W$
in $\mathbb{R}^{2m}$
asymptotic at infinity to some element of $\mathcal{C}'$ with multiplicity $1$
and satisfying $\mathcal{E}(W-b)>0$ for every $b\in\mathbb{R}^{2m}$.
\end{theorem}
\begin{remark*}
$W$ will automatically
be a special Lagrangian integral varifold
with $\partial{\overrightarrow{W}}=0$
in $(\mathbb{R}^{2m},\omega_0,J_0,\Omega_0)$.
\end{remark*}
\begin{proof}
We take positive real numbers $\rho$ and $\epsilon$,
which we shall make smaller if necessary.
By assumption we have $V_n\in\mathcal{V}_0\setminus\mathcal{X}'$.
By Corollary~\ref{energy small} therefore
we may suppose
$\mathcal{E}(V_n-y)\geq\epsilon$
for each $y\in B_{\rho\epsilon}$;
otherwise $V_n$ will be singular only at $y$
and asymptotic at $y$ to some element of $\mathcal{C}'$
with multiplicity $1$, which contradicts $V_n\in\mathcal{V}_0\setminus\mathcal{X}'$.
We put
\[\delta_n(y)
=\inf\left\{\delta\in(0,\rho):\mathcal{E}\bigl((V_n-y)
\llcorner\widetilde{A}_{\delta,\rho}\bigr)=\frac{\epsilon}{2}\right\}.\]
Since $\mathcal{E}(V_n-y)\geq\epsilon$ we see $\delta_n(y)>0$.
It is also easy to see that
$y\mapsto\delta_n(y)$ is lower semi-continuous.
Hence we can find
$y_n\in B_{\epsilon\rho}$ with
$\delta_n(y_n)=\inf_{y\in B_\epsilon}\delta_n(y)$.
We put
$\delta_n=\delta_n(y_n)>0$.
We have then:
\begin{lemma*}
$\delta_n$ tends to $0$ as $n\to\infty$.
\end{lemma*}
\begin{proof}
Making $\rho$ smaller if necessary
we may suppose $\mathcal{E}(X\llcorner\widetilde{B}_\rho)<\epsilon/2$.
We take $\delta>0$.
We have then $\mathcal{E}(X\llcorner\widetilde{A}_{\delta,\rho})<\epsilon/2$.
Since $V_n$ tends to $X$ as $n\to\infty$
we see that
for $n$ sufficiently large
we have $\mathcal{E}(V_n\llcorner\widetilde{A}_{\delta,\rho})<\epsilon/2$.
This implies $\delta_n(0)<\delta$
and so $\delta_n(0)$ tends to $0$ as $n\to\infty$.
Since $\delta_n\leq\delta_n(0)$
we see that $\delta_n$ also tends to $0$
as $n\to\infty$.
\end{proof}
We have also:
\begin{lemma*}
$\bigl(\delta_n^{-1}(V_n-y_n)\bigr)_{n=1}^{\infty}$
has a subsequence converging to
some varifold $W$ in
$\mathbb{R}^{2m}$.
\end{lemma*}
\begin{proof}
Let $R>0$. Then we have only to prove
\begin{equation}\label{bdd}
\sup_{n=1,2,3,\cdots}\|\delta_n^{-1}(V_n-y_n)\|(B_R)<\infty.
\end{equation}
Notice that
$\|\delta_n^{-1}(V_n-y_n)\|(B_R)=\delta_n^{-m}\|V_n-y_n\|(B_{\delta_nR})$.
Notice also that by Proposition~\ref{mono} we have
\[e^{k\epsilon_n\delta_nR}\frac{\|V_n-y_n\|(B_{\delta_nR})}{(\delta_nR)^m}
\leq e^{k\epsilon_n}\|V_n-y_n\|(B_1)\]
which tends to $\|X\|(B_1)$ as $n\to\infty$.
Then we get \eqref{bdd}.
\end{proof}


We take $W$ as above.
From the definition of $\delta_n$
it is easy to see
$\mathcal{E}(W\llcorner\widetilde{A}_{1,\infty})\leq\epsilon/2$.
In a way similar to the proof of Corollary~\ref{energy small},
therefore,
we can prove that
$W$ is asymptotic at infinity to a multiplicity $1$ smooth cone $C'$. 
Making $\epsilon$ smaller if necessary we may suppose $C'\in\mathcal{C}'$.
%
%

It remains to prove $\mathcal{E}(W-b)>0$
for every $b\in\mathbb{R}^{2m}$.
We put $V_n'=V_n-y_n-\delta_nb$.
By Proposition~\ref{mono} we have
\begin{equation}\label{V_n'}
\exp(k\epsilon_n\rho)
\frac{\|V_n'\|(B_\rho)}{\rho^m}
-\exp(k\epsilon_n\delta_n)\frac{\|V_n'\|(B_{\delta_n})}{\delta_n^m}
\geq\mathcal{E}(V_n'\llcorner\widetilde{A}_{\delta_n,\rho})\geq\frac{\epsilon}{2}.
\end{equation}
We note that $V_n'$ tends to $X$ and $\delta_n^{-1}V_n'$ tends to $W-b$
as $n\to\infty$.
By \eqref{V_n'} therefore we have
\begin{equation}\label{V_n''}
\frac{\|X\|(B_\rho)}{\rho^m}
-\|W-b\|(B_1)\geq\frac{\epsilon}{2}.
\end{equation}
Making $\rho>0$ smaller if necessary we may suppose
\[\frac{\|X\|(B_\rho)}{\rho^m}
\leq\area(C\cap B_1)+\frac{\epsilon}{4}.\]
This combined with \eqref{V_n''} implies
\[
\area(C\cap B_1)-\|W-b\|(B_1)>\frac{\epsilon}{4}.
\]
This combined with Proposition~\ref{CC'} implies
\begin{equation}\label{V_n3}
\area(C'\cap B_1)-\|W-b\|(B_1)>\frac{\epsilon}{4}.
\end{equation}
On the other hand by \eqref{flat version} we have
\[\frac{\|W-b\|(B_R)}{R^m}-\|W-b\|(B_1)
=\mathcal{E}\bigl((W-b)\llcorner\widetilde{A}_{\lambda,R}\bigr)\]
for each $R>1$. Letting $R\to\infty$ we get
\[
\area(C'\cap B_1)-\|W-b\|(B_1)
=\mathcal{E}\bigl((W-b)\llcorner\widetilde{A}_{\lambda,\infty}\bigr).
\]
This combined with \eqref{V_n3} implies
$\mathcal{E}\bigl((W-b)\llcorner\widetilde{A}_{1,\infty}\bigr)>\epsilon/4$
and so $\mathcal{E}(W-b)>0$, completing the proof.
\end{proof}
%



\section{Classification of Local Models}
\label{uniqueness section}
We define $C$ by \eqref{C} in \S\ref{intro}.
In this section we shall classify local models
for smoothing $C$.
We can describe the local models as follows.
We put
\begin{equation}\label{123}
\begin{split}
L_1=\{(z_1,z_2,z_3)\in\mathbb{C}^3:|z_1|^2-1=|z_2|^2=|z_3|^2,
z_1z_2z_3\in[0,\infty)\},\\
L_2=\{(z_1,z_2,z_3)\in\mathbb{C}^3:|z_1|^2=|z_2|^2-1=|z_3|^2,
z_1z_2z_3\in[0,\infty)\},\\
L_3=\{(z_1,z_2,z_3)\in\mathbb{C}^3:|z_1|^2=|z_2|^2=|z_3|^2-1,
z_1z_2z_3\in[0,\infty)\}.
\end{split}\end{equation}
If $L\in\{L_1,L_2,L_3\}$
then $L$ will be a
submanifold of $\mathbb{C}^3$
of dimension $3$ over $\mathbb{R}$.
We note that $L$ is non-singular
in spite of the condition $z_1z_2z_3\in[0,\infty)$.
Harvey and Lawson~\cite[Chapter~III.3.A, Theorem~3.1]{Harvey and Lawson} prove that
$L$ is a special Lagrangian submanifold of
$\mathbb{C}^3$.
It is easy to see that $L$
is asymptotic at infinity
to $C$ with multiplicity $1$.
Our classification result is:
\begin{theorem}\label{uniqueness}
Let $W$ be a special Lagrangian integral varifold
with $\partial\overrightarrow{W}=0$ in $(\mathbb{R}^6,\omega_0,J_0,\Omega_0)$
asymptotic at infinity to $C$ with multiplicity $1$.
Then we have $\|W\|=|sL+b|$
for some $s>0$, $L\in\{C,L_1,L_2,L_3\}$ and $b\in\mathbb{R}^6$.
\end{theorem}
\noindent
Here $|sL+b|$ denotes the Radon measure on $\mathbb{R}^6$
associated to $sL+b$
with multiplicity $1$ with respect to the Euclidean metric $g_0$.

In what follows we give a sketch of the proof of Theorem~\ref{uniqueness}.

First of all we analyse the asymptotic behaviour of
$W$ at infinity.
From the stability of $C$
we find $b\in\mathbb{R}^6$ such that
$W-b$ will decay rapidly to $C$ at infinity.
For the sake of simplicity, therefore,
we may suppose that
$W$ decays rapidly to $C$ at infinity.

We define a $T^2$-action on $\mathbb{R}^6=\mathbb{C}^3$ by setting
\[(e^{i\theta},e^{i\phi})\cdot(z_1,z_2,z_3)
=(e^{i\theta}z_1,e^{i\phi}z_2,e^{-i\theta-i\phi}z_3).\]
We note that $\omega_0$, $J_0$, $\Omega_0$ and $C$ are invariant under the
$T^2$-action.
We take a moment map $\mu:\mathbb{C}^3\to\mathbb{R}^2$.
We prove that $\mu$ is constant on $\spt\|W\|$.
The idea of the proof is as follows.

We suppose for a moment that $W$ is a submanifold of $\mathbb{C}^3$.
By the $T^2$-action, then, we can deform $W$
as a special Lagrangian submanifold of $\mathbb{C}^3$.
The infinitesimal deformation may be identified with $d\mu|_W$,
and so $d\mu|_W$ will be a harmonic $1$-form on $W$, by the theory of Mclean.
Therefore $\mu|_W$ will be a harmonic function on $W$.
Since $W$ decays rapidly to $C$ at infinity we shall see that
$\mu|_W$ decays rapidly at infinity to some constant.
By the maximum principle, therefore,
$\mu|_W$ will be a constant function.

Actually $W$ need not be a submanifold of $\mathbb{R}^6$.
Using some basic results on varifolds, however, we can modify the argument above.
Thus we see that
$\mu$ is constant on $\spt\|W\|$.

Harvey and Lawson
\cite[Chapter~III.3.A, Theorem~3.1]{Harvey and Lawson}
construct a special Lagrangian fibration
$F:\mathbb{C}^3\to\mathbb{R}^3$
invariant under the $T^2$-action.
From the proof of Harvey and Lawson
%or a more general result of Goldstein~\cite[Theorem~A]{Goldstein}
we see that every $T^2$-invariant special Lagrangian submanifold of $\mathbb{C}^3$
is contained in a fibre of $F$.

We suppose again that
$W$ is a submanifold of $\mathbb{C}^3$.
Since $\mu$ is constant on $W$ we see that
$W$ is then invariant under the $T^2$-action,
and so contained in a fibre of $F$. 
Actually $W$ need to be a submanifold of $\mathbb{C}^3$,
but we can again modify the argument and so
$\spt\|W\|$ is contained in a fibre of $F$.

For each fibre of $F$ we have an explicit description of
the topology and asymptotic behaviour at infinity,
and so we shall be able to
complete the proof by an elementary argument.

We begin now with a review of some basic properties of
Laplacians over cones.

We suppose that $\Sigma$ is a compact manifold of dimension $m-1$.
We put $C=(0,\infty)\times\Sigma$.
We denote by $r$ the projection of $C$ onto $(0,\infty)$.
We suppose that $g_\Sigma$ is a Riemannian metric on $\Sigma$.
We put $g_C=dr^2+r^2g_\Sigma$.
With respect to $g_C$
we can define the Laplacian $\Delta_C:C^\infty(C;\mathbb{R})
\to C^\infty(C;\mathbb{R})$.

We have also the Laplacian $\Delta_\Sigma:C^\infty(\Sigma;\mathbb{R})\to
C^\infty(\Sigma;\mathbb{R})$ with respect to $g_\Sigma$.
We put
$t=\log r:C\to\mathbb{R}$ and $\partial_t=\frac{\partial}{\partial t}$.
It is easy then to see
$-e^{2t}\Delta_C=\partial_t^2+(m-2)\partial_t-\Delta_\Sigma$.
We denote by
$0=\gamma_0\leq\gamma_1\leq\gamma_2\leq\cdots$ the eigenvalues of $\Delta_\Sigma$.
For each integer $i\geq0$
we consider the equation $x^2+(m-2)x-\gamma_i=0$
in $x$.
We denote by $\alpha_i$ and $\beta_i$
the two solutions with $\alpha_i\geq\beta_i$.
We suppose $m\geq2$.
Since $\gamma_0=0$ we get $\beta_0=2-m\leq0=\alpha_0$.
Since $\gamma_0\leq\gamma_1\leq\gamma_2\cdots$
we get $\alpha_0\leq\alpha_1\leq\alpha_2\leq\cdots$
and $\beta_0\geq\beta_1\geq\beta_2\geq\cdots$.
Since $\gamma_i$ tends to $\infty$
as $i\to\infty$ we see that $\alpha_i$ tends to $\infty$
and $\beta_i$ tends to $-\infty$ as $i\to\infty$.
We also put
$\Lambda=\{\alpha_0,\beta_0,\alpha_1,\beta_1,\alpha_2,\beta_2,\cdots\}$.
For each $\lambda\in\mathbb{R}$ we put
\[E_\lambda=\{f\in C^{\infty}(\Sigma;\mathbb{R}):
\Delta_{\Sigma}f=\lambda(\lambda+m-2)f\}.\]
By the definition of the eigenvalues we have
$E_\lambda\neq\{0\}$ if and only if $\lambda\in\Lambda$.

We can also take a complete orthonormal basis
$\{v_0,v_1,v_2,\cdots\}$ for $L^2(\Sigma;\mathbb{R})$ such that
$\Delta_{\Sigma}v_i=\gamma_iv_i$ for each integer $i\geq0$.
We have then:
\begin{proposition}\label{homogeneous}
Let $I$ be an open interval in $\mathbb{R}$,
let $u\in C^\infty(I\times \Sigma;\mathbb{R})$
and suppose $\Delta_Cu=0$.
Then there exist $a_0,b_0,a_1,b_1,a_2,b_2\cdots\in\mathbb{R}$
such that
$u=\sum_{i=0}^\infty(a_ir^{\alpha_i}+b_ir^{\beta_i})v_i$
where the series converge in the local $C^\infty$-topology.
\end{proposition}
We give a proof for the sake of clarity:
\begin{proof}
We put $\log I=\{t\in\mathbb{R}:e^t\in I\}$.
For each $t\in\log I$ and $i\in\{0,1,2,\cdots\}$ we define
$u_i(t)$ as the inner product of $u|_{\{e^t\}\times\Sigma}$
and $v_i$ in $L^2(\Sigma;\mathbb{R})$.
Then $t\mapsto u_i(t)$ is a smooth function on $\log I$.
Since $\Delta_Cu=0$ we get
$(\partial_t^2+(m-2)\partial_t-\lambda_i)u_i=0$.
Hence we find $a_i,b_i\in\mathbb{R}$ such that
$u_i=a_ie^{\alpha_it}+b_ie^{\beta_it}$.
It suffices therefore to prove that $\sum_{i=0}^\infty u_iv_i$
converges to $u$ in the local $C^\infty$-topology.
We denote by $d\mu_\Sigma$ the Riemannian measure on $\Sigma$ with respect to $g_\Sigma$,
and by $\|\bullet\|$ the norm of $L^2(\Sigma;\mathbb{R})$.
We have then
\[\int_{\log I\times\Sigma}|u|^2dtd\mu_\Sigma=\int_{\log I}\sum_{i=0}^\infty\|u_i(t)\|^2dt
=\sum_{i=0}^\infty\int_{\log I}\|u_i(t)\|^2dt<\infty.\]
Putting $w_n=\sum_{i=0}^nu_iv_i$ we get
$\int_{\log I\times\Sigma}|u-w_n|^2dtd\mu_\Sigma=\sum_{i=n+1}^\infty\int_{\log I}
\|u_i(t)\|^2dt$
which tends to $0$ as $n\to\infty$.
Thus $w_n$ tends to $u$ in the $L^2$-topology.
Applying elliptic regularity to $u-w_n$
we see that $w_n$ tends to $u$ in
the local $C^\infty$-toplogy.
\end{proof}


Let $u\in C^\infty(C;\mathbb{R})$
and $\alpha\in\mathbb{R}$.
%we shall write $u=O(r^\alpha)$
%as $r\to0$ if we have
%\[\lim_{\epsilon\to+0}\sup_{C\cap B_\epsilon}|r^{-\alpha+k}\nabla^ku|<\infty
%\text{ for every }k=0,1,2,3,\cdots\]
%where $\nabla$ denotes the Levi-Civita connexion
%over the Riemannian manifold $(C,i_C^*g_0)$,
%and $|\bullet|$ denotes the norm with respect to $i_C^*g_0$.
Then we shall write $u=O(r^\alpha)$ as $r\to\infty$ if
there exists $R>0$ such that
\[\sup_{(R,\infty)\times\Sigma}|r^{-\alpha+k}\nabla^ku|<\infty
\text{ for every }k=0,1,2,3,\cdots\]
where $\nabla$ denotes the Levi-Civita connexion with respect to $g_C$,
and $|\bullet|$ denotes the norm with respect to $g_C$.


By a result of Simon~\cite[Part~I, Lemma~5.9]{Simon2}
we have:
\begin{proposition*}
Suppose
$q\in(2-m,\infty)\setminus\Lambda$ and
$f\in C^\infty\bigl((R,\infty)\times\Sigma;\mathbb{R}\bigr)$
with $f=O(r^{q-2})$.
Then there exists $u\in C^\infty\bigl((R,\infty)\times\Sigma;\mathbb{R}\bigr)$
with $u=O(r^q)$ such that $\Delta_Cu=f$.
\end{proposition*}
We give a corollary to this:
\begin{corollary}
\label{linearized equation}
Suppose $R>0$, $u\in C^{\infty}\bigl((R,\infty)\times\Sigma;\mathbb{R}\bigr)$,
$p,q\in\mathbb{R}$, $2-m<q<p$, $q\notin\Lambda$,
$u=O(r^p)$ and
$\Delta_Cu=O(r^{q-2})$.
Then
there exists
$(f_\lambda)_{\lambda\in\Lambda}\in\bigoplus_{\lambda\in\Lambda}
E_\lambda$
such that $u=
\sum_{\lambda\in\Lambda\cap(q,p]}f_\lambda r^\lambda+O(r^q)$
where $\Lambda\cap(q,p]$ is possibly empty
and in that case we set
$\bigoplus_{\lambda\in\emptyset}E_\lambda=\{0\}$.
\end{corollary}
\begin{proof}
Applying the proposition above to $\Delta_Cu$ in place of $f$
we find $u'\in C^\infty\bigl((R,\infty)\times\Sigma;\mathbb{R}\bigr)$
with $u'=O(r^q)$ such that $\Delta_Cu'=\Delta_Cu$.
By Proposition~\ref{homogeneous}
we can find some $a_0,b_0,a_1,b_1,a_2,b_2,\cdots\in\mathbb{R}$ such that
$u-u'=\sum_{i=0}^\infty(a_ir^{\alpha_i}+b_ir^{\beta_i})v_i$.

Since $u'=O(r^q)$ we get
$u=\sum_{i=0}^\infty(a_ir^{\alpha_i}+b_ie^{\beta_i})v_i+O(r^q)$.
Since $u=O(r^p)$ and $p>q$
we get $a_i=0$ if $\alpha_i>p$.
Since $\alpha_0\leq\alpha_1\leq\alpha_2\leq\cdots$
tend to $\infty$
we can find a unique integer $i(p)$ such that $\alpha_{i(p)+1}>p$
and $\alpha_{i(p)}\leq p$.
Since $a_i=0$ for every $i>i(p)$
we see that
$\sum_{i=0}^\infty b_ir^{\beta_i}v_i$
converges in the local $C^\infty$-topology.
We put $w=\sum_{i=0}^\infty b_ir^{\beta_i}v_i$.

Since $0\geq2-m=\beta_0\geq\beta_1\geq\beta_2\geq\cdots$
it follows that for every $\rho>R$ we have
\[\int_\rho^{2\rho}\int_\Sigma
|w|^2\frac{dr}{r}d\mu_\Sigma\leq\left(\frac{\rho}{R}\right)^{2-m}
\int_R^{2R}\int_\Sigma|w|^2\frac{dr}{r}d\mu_\Sigma.\]
Applying elliptic regularity to $w$
we get $w=O(r^{2-m})$.
Since
$u=\sum_{i=0}^{i(p)}a_ir^{\alpha_i}v_i
+w+O(r^q)$ and $q>2-m$
we get $u=\sum_{i=0}^{i(p)}a_ir^{\alpha_i}v_i
+O(r^q)$, completing the proof.
\end{proof}
We suppose now
that $C$ is a smooth special Lagrangian cone in
$(\mathbb{R}^{2m}\setminus\{0\},\omega_0,J_0,\Omega_0)$
and $\Sigma=C\cap S^{2m-1}$.
For each $\rho>0$ we put $B_\rho=\{y\in\mathbb{R}^n:|y|<\rho\}$.
We denote by $i_C:C\to\mathbb{R}^{2m}$ the inclusion map of $C$ into $\mathbb{R}^{2m}$.

Joyce~\cite[Definition~7.1]{Joyce1} defines
special Lagrangian submanifolds of $(\mathbb{R}^{2m},\omega_0,J_0,\Omega_0)$
asymptotic to $C$ with multiplicity $1$ at infinity with some rate $<2$.
We extend it to varifolds as follows.
We denote by $\mathcal{W}$ the set of all
special Lagrangain integral varifolds $W$
with $\partial\overrightarrow{W}=0$ in
$(\mathbb{R}^{2m},\omega_0,J_0,\Omega_0)$
asymptotic at infinity to $C$ with multiplcity $1$.
For each $\lambda<2$ we denote by $\mathcal{W}_\lambda$
the set of all $W\in\mathcal{W}$
such that we can find a compact subset $K$ of $W$,
an $R>0$ and a diffeomorphism $f:C\setminus\overline{B_R}\to
\spt\|W\|\setminus K$ such that $f-i_C=O(r^{\lambda-1})$.

We suppose that $C$ is Jacobi-integrable in the sense of
Joyce~\cite[Definition~6.7]{Joyce1}.
In a way similar to Simon~\cite[Part~II, \S\S5 and 6]{Simon2},
then, we can prove that if $W\in\mathcal{W}$ then
there exists $\lambda<2$ such that $W\in\mathcal{W}_\lambda$.

Joyce~\cite[Theorem~4.3]{Joyce1} proves
a version of Weinstein's theorem~\cite[Corollary~6.2]{Weinstein},
which we shall recall next.
We denote by $T^*C$ the cotangent bundle over $C$,
by $0_C$ the zero-section of $T^*C$, and by $\omega_C$
the canonical symplectic form on $T^*C$.
We regard $(0,\infty)$ as a multiplicative group
acting upon $C$ and $\mathbb{R}^{2m}$ as re-scaling.
We can lift the $(0,\infty)$-action
uniquely to $T^*C$
so that for each $t\in(0,\infty)$ we shall have
$t^*\omega_C=t^2\omega_C$ (on the left-hand side
we regard $t$ as a map of $C$ into itself).
We have then:
\begin{lemma}\label{UC}
There exist a neighbourhood $UC$ of $\im0_C$ in $T^*C$
invariant under $(0,\infty)$, 
and a diffeomorphism $\Phi_C$ of $UC$ into $\mathbb{R}^{2m}$
equivariant under $(0,\infty)$
with $\Phi_C\circ0_C=i_C$ and $\Phi_C^*\omega_0=\omega_C$.
\end{lemma}


Let $W\in\mathcal{W}_\lambda$.
Then we can take a compact subset $K$ of $W$
and a closed $1$-form $w$ on $C\setminus\overline{B_R}$
such that $\spt\|W\|\setminus K_W=\Phi_C(\graph w)$.
We denote by $\pi_\Sigma$ denotes the projection of $C$ onto $\Sigma$.
We may write $w=\pi_\Sigma^*\eta_W+dh_W$
for some $1$-form $\eta_W$ on $\Sigma$
and some $h_W:C\setminus\overline{B_R}\to\mathbb{R}$.
By results of Joyce~\cite[Equations~(7.7) and (7.8)]{Joyce1} we have:
\begin{lemma}\label{delta h}
If $\alpha<2$ and $h_W=O(r^\alpha)$ then
we have $\Delta_Ch_W=O(r^{2(\alpha-2)})$.
\end{lemma}
We can extend a result of Joyce~\cite[Theorem~7.11]{Joyce1}
as follows:
\begin{lemma}\label{improvement}
Let $W\in\mathcal{W}_\lambda$,
$\lambda'<\lambda<2$ and $[\lambda',\lambda)\cap\Lambda=\emptyset$.
Then we have $W\in\mathcal{W}_{\lambda'}$.
\end{lemma}
\begin{proof}
For each integer $n\geq0$ we put
$\lambda(n)=2^n(\lambda-2)+2$.
We can take a unique integer $\nu$ such that
$\lambda(\nu+1)<\lambda'\leq\lambda(\nu)$.
By an induction on $n=0,1,\cdots,\nu$,
we shall prove
$h_W=O(r^{\lambda(n)})$.

By the property of $h_W$ we have $h_W=O(r^\lambda)=O(r^{\lambda(0)})$.
If $\nu=0$ we can then complete the induction automatically.
We suppose therefore $\nu>0$.
Suppose also that we have $h_W=O(r^{\lambda(n)})$
for some $n=0,1,\cdots,\nu-1$.
By Lemma~\ref{delta h} we have $\Delta h_W=O(r^{2(\lambda(n)-2)})=O(r^{\lambda(n+1)-2})$.
Since $n<\nu$ we get $\lambda(n+1)\geq\lambda'$ and so
$[\lambda(n+1),\lambda)\cap\Lambda=\emptyset$.
Applying Corollary~\ref{linearized equation} to
$h_W,\lambda,\lambda(n+1)$ in place of $u,p,q$ respectively, we get
$h_W=O(r^{\lambda(n+1)})$, completing the induction.

We have thus proved
$h_W=O(r^{\lambda(n)})$ for every $n=0,1,\cdots,\nu$.
Putting $n=\nu$ we get $h_W=O(r^{\lambda(\nu)})$.
By Lemma~\ref{delta h} we have $\Delta_Ch_W=O(r^{\lambda(\nu+1)-2})$.
By the definition of $\nu$ we have $\lambda(\nu+1)\leq\lambda'$
and so $\Delta_Ch_W=O(r^{\lambda'-2})$.
Applying Corollary~\ref{linearized equation} to
$h_W,\lambda,\lambda'$ in place of $u,p,q$ respectively, we get
$h_W=O(r^{\lambda'})$, completing the proof.
\end{proof}
Joyce~\cite[Definition~3.6]{Joyce2} defines the stability of $C$.
We have:
\begin{theorem}\label{decay order estimate}
Let $C$ be stable in the sense of Joyce.
Then there exists $b\in\mathbb{R}^{2m}$ such that
$W-b\in\mathcal{W}_0$.
\end{theorem}
\begin{proof}
By the stability of $C$
we have $\Lambda\cap(1,2)=\emptyset$ and
$E_1=\{b\cdot x:b\in\mathbb{R}^{2m}\}$.
We take $\lambda\in(1,1+\epsilon)$.
Let $h_W$ be as above. Then by Lemma~\ref{improvement} we have
$h_W=O(r^{\lambda})$.
By Lemma~\ref{delta h}, therefore, we have
$\Delta_Ch_W=O(r^{2(\lambda-2)})$.
Applying Corollary~\ref{linearized equation} to $h_W,\lambda,2(\lambda-2)$
in place of $u,p,q$
we get $h_W=b\cdot x|_C+O(r^{2(\lambda-2)+2})$.

We may suppose that for each $t\in[0,1]$
there exist a compact subset $K_t$ of $\mathbb{R}^{2m}$ and
a $1$-form $w_t$ on $C\setminus\overline{B_R}$ such that
$(\spt\|W\|-tb)\setminus K_t
=\Phi_C(\graph w_t)$.
We put $\beta=(b\cdot x)\circ\Phi_C$.
We have then a function $\beta:UC\to\mathbb{R}$.
Notice that $\Phi_C^{-1}(W-tb)$ is the image of the time-one map of
the flow generated by $d\beta\lrcorner\omega_C$. 
Then we have
$\partial w_t/\partial t=-w_t^*d\beta$
and so $w_1=w_0-d\int_0^1w_t^*\beta dt
=\pi_\Sigma^*\eta_W+d(h_W-\int_0^1w_t^*\beta dt)$.
Hence we get
$h_{W-b}=h_W-\int_0^1w_t^*\beta dt=h_W-\beta|_C-
\int_0^1(w_t^*\beta-\beta|_C)dt=O(r^{2\lambda-2})+O(r^{\lambda-1})
=O(r^{2\lambda-2})$.
By results of Joyce~\cite[Equations~(7.7) and (7.8)]{Joyce1}
we have
$\Delta_Ch_{W-b}=O(r^{4\lambda-8})$.
Applying Corollary~\ref{linearized equation} to $h_{W-b},2\lambda-2,4\lambda-8$
in place of $u,p,q$
we get $c\in\mathbb{R}$ such that
$h_{W-b}=c+O(r^{4\lambda-6})=O(r^0)$
as we may suppose $4\lambda-6<0$.
This completes the proof.
\end{proof}
Let $\lambda<2$ and $W\in\mathcal{W}_\lambda$.
Take a compact subset $K_W$ of $\mathbb{R}^{2m}$,
an $R>0$ and a diffeomorphism $f_W:C\setminus\overline{B_R}\to\spt\|W\|\setminus K_W$
with $f_W^*i_C=O(r^{\lambda-1})$.
Then we have a Riemannian metric $f_W^*g_0$ over $C\setminus\overline{B_R}$.
With respect to $f_W^*g_0$ we can define the Laplacian
$\Delta_W:C^\infty(C\setminus\overline{B_R};\mathbb{R})
\to C^\infty(C\setminus\overline{B_R};\mathbb{R})$.
We have then:
\begin{proposition*}
If $u\in C^\infty(C\setminus\overline{B_R};\mathbb{R})$ and $u=O(r^\alpha)$
then we have
$\Delta_Wu=\Delta_Cu+O(r^{\alpha+\lambda-4})$.
\end{proposition*}
\begin{proof}
Since $f_W-i_C=O(r^{\lambda-1})$ we get
$f_W^*g_0=i_C^*g_0+O(r^{\lambda-2})$.
We denote by $\nabla_W$ and $\nabla_C$
the Levi-Civita connexions over $C\setminus\overline{B_R}$ with respect to $f_W^*g_0$
and $i_C^*g_0$ respectively.
We have then $\nabla_W=\nabla_C+O(r^{\lambda-3})$
and so $\Delta_Wu=\Delta_Cu+O(r^{\alpha+\lambda-4})$
as we want.
\end{proof}
We give a corollary to this:
\begin{corollary}\label{improvement2}
If $u\in C^\infty(C\setminus\overline{B_R};\mathbb{R})$,
$\Delta_Wu=0$ and $u=O(r^0)$
then we have $u=c+O(r^{2-m})$ for some $c\in\mathbb{R}$.
\end{corollary}
\begin{proof}
By the proposition above we have
$\Delta_Cu=O(r^{0+\lambda-4})=O(r^{\lambda-4})$.
If $\lambda-2<2-m$ then we can complete the proof
by applying Corollary~\ref{linearized equation} to $0,\lambda-2$
in place of $p,q$ respectively.

We suppose therefore $\lambda-2\geq2-m$.
We take $\lambda'\in(\lambda,2)$ such that $\lambda'-2>2-m$.
Applying Corollary~\ref{linearized equation} to $0,\lambda'-2$
in place of $p,q$ respectively, we get $u=c+O(r^{\lambda'-2})$
for some $c>0$.
We have thus improved the decay order estimate for $u$,
and so we can complete the proof in a way similar to the proof of
Lemma~\ref{improvement}.
\end{proof}





We suppose now
\[C=\{(z_1,\cdots,z_m)\in\mathbb{C}^m\setminus\{0\}:
|z_1|=\cdots=|z_m|,z_1\cdots z_m\in(0,\infty)\}.\]
This is an extension of \eqref{C} in \S\ref{intro} to dimension $m$.
Harvey and Lawson~\cite[Chapter~III.3.A, Theorem~3.1]{Harvey and Lawson}
prove that $C$ is a special Lagrangian submanifold of $(\mathbb{R}^{2m}\setminus\{0\},
\omega_0,J_0,\Omega_0)$.

We define a $T^{m-1}$-action on $\mathbb{C}^m$ as follows.
We write $T^{m-1}=S^1\times\cdots\times S^1$ and $S^1=\{t\in\mathbb{C}:|t|=1\}$.
For each $j\in\{2,\cdots,m\}$
we define the $j$-th $S^1$-action on $\mathbb{C}^m$ by setting
$t\cdot(z_1,\cdots,z_j,\cdots,z_m)=(tz_1,\cdots,t^{-1}z_j,\cdots,z_m)$
for each $(z_1,\cdots,z_m)\in\mathbb{C}^m$.

We also define a map $\mu_j:\mathbb{C}^m\to\mathbb{R}$ by
$2\mu_j(z_1,\cdots,z_m)=|z_1|^2-|z_j|^2$.
We shall identify $\mathbb{R}$ with the Lie algebra of $S^1$
so that $\mu_j$ will be the moment map on $(\mathbb{C}^m,\omega_0)$
with respect to the $S^1$-action.
We have then:
\begin{theorem*}
If $W\in\mathcal{W}_0$
and $j\in\{2,\cdots,m\}$ then
we have $\grad_{TW}\mu_j=0$
almost everywhere on $\mathbb{R}^{2m}$
with respect to $\|W\|$.
\end{theorem*}
\begin{proof}
We can take a compact subset $K_W$ of $\mathbb{R}^{2m}$,
an $R>0$ and a diffeomorphism $f_W:C\setminus\overline{B_R}\to
\spt\|W\|\setminus K_W$ with $f_W-i_C=O(r^{-1})$.
Since $\mu_j=O(r^2)$ and $\mu_j\circ i_C=0$ we get
$f_W^*\mu_j=\mu_j\circ f_W=\mu_j\circ f_W-\mu_j\circ i_C
=\int_0^1d\mu_j|_{(1-t)f_W+t i_C}(f_W-i_C)dt
=O(r^{2-1}r^{-1})=O(r^0)$ and
so $f_W^*\mu_j=O(r^0)$.
By a result of Joyce~\cite[Lemma~3.4]{Joyce2} we have
$\Delta_Wf_W^*\mu_j=0$ for each $j=2,\cdots,m$.
By Corollary~\ref{improvement2} we can find
$c_j\in\mathbb{R}$ such that
$f_W^*\mu_j-c_j=O(r^{2-m})$.
Putting $\mu_j'=\mu_j-c_j$ we get
$f_W^*\mu_j'=O(r^{2-m})$.
We have clearly $\grad\mu_j'=\grad\mu_j$.

Take a smooth function
$\chi:\mathbb{R}\to[0,1]$ with $\chi=1$ on $B_1$ and $\chi=0$ on $B_2$.
Let $R>0$, and define a function $\chi_R:\mathbb{R}^{2m}\to[0,1]$
by setting $\chi_R(x)=\chi(|x|/R)$.
Since $W$ has first variation $0$ in $B_{2R}$ 
we get then
\[\int_{B_{2R}}\di_{TW}(\chi_R\mu_j'\grad\mu_j)d\|W\|=0.\]
Also by a result of Joyce~\cite[Lemma~3.4]{Joyce2} we have
$\di_{TW}\grad\mu_j=0$ and so
\begin{equation}\label{mu grad mu}
\int_{B_{2R}}\mu_j'(\grad\chi_R,\grad_{TW}\mu_j)
+\chi_R|{\grad_{TW}\mu_j}|^2d\|W\|=0.
\end{equation}
Notice that
$(\grad\chi_R,\grad_{TW}\mu_j)
=(d\chi_R,d\mu_j|_{\spt\|W\|\setminus K_W})$
on $\spt\|W\|\setminus K_W$.
Take $R$ sufficiently large so that
$K_W\subset B_R$.
Then we have $\grad\chi_R=0$ on $K_W$
and so
$(\grad\chi_R,\grad_{TW}\mu_j)
=(d\chi_R,d\mu_j|_{\spt\|W\|\setminus K_W})$
on $\spt\|W\|$.
We have therefore
\[\int_{B_{2R}}-\mu_j'(\grad\chi_R,\grad_{TW}\mu_j)d\|W\|
\leq\sup_{\spt\|W\|\setminus K_W}|\mu_j'(d\chi_R,d\mu_j)|
\int_{B_{2R}}d\|W\|\]
and so by \eqref{mu grad mu} we have
\begin{equation}\label{mu grad}
\int_{B_{2R}}\chi_R|{\grad_{TW}\mu_j}|^2d\|W\|
\leq\sup_{\spt\|W\|\setminus K_W}|\mu_j'(d\chi_R,d\mu_j)|
\int_{B_{2R}}d\|W\|.
\end{equation}
Since $f_W:C\setminus\overline{B_R}\to\spt\|W\|\setminus K_W$
is a diffeomorphism we get
\begin{equation}
\sup_{\spt\|W\|\setminus K_W}|\mu_j'(d\chi_R,d\mu_j)|=
\sup_{C\setminus\overline{B_R}}|f_W^*\mu_j'(d\chi_R,df_W^*\mu_j)|.
\end{equation}
Since $\chi_R=\chi(r/R)$
we get $|d\chi_R|\leq kR^{-1}$ for some $k>0$
independent of $R$.
Since $f_W^*\mu_j'=O(r^{2-m})$ we get
\begin{equation}\label{grad mu2}
\sup_{C\setminus\overline{B_R}}|f_W^*\mu_j'(d\chi_R,df_W^*\mu_j)|
\leq kR^{2-m}R^{-1}R^{1-m}=kR^{2-2m}
\end{equation}
for some $k>0$ independent of $R$.
By \eqref{mu grad}--\eqref{grad mu2} we have
\[\int_{B_{2R}}\chi_R|{\grad_{TW}\mu_j}|^2d\|W\|
\leq kR^{2-2m}
\int_{B_{2R}}d\|W\|.\]
On the other hand, by the monotonicity formula,
we have $\int_{B_{2R}}d\|W\|\leq kR^m$
for some $k>0$ depending only on $m$ and $T_\infty W$.
We have therefore
\[\int_{B_{2R}}\chi_R|{\grad_{TW}\mu_j}|^2d\|W\|\leq kR^{2-m},
k>0\text{ independent of }R.\]
Letting $R\to\infty$ we get $\int_{\mathbb{R}^{2m}}|{\grad_{TW}\mu_j}|^2d\|W\|=0$,
completing the proof.
\end{proof}
We give a corollary to the theorem above.
We define a map $f:\mathbb{C}^m\to\mathbb{C}$ by
setting $f(z_1,\cdots,z_m)=i^{m+1}z_1\cdots z_m$
for each $(z_1,\cdots,z_m)\in\mathbb{C}^m$.
We put $\im f=(f-\bar{f})/2i:\mathbb{C}^m\to\mathbb{R}$
and $F=(\mu_2,\cdots,\mu_m,\im f):
\mathbb{C}^m\to\mathbb{R}^m$.
We have then:
\begin{corollary}
\label{T_zV}
If $W\in\mathcal{W}_0$ then we have
$T_yW=\Ker dF|_y$
for $\|W\|$-almost every $y\in\mathbb{R}^{2m}$.
\end{corollary}
\noindent
This follows readily from the proof of
Harvey and Lawson
\cite[Chapter~III.3.A, Theorem~3.1]{Harvey and Lawson}.
%or a more general result of Goldstein~\cite[Theorem~A]{Goldstein}.

We have moreover:
\begin{corollary}\label{F}
For every $W\in\mathcal{W}_0$ there exists $c\in\mathbb{R}^m$
such that $F=c$ on $\spt\|W\|$.
\end{corollary}
\begin{proof}
By Corollary~\ref{T_zV} we have
$dF|_{\spt\|W\|\setminus K_W}=0$,
and $F|_{\spt\|W\|\setminus K_W}$ is therefore locally constant.
Since $\spt\|W\|\setminus K_W\cong C\setminus\overline{B_R}\cong(R,\infty)\times T^{m-1}$
we see that $\spt\|W\|\setminus K_W$ is connected, and
$F|_{\spt\|W\|\setminus K_W}$ is therefore constant;
i.e. we have $F|_{\spt\|W\|\setminus K_W}=c$ for some $c\in\mathbb{R}^m$.

Put $\phi=|F-c|^2$.
Then we have $\phi=0$ on $\spt\|W\|\setminus K_W$
and so $\spt\phi\cap\spt\|W\|\subset K_W$.
By Corollary~\ref{T_zV} we have $\grad_{TW}\phi=0$
almost everywhere on $\mathbb{R}^{2m}$ with respect to $\|W\|$.

We shall now use a result of
Michael and Simon~\cite[Theorem~2.1]{Michael Simon},
who prove a
Poincar\'e--Sobolev inequality for varifolds;
we refer also to Simon~\cite[Theorem~18.6]{Simon3},
who uses varifolds more explicitly.
We are going to use the following version:
\begin{lemma*}
Let $W$ be a stationary integral varifold of dimension $m$ in $(\mathbb{R}^n,g_0)$.
Suppose that we have a smooth function $\phi:\mathbb{R}^n\to[0,\infty)$
with $\spt\phi\cap\spt\|W\|$ compact and $\grad_{TW}\phi=0$
almost everywhere on $\mathbb{R}^n$
with respect to $\|W\|$.
Then we have $\phi=0$ on $\spt\|W\|$.
\end{lemma*}
We give a proof for the sake of clarity:
\begin{proof}
We define $r:\mathbb{R}^n\to[0,\infty)$
by setting $r(y)=|y|$.
We can then define
$\partial_r=\partial/\partial r$ as a smooth vector field on $\mathbb{R}^n\setminus\{0\}$.
It is easy to see that $r\partial_r$
extends smoothly to $\mathbb{R}^n$.
Let $\phi$ be as above, and let $\chi$ be a compactly-supported
smooth function on $\mathbb{R}^n$ with $\chi=1$ on $\spt\phi\cap\spt\|W\|$.
Then we can define
$\phi r\partial_r$ as a smooth vector field on $\mathbb{R}^n$.
Since $W$ has first variation $0$ we get
\[\int_{\mathbb{R}^n}
\di_{TW}\chi\phi r\partial_rd\|V\|=0.\]
Since $\chi=1$ on $\spt\phi\cap\spt\|W\|$ we get
$\int_{\spt\phi\cap\spt\|W\|}\di_{TW}\phi r\partial_rd\|W\|=0$.
Since $\grad_{TW}\phi=0$ on $\spt\|W\|$ we get
$\int_{\spt\phi\cap\spt\|W\|}\phi\di_{TW}r\partial_rd\|W\|=0$.
Since $\di_{TW}r\partial_r=m$ we get
$\int_{\spt\phi\cap\spt\|W\|}m\phi d\|W\|=0$.
Since $\phi\geq0$ we get $\phi=0$ on $\spt\phi\cap\spt\|V\|$,
completing the proof.
\end{proof}
Hence we get $\phi=|F-c|^2=0$ on $\spt\|W\|$,
completing the proof of Corollary~\ref{F}.
\end{proof}

We suppose now $m=3$.
For the fibres of $F:\mathbb{C}^3\to\mathbb{R}^3$
we have an explicit description of
the topology and asymptotic behaviour at infinity,
which we shall use next.
The behaviour of $F:\mathbb{C}^m\to\mathbb{R}^m$
is rather complicated if $m>3$, which we shall not discuss.

We put $Y=\{(a,0,0)\in\mathbb{R}^3:a\geq0\}\cup\{
(0,a,0)\in\mathbb{R}^3:a\geq0\}\cup\{(-a,-a,0)\in\mathbb{R}^3:a\geq0\}$.
We note that if $c\in\mathbb{R}^3\setminus Y$
then $F^{-1}(c)$ has no fixed point with respect to the $T^2$-action
\[(e^{i\theta},e^{i\phi})\cdot(z_1,z_2,z_3)=(e^{i\theta}z_1,
e^{i\phi}z_2,e^{-i\theta-i\phi}z_3).\]

We have:
\begin{proposition}\label{c in Y}
Let $W\in\mathcal{W}$
and suppose $\spt\|W\|\subset F^{-1}(c)$ for some $c\in\mathbb{R}^3$.
Then we have $c\in Y$.
\end{proposition}

For the proof we shall use:
\begin{lemma}\label{RS^1S^1}
If $c\in\mathbb{R}^3\setminus Y$
then $F^{-1}(c)$ is a submanifold of $\mathbb{R}^6$
diffeomorphic to $\mathbb{R}\times S^1\times S^1$
and asymptotic to $C\cup-C$ with multiplicity $1$ at infinity.
\end{lemma}
\begin{proof}
We put $c=(c_1,c_2,c_3)$.
We may suppose $c_1\geq c_2\geq0$ without loss of generality.
Since $(c_1,c_2,c_3)\in\mathbb{R}^3\setminus Y$ we get $c_3\neq0$ or $c_2>0$.
For each $t\in\mathbb{R}$ we can find a unique $\phi_c(t)\in[0,\infty)$
such that
\[(\phi_c(t)+c_1)
(\phi_c(t)+c_2)\phi_c(t)=|t+ic_3|^2.\]
It is easy to see that $\phi_c(t)$ depends smoothly on $t$.
We put $\psi_c(t)=\sqrt{\phi_c(t)+c_1}
\sqrt{\phi_c(t)+c_2}$.
Since $c_3\neq0$ or $c_2>0$ we get
$\psi_c(t)>0$ for every $t\in\mathbb{R}$,
and so we can define $\psi_c^{-1}(t)>0$
for every $t\in\mathbb{R}$.
Define a smooth map $\Phi_c:\mathbb{R}\times S^1\times S^1\to f^{-1}(c)$
by setting
\[\Phi_c(t,u,v)=(\sqrt{\phi_c(t)+c_1}u,\sqrt{\phi_c(t)+c_2}v,
\frac{t+ic_3}{\psi_c(t)uv}),\]
and a smooth map $\Psi_c(t):F^{-1}(c)\to\mathbb{R}\times S^1\times S^1$
by setting
\[\Psi_c(z_1,z_2,z_3)=(\re z_1z_2z_3,\frac{z_1}{\phi_c(\re z_1z_2z_3)},
\frac{z_2}{\phi_c(\re z_1z_2z_3)}).\]
Then $\Psi_c\circ\Phi_c$ is clearly the identity map of
$\mathbb{R}\times S^1\times S^1$.
It is also easy to see that $\Phi_c\circ\Psi_c$
is the identity map of $F^{-1}(c)$,
and so $F^{-1}(c)$ is a submanifold of $\mathbb{C}^3$
diffeomorphic to $\mathbb{R}\times S^1\times S^1$.
It is also easy to see that $\Phi_c\circ\Psi_c$
is asymptotic to $C\cup-C$ with multiplicity $1$ at infinity.
\end{proof}


\begin{proof}[Proof of Proposition~$\ref{c in Y}$]
We give a proof by contradiction,
and so suppose $c\notin Y$.
By Lemma~\ref{RS^1S^1}, then,
$F^{-1}(c)$ will be a connected submanifold of $\mathbb{R}^6$.
By a constancy theorem of Allard~\cite[Theorem~4.6(3)]{Allard}
or Simon~\cite[Theorem~41.1]{Simon3},
therefore,
$\Theta_W$ will be constant on $F^{-1}(c)$.
On the other hand we have $\Theta_W=1$ near infinity on $\spt\|W\|$
and so we shall have $\Theta_W=1$ on $F^{-1}(c)$,
which implies $\|W\|=|F^{-1}(c)|$.
By Lemma~\ref{RS^1S^1}, however,
$F^{-1}(c)$ is asymptotic at infinity to $C\cup-C$
with multiplicity $1$,
which contradicts that $W$ is asymptotic at infinity to $C$
with multiplicity $1$.
This completes the proof.
\end{proof}

Let $c\in Y\subset\mathbb{R}^2\times\{0\}$.
Then we have $z_1z_2z_3\in\mathbb{R}$ on $F^{-1}(c)$.
For each $\epsilon>0$ we put
$L_c^\epsilon=F^{-1}(c)\cap\{z_1z_2z_3>\epsilon\}$.
We also put $L_c=F^{-1}(c)\cap\{z_1z_2z_3\geq0\}$.
By the definition of $C$ we have $L_0=C$.
By \eqref{123} we have
$L_1=L_{(2,0,0)}$, $L_2=L_{(0,2,0)}$ and $L_3=L_{(0,-2,-2)}$.

We have:
\begin{proposition}\label{fibre}
Let $W\in\mathcal{W}$ and suppose
$\spt\|W\|\subset F^{-1}(c)$ for some $c\in Y$.
Then we have
$\|W\|=|sL|$
for some $s>0$ and $L\in\{C,L_1,L_2,L_3\}$.
\end{proposition}

For the proof we shall use:
\begin{lemma}\label{R^2S^1}
$L_1,L_2,L_3$ are
submanifolds of $\mathbb{C}^3$ 
diffeomorphic to $S^1\times\mathbb{R}^2$
and asymptotic to $C$ with multiplicity $1$ at infinity.
\end{lemma}
\begin{proof}[Proof of Proposition~$\ref{fibre}$]
We give the proof only for $L_1$;
we can do it likewise for $L_2$ and $L_3$.
By definition we have
$L_1=\{(z_1,z_2,z_3)\in\mathbb{C}^3:|z_1|^2-1=|z_2|^2=|z_3|^2,
z_1z_2z_3\in[0,\infty)\}$.
Define a smooth map $\phi:L_1\to S^1\times\mathbb{C}$
by setting $\phi(z_1,z_2,z_3)=(z_1/\sqrt{|z_2|^2+1},z_2)$,
and a smooth map $\psi:S^1\times\mathbb{C}\to L_1$ by setting
$\psi(e^{i\theta},z)=(\sqrt{|z|^2+1}e^{i\theta},z,e^{-i\theta}\bar{z})$.
Then $\phi\circ\psi$ is clearly the identity map of $S^1\times\mathbb{C}$.
It is also easy to see that $\psi\circ\phi$ is the identity map of $L_1$.
This implies that $L_1$ is a submanifold of $\mathbb{C}^3$
diffeomorphic to $S^1\times\mathbb{C}$.
It is also easy to see that $L_1$ is asymptotic to $C$
with multiplicity $1$ at infinity.
\end{proof}
\begin{proof}[Proof of Prosposition~$\ref{fibre}$]
We have only to prove $\|W\|=|L_c|$.
For each $\epsilon>0$ we put $U_\epsilon^+=\{\re z_1z_2z_3>\epsilon\}$
and $U_\epsilon^-=\{\re z_1z_2z_3<-\epsilon\}$.
In a way similar to the proof of Lemma~\ref{R^2S^1}
we can take $\epsilon_0>0$ such that if $\epsilon\in(0,\epsilon_0)$
then $F^{-1}(c)\cap U_\epsilon^+$ and $F^{-1}(c)\cap U_\epsilon^-$
will be connected submanifolds of $\mathbb{R}^6$.
By the constancy theorem, therefore,
%Applying Lemma~\ref{constancy} to $F^{-1}(c)\cap U_\epsilon^+$
%and $F^{-1}(c)\cap U_\epsilon^-$ in place of $L$
$\Theta_W$ is constant on
$F^{-1}(c)\cap U_\epsilon^+$ and $F^{-1}(c)\cap U_\epsilon^-$,
respectively.

In a way similar to the proof of Lemma~\ref{R^2S^1}
we can prove that $F^{-1}(c)\cap U_\epsilon^+$ is asymptotic at infinity
to $C$ with multiplicity $1$
and $F^{-1}(c)\cap U_\epsilon^-$ is asymptotic at infinity to
$-C$ with multiplicity $1$.
Notice also that $W$ is asymptotic at infinity to $C$
with multiplicity $1$.
Then we see that on $F^{-1}(c)\cap U_\epsilon^+$
we have $\Theta_W=1$
and on $F^{-1}(c)\cap U_\epsilon^-$ we have
$\Theta_W=0$.
This holds for every $\epsilon>0$, and so
$\|W\|=|F^{-1}(c)\cap\{\re z_1z_2z_3\geq0\}|=|L_c|$
as we want.
\end{proof}

We shall now complete the proof of Theorem~\ref{uniqueness}.
We suppose that $W$ is as in Theorem~\ref{uniqueness}.
By Theorem~\ref{decay order estimate}, then,
we can find $b\in\mathbb{R}^6$ such that $W-b\in\mathcal{W}_0$.
By Corollary \ref{F}, therefore,
we can find $c\in\mathbb{R}^3$
such that $\spt\|W-b\|\subset F^{-1}(c)$.
By Proposition~\ref{c in Y}, therefore, we have $c\in Y$.
By Proposition~\ref{fibre}, therefore, we have
$\|W-b\|=|sL|$ for some $a>0$ and $L\in\{C,L_1,L_2,L_3\}$
as we want.


\section{Proof of Theorem~\ref{main}}
\label{completion section}
%We begin with a review of the theory of
%Joyce~\cite{Joyce1,Joyce2,Joyce3,Joyce4,Joyce5}.
%As in \S\ref{intro} we define $C$ by \eqref{C}.
%We put $\Sigma=C\cap S^5$.
%Let $u\in C^\infty(C;\mathbb{R})$
%and $\alpha\in\mathbb{R}$.
%we shall write $u=O(r^\alpha)$
%as $r\to0$ if we have
%\[\lim_{\epsilon\to+0}\sup_{C\cap B_\epsilon}|r^{-\alpha+k}\nabla^ku|<\infty
%\text{ for every }k=0,1,2,3,\cdots\]
%where $\nabla$ denotes the Levi-Civita connexion
%over the Riemannian manifold $(C,i_C^*g_0)$,
%and $|\bullet|$ denotes the norm with respect to $i_C^*g_0$.
%Then we shall write $u=O(r^\alpha)$ as $r\to0$ if
%there exists $\delta>0$ such that
%\[\sup_{(0,\delta)\times\Sigma}|r^{-\alpha+k}\nabla^ku|<\infty
%\text{ for every }k=0,1,2,3,\cdots\]
%and write $u=O(r^\alpha)$ as $r\to0$ if
%there exists $R>0$ such that
%\[\sup_{(R,\infty)\times\Sigma}|r^{-\alpha+k}\nabla^ku|<\infty
%\text{ for every }k=0,1,2,3,\cdots\]
%where $\nabla$ denotes the Levi-Civita connexion induced from $g_0$,
%and $|\bullet|$ denotes the norm induced from $g_0$.
As in \S\ref{intro} the main steps in the
proof of Theorem~\ref{main} are the bubbling-off analysis
in \S\ref{bubble section} and the classification result in
\S\ref{uniqueness section}.
In order to complete the proof, however,
we have to do some technical things in addition.
We begin with a review of the theory of Joyce~\cite{Joyce1,Joyce2,Joyce3,Joyce4,Joyce5}.

We suppose that $(M,\omega,J,\Omega)$
is an almost Calabi--Yau manifold of dimension $3$ over $\mathbb{C}$.
We suppose that $M$ is compact.
We take $a\in H_3(M;\mathbb{Z})$.
As in \S\ref{SL section}
we denote by $\mathcal{V}$ the space of all special Lagrangian varifolds
$V$ in $(M,\omega,J,\Omega)$ with $\spt\|V\|$ compact,
$\partial\overrightarrow{V}=0$ and
$[\overrightarrow{V}]=a$.

We denote by $\mathcal{X}$ the set of all
elements of $\mathcal{V}$ with support connected,
and singularity
only at one point and modelled on $C$ with multiplicity $1$.
We take $X\in\mathcal{X}$.
We denote by $x$ the singular point of $X$.
We take a linear isomorphism
$\gamma:\mathbb{R}^6\to T_xM$
with $\gamma^*g|_x=g_0$, $\gamma^*J|_x=J|_0$,
$\gamma^*\Omega|_x=\Omega_0$
and $\gamma(C)$ a multiplicity $1$ smooth tangent cone
to $X$ at $x$.
We may regard $X$ as a submanifold of $M\setminus\{x\}$,
which we shall denote by $X'$.

As in \S\ref{SL section} we define
$\psi:M\to(0,\infty)$ by \eqref{psi}.
We have then
$\omega_0=\psi^2(x)\gamma^*\omega|_x$.
By a theorem of Darboux
we can find a real number $\delta>0$ and a diffeomorphism $\Gamma$ of
$B_\delta$ into $M$ with $\Gamma(0)=x$, $d\Gamma|_0=\psi(x)\gamma$
and $\Gamma^*\omega|_x=\omega_0$.

Joyce~\cite[\S3.2]{Joyce2}
proves that $C$ is rigid and so special Lagrangian Jacobi-integrable,
which implies that $x$ is an
isolated conical singularity
in the sense of Joyce~\cite[Definition~3.6]{Joyce1};
for the details we refer to
Joyce~\cite[Theorem~6.8]{Joyce1},
Allard--Almgren~\cite{AA},
Simon~\cite[Part~II, \S6]{Simon2}
and Adams--Simon~\cite{AS}.

We define $T^*C$, $\omega_C$, $U_C$ and $\Phi_C$ as in Lemma~\ref{UC}.
From results of Joyce~\cite[Theorem~4.4 and Lemma~4.5]{Joyce1}
we see that making $\delta>0$ smaller if necessary
we can take an embedding $f_X$ of $C\cap B_\delta$ into $X'$,
a function $h_X:C\cap B_\delta\to\mathbb{R}$ and an $\alpha>2$
with $h_X=O(r^\alpha)$ such that
$f_X^{-1}(X')=\Phi_C(\graph dh_X)$.
We put $Z=X'\setminus\im f_X$
where $\im f_X=f_X(C\cap B_\delta)$.
It is clear that $Z$ is
a compact submanifold of $X'$ of co-dimension $0$
with boundary diffeomorphic to $\Sigma$.

We denote by $T^*X'$ the cotangent bundle over $X'$,
by $0_C$ the zero-section of $T^*X'$, and by $\omega_X$
the canonical symplectic form on $T^*X'$.
Since $f_X$ maps $C\cap B_\delta$ diffeomorphically onto $X'\setminus Z$
we get a vector-bundle isomorphism
$T^*f_X:T^*(X'\setminus Z)\to T^*(C\cap B_\delta)$
covering $f_X^{-1}:X'\setminus Z\to C\cap B_\delta$.
By a result of Joyce~\cite[Theorem~4.6]{Joyce1}
we can find a neighbourhood $UX$ of $\im0_X$ in $T^*X'$
and a diffeomorphism $\Phi_X$ of $UX$ into $M$
with
$\Phi_X\circ0_X=i_X$,
$\Phi_X^*\omega=\omega_X$
and
\[\Phi_X|_{UX\setminus T^*Z}
=\Gamma\circ\Phi_C\circ(+dh_X)\circ T^*f_X\]
where $+dh_X$ denotes the fibrewise translation of $T^*C$
by $dh_X$.

For each $y\in M$
we denote by $P|_y$ the set of all
linear isomorphisms $\phi:\mathbb{R}^6\to T_yM$
with $\phi^*g|_y=g_0$,
$\phi^*J|_y=J_0$ and $\phi^*\Omega|_y=\Omega_0$.
We put $P=\bigcup_{y\in M}P|_y$.
It is clear that $P$ is a principal bundle over $M$
with structure group $SU_3$.
We have $(x,\gamma)\in P$.
By a result of Joyce~\cite[Theorem~5.2]{Joyce2}
we can take a neighbourhood $U_{x,\gamma}$ of $(x,\gamma)$ in $P$
such that for all $p=(y,\phi)\in U_{x,\gamma}$
we can construct
embeddings $\Gamma_p$ of $B_\delta$ into $M$
depending smoothly on $p$ with
$\Gamma_p(0)=y$, $d\Gamma_p|_0=\phi$ and
$\psi^2(y)\Gamma_p^*\omega=\omega_0$, and
embeddings $\Phi_p$ of $UX$ into $M$
depending smoothly on $p$ with
$\Phi_p\circ0_X=i_X$, $\Phi_p^*\omega=\omega_X$ and
\[\Phi_p|_{UX\setminus T^*Z}=\Gamma_p\circ\Phi_C\circ(+dh_X)\circ T^*f_X.\]

Joyce~\cite[Definition~5.6]{Joyce2} defines a topology on $\mathcal{X}$.
On the other hand
$\mathcal{X}$ has also the topology induced from $\mathcal{V}$.
By a result of Joyce~\cite[Theorem~5.3]{Joyce2},
Allard's regularity theorem and Theorem~\ref{X top},
the two topologies on $\mathcal{X}$ are the same.

By the stability of $C$ we can apply a result of
Joyce~\cite[Corollary~6.11]{Joyce2},
who proves that
we can take a neighbourhood $\mathcal{Y}$ of $X$ in $\mathcal{X}$
which has a structure of manifold, and
$T_X\mathcal{Y}$ is isomorphic
to the image of $H^1_c(X';\mathbb{R})\to H^1(X';\mathbb{R})$.
We can parametrize $\mathcal{Y}$ as follows.
Since $\spt\|X\|$ and $\Sigma$ are connected
we see that $X'$ is also connected,
and so $H^1_c(X';\mathbb{R})\to H^1(X';\mathbb{R})$
is injective.
We denote by $\Omega^1_Z$ the space of all $1$-forms on $X'$ supported in $Z$;
we can then define the natural projection
$\pi_Z:\Omega^1_Z\to H^1_c(X';\mathbb{R})$;
we can take a linear subspace $H^1_Z$ of $\Omega^1_Z$
such that $\pi_Z$ maps $H^1_Z$ isomorphically
onto $H^1_c(X';\mathbb{R})$.
We take a real number $\mu\in(2,3)$ as in Joyce~\cite[Definition~3.7]{Joyce2}
and define a function space
$C^\infty_\mu(X';\mathbb{R})$ as in \cite[Definition~4.2]{Joyce2}.
From the proof of Joyce~\cite[Corollary~6.11]{Joyce2}
we get:
\begin{lemma}\label{zeta}
There exist a neighbourhood $\mathcal{Z}$ of $0$ in $H^1_Z$
and an embedding $\mathcal{H}$ of $\mathcal{Z}$ into $\mathcal{X}$
with the following property:
for all $\zeta\in\mathcal{Z}$
there exist $p(\zeta)\in P$ and $h_\zeta\in C^\infty_\alpha(X';\mathbb{R})$
depending smoothly on $\zeta$ with
$\mathcal{H}(\zeta)=\Phi_{p(\zeta)}\bigl(\graph(\zeta+dh_\zeta)\bigr)$.
\end{lemma}

We define $L_1$, $L_2$ and $L_3$ by \eqref{123} in \S\ref{uniqueness section}.
We take $L\in\{L_1,L_2,L_3\}$.
We can then take a compact submanifold $K$ of $L$
of co-dimension $0$ with boundary diffeomorphic to $\Sigma$,
a real number $R>0$ and a diffeomorphism $f_L$ of $C\setminus\overline{B_R}$ onto
$L\setminus K$ with
$f_L-i_C=O(r^{-1})$.
Making $R>0$ larger if necessary
we can find a $1$-form $\eta_L$ on $C\setminus\overline{B_R}$
with $\Phi_C(\graph \eta_L)=L\setminus K$.

We denote by $T^*L$ the cotangent bundle over $L$,
by $0_L$ the zero-section of $T^*L$, and by $\omega_L$
the canonical symplectic form on $T^*L$.
Since $f_L$ maps $C\setminus\overline{B_R}$ diffeomorphically onto $L\setminus K$
we get a vector-bundle isomorphism
$T^*f_L:T^*(L\setminus {K})\to T^*(C\setminus\overline{B_R})$
covering $f_L^{-1}:L\setminus K\to C\cap\overline{B_R}$.
By a result of Joyce~\cite[Theorem~7.5]{Joyce1}
we can find a neighbourhood $UL$ of $\im0_L$ in $T^*L$
and a diffeomorphism $\Phi_L$ of $UL$ into $\mathbb{R}^6$ with
$\Phi_L^*\omega_0=\omega_L$,
$\Phi_L\circ0_L=i_L$ and
\[\Phi_L|_{UL\setminus T^*{K}}
=\Phi_C\circ(+\eta_L)\circ T^*f_L\]
where $+\eta_L$ denotes the fibrewise translation of $T^*C$
by $\eta_L$.

We have a homomorphism $f_X^*:H^1(X';\mathbb{R})\to H^1(C\cap B_\delta;\mathbb{R})
\cong H^1(\Sigma;\mathbb{R})$.
We denote by $[\eta_L]$ the de Rham cohomology class of $\eta_L$.
We have then $[\eta_L]\in H^1(C\setminus\overline{B_R};\mathbb{R})
\cong H^1(\Sigma;\mathbb{R})$.
By Hodge theory we can take a harmonic $1$-form $\eta_L'$ on $\Sigma$
with $[\eta_L']=[\eta_L]$.
We denote by $\pi_\Sigma$ the projection of $C$ onto $\Sigma$.
We put $\widetilde{\eta}_L=\pi_\Sigma^*\eta_L'$.

We denote by $\im f_X^*$ the image of the linear map
$f_X^*:H^1(X';\mathbb{R})\to H^1(C\cap B_\rho;\mathbb{R})
\cong H^1(\Sigma;\mathbb{R})$.
Joyce~\cite[Lemma~10.1]{Joyce5} proves that
$\im f_X^*$ is of dimension $1$ over $\mathbb{R}$.
Joyce~\cite[Equation (77)]{Joyce5} proves also that
if $L\neq L'\in\{L_1,L_2,L_3\}$ then $[\eta_L]$ and $[\eta_L']$
are linearly independent.
Hence we get:
\begin{lemma}\label{unique L}
There exists at most one $L\in\{L_1,L_2,L_3\}$
with $[\eta_L]\in\im f_X^*$.
\end{lemma}

As in \S\ref{intro} we suppose $X\in\partial{\mathcal{N}}$.
By Corollary~\ref{gluing condition} below, then,
we shall have $L\in\{L_1,L_2,L_3\}$
with $[\eta_L]\in\im f_X^*$.
Hence we can find
a closed $1$-form $\xi_L$ on $X'$ with $f_X^*[\xi_L]=[\eta_L]$.
By a result of Joyce~\cite[Theorem~10.4]{Joyce5}
there exist a real number $\tau>0$.
an open submanifold $\widehat{K}$ of $L$
with $K\subset\widehat{K}$ and
an open submanifold $\widehat{Z}$ of $X$
with $Z\subset\widehat{Z}$
such that for all $t\in(0,\tau)$
we can find $u_C:C\cap A_{tR,\delta}\to\mathbb{R}$
with $|du_C|^{1,\cyl}_{[tR,\delta]}=o(t)$,
$u_L:\widehat{K}\to\mathbb{R}$ with $|du_L|_{C^1(\widehat{K})}=o(t)$,
and $u_X:\widehat{Z}\to\mathbb{R}$ with
$|du_X|_{C^1(\widehat{Z})}=o(t)$
such that
on $f_L^{-1}(\widehat{K})\cap A_{tR,\delta}$
we have $t^2(f_L^*du_L+\eta_L)=du_C+t^2\widetilde{\eta}_L$,
on $f_X^{-1}(\widehat{Z})\cap A_{tR,\delta}$
we have $f_X^*(du_X+t^2\xi_L)+dh_X=du_C+t^2\widetilde{\eta}_L$,
and
\[\Gamma\bigl(t\Phi_L(\graph du_L)\bigr)
\cup\Gamma\circ\Phi_C\bigl(\graph(du_C+t^2\widetilde{\eta}_L)\bigr)
\cup\Phi_X\bigl(\graph(du_X+t^2\xi_L)\bigr)\]
will be a compact special Lagrangian submanifold of $(M,\omega,J,\Omega)$.
From the proof Joyce~\cite[Theorem~10.4]{Joyce5}
it follows readily
that we can extend the result as follows:
\begin{lemma}\label{zeta gluing}
There exist a real number $\tau>0$,
a neighbourhood $U_{x,\gamma}$ of $(x,\gamma)$ in $P$,
a neighbourhood $\mathcal{Z}$ of $0$ in $H^1_Z$,
an open submanifold $\widehat{K}$ of $L$
with $K\subset\widehat{K}$ and
an open submanifold $\widehat{Z}$ of $X$
with $Z\subset\widehat{Z}$
such that
for all $(t,p,\zeta)\in(0,\tau)\times U_{x,\gamma}\times\mathcal{Z}$
we can find
$u_C\in C^\infty(C\cap A_{tR,\delta};\mathbb{R})$
with $|u_C|^{1,\cyl}_{[tR,\delta]}=o(t)$,
$u_L\in C^\infty(\widehat{K};\mathbb{R})$
with $|u_L|_{C^1(\widehat{K})}=o(t)$,
and $u_X\in C^\infty(\widehat{Z};\mathbb{R})$
with $|u_X|_{C^1(\widehat{Z})}=o(t)$
such that
on $f_L^{-1}(\widehat{K})\cap A_{tR,\delta}$
we have $t^2(f_L^*du_L+\eta_L)=du_C+t^2\widetilde{\eta}_L$,
on $f_X^{-1}(\widehat{Z})\cap A_{tR,\delta}$
we have $f_X^*(du_X+\zeta+dh_\zeta+t^2\xi_L)+dh_X=du_C
+t^2\widetilde{\eta}_L$,
and if we put
\begin{equation*}\begin{split}
G(t,p,\zeta)=\Gamma_p\bigl(t\Phi_L(\graph du_L)\bigr)
\cup\Gamma_p\circ\Phi_C\bigl(\graph(du_C+t^2\widetilde{\eta}_L)\bigr)\\
\cup\Phi_p\bigl(\graph(du_X+\zeta+dh_\zeta+t^2\xi_L)\bigr)
\end{split}\end{equation*}
then $G(t,p,\zeta)$
will be a compact special Lagrangian submanifold of $(M,\omega,J,\Omega)$.
\end{lemma}
\noindent
Thus we get a map $G:(0,\tau)\times U_{x,\gamma}\times\mathcal{Z}\to\mathcal{N}$.

We take $(t,p,\zeta)\in(0,\tau)\times U_{x,\gamma}\times\mathcal{Z}$.
We denote by $i_{t,p,\zeta}$ the inclusion map of $G(t,p,\zeta)$ into $M$.
We denote by $T^*G(t,p,\zeta)$ the cotangent bundle over $G(t,p,\zeta)$,
and by $0_{t,p,\zeta}$ the zero-section of $T^*G(t,p,\zeta)$.
We shall define a neighbourhood $U_{t,p,\zeta}$ of $\im 0_{t,p,\zeta}$
and an embedding $\Phi_{t,p,\zeta}$ of $U_{t,p,\zeta}$ into $M$
with $\Phi_{t,p,\zeta}\circ0_{t,p,\zeta}=i_{t,p,\zeta}$.

We put $\phi_L^{t,p,\zeta}=\Gamma_p\circ t\circ\Phi_L\circ du_L$
where $du_L$ is regarded as a map of $\widehat{K}$ into $T^*\widehat{K}$.
Thus we get an embedding $\phi_L^{t,p,\zeta}:\widehat{K}\to G(t,p,\zeta)$.
We put $G_L^{t,p,\zeta}=\im\phi_L^{t,p,\zeta}$.
Since $\phi_L^{t,p,\zeta}$ maps $\widehat{K}$ diffeomorphically
onto $G_L^{t,p,\zeta}$ we get a vector bundle isomorphism
$T^*\phi_L^{t,p,\zeta}:T^*G_L^{t,p,\zeta}\to T^*\widehat{K}$.
On a neighbourhood of $\im(0_{t,p,\zeta}\circ\phi_L^{t,p,\zeta})$ in $T^*G(t,p,\zeta)$
we set
$\Phi_{t,p,\zeta}=\Gamma_p\circ t\circ\Phi_L\circ(+du_L)\circ
T^*\phi_L^{t,p,\zeta}$.

We put $\phi_C^{t,p,\zeta}=\Gamma_p\circ\Phi_C\circ(du_C
+t^2\widetilde{\eta}_L)$
where $du_C+t^2\widetilde{\eta}_L$
is regarded as a map of $C\cap A_{tR,\delta}$ into $T^*C$.
Thus we get an embedding $\phi_C^{t,p,\zeta}:C\cap A_{tR,\delta}\to G(t,p,\zeta)$.
We put $G_C^{t,p,\zeta}=\im\phi_C^{t,p,\zeta}$.
Since $\phi_C^{t,p,\zeta}$ maps $C\cap A_{tR,\delta}$ diffeomorphically
onto $G_C^{t,p,\zeta}$ we get a vector bundle isomorphism
$T^*\phi_C^{t,p,\zeta}:T^*G_C^{t,p,\zeta}\to T^*(C\cap A_{tR,\delta})$.
On a neighbourhood of $\im(0_{t,p,\zeta}\circ\phi_C^{t,p,\zeta})$ in $T^*G(t,p,\zeta)$
we set
$\Phi_{t,p,\zeta}=\Gamma_p\circ\Phi_C\circ(+du_C)\circ
T^*\phi_C^{t,p,\zeta}$.

We put $\phi_X^{t,p,\zeta}=\Phi_p\circ(du_X+\zeta+dh_\zeta+t^2\xi_L)$
where $du_X+\zeta+dh_\zeta+t^2\xi_L$
is regarded as a map of $\widehat{Z}$ into $T^*\widehat{Z}$.
Thus we get an embedding $\phi_X^{t,p,\zeta}:\widehat{Z}\to G(t,p,\zeta)$.
We put $G_X^{t,p,\zeta}=\im\phi_X^{t,p,\zeta}$.
Since $\phi_X^{t,p,\zeta}$ maps $\widehat{Z}$ diffeomorphically
onto $G_X^{t,p,\zeta}$ we get a vector bundle isomorphism
$T^*\phi_X^{t,p,\zeta}:T^*G_X^{t,p,\zeta}\to T^*\widehat{Z}$.
On a neighbourhood of $\im(0_{t,p,\zeta}\circ\phi_X^{t,p,\zeta})$ in $T^*G(t,p,\zeta)$
we set
$\Phi_{t,p,\zeta}=\Phi_p\circ(+du_X)\circ
T^*\phi_X^{t,p,\zeta}$.
Thus we get:
\begin{lemma}\label{Utp}
There exists a neighbourhood $U_{t,p,\zeta}$ of $\im 0_{t,p,\zeta}$
and an embedding $\Phi_{t,p,\zeta}$ of $U_{t,p,\zeta}$ into $M$
with $\Phi_{t,p,\zeta}\circ0_{t,p,\zeta}=i_{t,p,\zeta}$.
\end{lemma}
We shall now take a neighbourhood $\mathcal{U}$ of $X$ in $\mathcal{V}$
and define a map $F:\mathcal{U}\cap\mathcal{N}\to(0,\tau)\times\mathcal{Z}$.
By Allard's regularity theorem we may suppose that for every
$N\in\mathcal{U}\cap\mathcal{N}$ we can find a unique $1$-form $\beta_X$
on $Z$ such that $N\setminus\Phi_X\bigl(T^*(C\cap B_\delta)\bigr)
=\Phi_X(\graph\beta_X)$.
Since $0\neq[\eta_L]\in\im f_X^*$
and $\im f_X^*$ is of dimension $1$
we can find a unique $s\in\mathbb{R}$ with $s[\eta_L]=f_X^*[\beta_X]$.
We have then a compactly-supported $1$-form $\beta_X-s\xi_L$ on $Z$.
As $[\beta_X-s\xi_L]|_{\partial Z}=0$
we may write $[\beta_X-s\xi_L]\in H^1_c(X';\mathbb{R})$.
As $\pi_Z$ maps $H^1_Z$ isomorphically onto $H^1_c(X';\mathbb{R})$
we may write $[\beta_X-s\xi_L]\in H^1_Z$.
We set $F(N)=(s,[\beta_X-s\xi_L])$.

It is easy to see that for each $(t,\zeta)\in(0,\tau)\times\mathcal{Z}$
we have
$F\circ G(t,\zeta)=(t^2,\zeta)$.
Hence we see that
$G:(0,\tau)\times\mathcal{Y}\to\mathcal{N}$ is injective.
By an argument of Joyce~\cite[\S8.2]{Joyce5}, moreover,
$F:\mathcal{U}\cap\mathcal{N}
\to(0,\tau)\times\mathcal{Z}$ is a local diffeomorphism.
Hence we see that $G(t,p,\zeta)$ is independent of $p\in U_{x,\gamma}$
as we may suppose that $U_{x,\gamma}$ is connected.
We may therefore write $G:(0,\tau)\times\mathcal{Z}\to\mathcal{N}$.

We can extend $G$ continuously to $[0,\tau)\times\mathcal{Z}$
by setting $G(0,0)=0$ for each $0\in\mathcal{Z}$.
We can re-state Theorem~\ref{main} as follows:
\begin{theorem*}
$G$ maps $[0,\tau)\times\mathcal{Z}$ onto a neighbourhood of $X$ in $\mathcal{V}$.
\end{theorem*}

We are going now to prove the theorem above.

Using the results of \S\S\ref{bubble section}--\ref{uniqueness section}
we shall determine a neighbourhood of $X$ in $\mathcal{V}$:
\begin{theorem}\label{bubble and glue}
Let $\epsilon>0$,
let $\widehat{K}$
be an open submanifold of $L$ with $K\subset\widehat{K}$,
and let $\widehat{Z}$
be an open submanifold of $X$ with $Z\subset\widehat{Z}$.
Then
there exists a neighbourhood $\mathcal{U}$ of $X$ in $\mathcal{V}$
such that for every $V\in\mathcal{U}\setminus\mathcal{X}$
we can find
an $L\in\{L_1,L_2,L_3\}$,
a real number $s>0$,
a point $p\in P$,
a $1$-form $\beta_C$ on $C\cap A_{sR,\delta}$ with
$|\beta_C|^{1,\cyl}_{[sR,\delta]}<\epsilon$,
a $1$-form $\beta_L$ on $\widehat{K}$
with $|\beta_L|_{C^1(\widehat{K})}<\epsilon$,
and a $1$-form $\beta_X$ on $\widehat{Z}$
with $|\beta_X|_{C^1(\widehat{Z})}<\epsilon$ such that
on $f_L^{-1}(\widehat{K})\cap A_{sR,\delta}$ we have
$s^2(f_L^*\beta_L+\eta_L)=\beta_C$,
on $f_X^{-1}(\widehat{Z})\cap A_{sR,\delta}$ we have
$f_X^*\beta_X+dh_X=\beta_C$,
and if we put
\[N=\bigl|\Gamma_p\bigl(s\Phi_L(\graph\beta_L)\bigr)
\cup\Gamma_p\circ\Phi_C(\graph\beta_C)
\cup\Phi_X(\graph\beta_X)\bigr|\]
then we have $\|V\|=|N|$ where $|N|=\mathcal{H}^3\llcorner N$
where $\mathcal{H}^3$ denotes the Hausdorff $3$-dimensional measure on
$(M,g)$.
\end{theorem}
\begin{remark*}
Since $s^2(f_L^*\beta_L+\eta_L)=\beta_C$ 
on $f_L^{-1}(\widehat{K})\cap A_{sR,\delta}$ and
$f_X^{-1}(\widehat{Z})\cap A_{sR,\delta}$
on $f_X^*\beta_X+dh_X=\beta_C$
we see that
$N$ will be a compact submanifold of $M$.
\end{remark*}
\begin{proof}
Let $(V_n)_{n=1}^\infty$ be a sequence in $\mathcal{V}\setminus\mathcal{X}$
converging to $X$.
Then we have only to prove that
for each $\nu=1,2,3,\cdots$ we can find an integer $n>\nu$ such that
with $V_n$ in place of $V$
the property above will hold.

We consider $\gamma^*\exp_x^*V_n$.
To $\gamma^*\exp_x^*V_n$ in place of $V_n$
we can apply Theorem~\ref{bubble-off},
and so we can find a sequence
$(\delta_n)_{n=1}^\infty$
in $(0,\infty)$ converging to $0$,
a sequence $(y_n)_{n=1}^\infty$ in $B_\rho$ converging to $0$,
and a subsequence of $\bigl(\delta_n^{-1}(V_n-y_n)\bigr)_{n=1}^\infty$
converging to some special Lagrangian varifold $W$ in
$(\mathbb{R}^6,\omega_0,J_0,\Omega_0)$
asymptotic at infinity to some $C'\in\mathcal{C}_\epsilon$
with multiplicity $1$
and satisfying
$\mathcal{E}(W-b)>0$ for each $b\in\mathbb{R}^6$.
We may suppose that
$\bigl(\delta_n^{-1}(V_n-y_n)\bigr)_{n=1}^\infty$
itself converges to $W$.

Since $C$ is rigid in the sense of Joyce
we can find $a\in SU_3$ with $aC=C'$.
To $a^{-1}W$ we can apply Theorem~\ref{uniqueness}
and so we can find $\lambda>0$,
$L\in\{C,L_1,L_2,L_3\}$ and $b\in\mathbb{R}^6$
such that $\|W\|=|\lambda L+b|$.
Since $\mathcal{E}(W-b)>0$ we get
$\mathcal{E}(L)>0$ and so $L\neq C$, which implies $L\in\{L_1,L_2,L_3\}$.

We define an embedding $E_n$ of $B_\rho$ into $M$ by setting
$E_n(y)=\exp_x\gamma a(y+\delta_nb)$.
We put $y(n)=E_n(0)=\exp_x\gamma a(\delta_nb)$,
It is easy to see that $y(n)$ tends as $n\to\infty$
to $\exp_x\gamma a(0)=x$.
We put also $\phi(n)=dE_n|_0=\gamma\circ a$.
It is easy to see that $\phi(n)$ is close to $\gamma$.

We put $s_n=\delta_n\lambda$.
From Allard's regularity theorem
and Theorem~\ref{AAS} we see that
for $n$ sufficiently large
we can find an embedding $e_{L,n}$ of $\widehat{K}$
into $\mathbb{R}^6$
with $|e_{L,n}-i_L|_{C^1(\widehat{K})}<\epsilon$,
an embedding $e_{C,n}$ of $C\cap A_{s_nR,\sigma}$ into $\mathbb{R}^6$,
with $|e_{C,n}-i_C|^{1,\cyl}_{[s_nR,\sigma]}<\epsilon$
and an embedding of $\widehat{Z}$ into $X$ with
$|e_{X,n}-i_X|_{C^1(\widehat{Z})}<\epsilon$
such that
\[\|V_n\|=\bigl|E_n(s_n\im e_{L,n})
\cup E_n(\im e_{C,n})
\cup\im e_{X,n}\bigr|.\]
We put $p(n)=(y(n),\phi(n))$.
We have then $\Gamma_{p(n)}(0)=E_n(0)$ and
$d\Gamma_{p(n)}|_0=dE_n|_0$.
Hence we can find
an embedding $f_{L,n}$ of $\widehat{K}$
into $\mathbb{R}^6$
with $|f_{L,n}-i_L|_{C^1(\widehat{K})}<\epsilon$,
an embedding $f_{C,n}$ of $C\cap A_{s_nR,\sigma}$ into $\mathbb{R}^6$,
with $|f_{C,n}-i_C|^{1,\cyl}_{[s_nR,\sigma]}<\epsilon$
and an embedding of $\widehat{Z}$ into $X$ with
$|f_{X,n}-i_X|_{C^1(\widehat{Z})}<\epsilon$
such that
\[\|V_n\|=\bigl|\Gamma_{p(n)}(s_n\im f_{L,n})
\cup\Gamma_{p(n)}(\im f_{C,n})
\cup\im f_{X,n}\bigr|.\]
Hence we can find
a $1$-form $\beta_{L,n}$ on $\widehat{K}$
with $|\beta_{L,n}|_{C^1(\widehat{K})}<\epsilon$,
a $1$-form $\beta_{C,n}$ on $C\cap A_{s_nR,\delta}$ with
$|\beta_{C,n}|^{1,\cyl}_{[s_nR,\delta]}<\epsilon$,
and a $1$-form $\beta_{X,n}$ on $\widehat{Z}$
with $|\beta_{X,n}|_{C^1(\widehat{Z})}<\epsilon$ such that
\[\|V\|=\bigl|\Gamma_p\bigl(s_n\Phi_L(\graph\beta_L)\bigr)
\cup\Gamma_p\circ\Phi_C(\graph\beta_C)
\cup\Phi_X(\graph\beta_X)\bigr|.\]
This implies that
on $f_L^{-1}(\widehat{K})\cap A_{s_nR,\delta}$ we have
$s^2(f_L^*\beta_{L,n}+\eta_L)=\beta_{C,n}$,
on $f_X^{-1}(\widehat{Z})\cap A_{s_nR,\delta}$ we have
$f_X^*\beta_{X,n}+dh_X=\beta_{C,n}$.
This contradicts our hypothesis, and so completes the proof.
\end{proof}


We give a corollary to Theorem~\ref{bubble and glue}:
\begin{corollary}\label{gluing condition}
Suppose that for every neighbourhood $\mathcal{U}$ of $X$ in $\mathcal{V}$
we have $\mathcal{U}\setminus\mathcal{X}\neq\emptyset$.
Then there exists an $L\in\{L_1,L_2,L_3\}$
with $[\eta_L]\in\im f_X^*$.
\end{corollary}
\begin{proof}
Let $\epsilon>0$, and let $\mathcal{U}$ be as in Theorem~\ref{bubble and glue}.
Then we have by assumption $\mathcal{U}\setminus\mathcal{X}\neq\emptyset$,
and so we can find some $V\in\mathcal{U}\setminus\mathcal{X}$.
Let $s$, $\beta_L$, $\beta_C$ and $\beta_X$ be as in Theorem~\ref{bubble and glue}.
Then we have
$s^2[f_L^*\beta_L+\eta_L]=[f_X^*\beta_X+dh_X]\in H^1(\Sigma;\mathbb{R})$
and so $[f_L^*\beta_L+\eta_L]\in\im f_X^*$.
Notice also that $[\beta_L]$ tends to $0$ as $\epsilon\to0$.
Then we get $[\eta_L]\in\im f_X^*$,
completing the proof.
\end{proof}

We shall give another corollary to Theorem~\ref{bubble and glue}.
By assumption we have $X\in\partial\mathcal{N}$.
By Corollary~\ref{gluing condition}, therefore,
we can find $L\in\{L_1,L_2,L_3\}$
with $[\eta_L]\in\im f_X^*$.
We take $\epsilon>0$ and suppose that
$\widehat{K}$ and $\widehat{Z}$ are as in Theorem~\ref{bubble and glue}.
Making $\mathcal{K}$ and $\widehat{Z}$ smaller if necessary
we can construct $G:(0,\tau)\times U_{x,\gamma}\times\mathcal{Z}\to
\mathcal{N}$ as in Lemma~\ref{zeta gluing}.
We define $\Phi_{t,p,\zeta}$ as in Lemma~\ref{Utp}.
We have then:
\begin{corollary}\label{vC}
There exists a neighbourhood $\mathcal{U}$ of $X$ in $\mathcal{V}$
such that
for every $V\in\mathcal{U}\setminus\mathcal{X}$
there exist $t>0$,
$p\in P$, $\zeta\in H^1_Z$
and $w\in C^\infty\bigl(G(t,p,\zeta);\mathbb{R}\bigr)$
with $|dw|_{C^1(G(t,p,\zeta))}<\epsilon$
such that $\|V\|=|\Phi_{t,p,\zeta}(\graph dw)|$.
\end{corollary}
\begin{proof}
Let $\mathcal{U}$ be as in Theorem~\ref{bubble and glue},
and let $V\in\mathcal{U}\setminus\mathcal{X}$.
Let $\beta_C$ and $\beta_X$ be as in Theorem~\ref{bubble and glue}.
Then we have $[\beta_C]=[dh_X+f_X^*\beta_X]=[f_X^*\beta_X]$ and so
$[\beta_C]\in\im f_X^*$.
Let $s$ and $\beta_L$ be as in Theorem~\ref{bubble and glue}.
Then we have $s^2[f_L^*\beta_L+\eta_L]=[\beta_C]$.
We have also $|\beta_L|_{C^1(\widehat{K})}<\epsilon$ and $[\eta_L]\neq0$.
We may therefore suppose $[\beta_C]\neq0$.
Notice that $\im f_X^*$ is generated by $[\eta_L]$
and $[\beta_C]$ is sufficiently close to $s^2[\eta_L]$.
Then we can find a unique $t>0$ with $t^2[\eta_L]=[\beta_C]$
and $t$ sufficently close to $s$.

Notice that $f_X^*[\beta_X-t^2\xi_L]=[\beta_C]-t^2[\eta_L]=0$.
Notice also that we have an exact sequence $H^1_c(X';\mathbb{R})\to H^1(X';\mathbb{R})\to
H^1(\Sigma;\mathbb{R})$.
Then we get
$\beta_X-t^2\xi_L\in\im\bigl(H^1_c(X';\mathbb{R})\to H^1(X';\mathbb{R})\bigr)$.
Hence we can find $a:X'\to\mathbb{R}$
with $\beta_X-t^2\xi_L-da$ compactly-supported.
We denote its compactly-supported de Rham cohomology class
by $[v_X-t^2\xi_L-da]_c$.
Let $\pi_Z$ and $H^1_Z$ be as in Lemma~\ref{zeta}.
Then $\pi_Z$ maps $H^1_Z$ isomorphically onto $H^1_c(X';\mathbb{R})$,
Hence we can find a unique $\zeta\in H^1_Z$
with $[\zeta]_c=[\beta_X-t^2\xi_L-da]_c$
where $[\zeta]_c$ denotes compactly-supported de Rham cohomology class of $\zeta$.
Let $\phi_X^{t,p,\zeta}$ be as in Lemma~\ref{Utp}.
Then we can find an exact $1$-form $\alpha_X$ on $\widehat{Z}$
such that $\spt\|V\|\cap \Phi_p(T^*Z)=\Phi_{t,p,\zeta}(\graph\alpha_X)$.

Let $p$ be as in Theorem~\ref{bubble and glue}.
Then we can find a $1$-form $\alpha$ on $G(t,p,\zeta)$
with $|\alpha|_{C^1(G(t,p,\zeta))}<\epsilon$ such that
$\|V\|=|\Phi_{t,p,\zeta}(\graph w)|$.
We have to prove that $\alpha$ is an exact $1$-form.

Since $L$ and $X'$ are connected we get
an exact sequence $0\to H^1\bigl(G(t,p,\zeta)\bigr)\to
H^1\bigl(\widehat{K}\bigr)\oplus H^1\bigl(\widehat{Z}\bigr)$.
It suffices therefore to prove $[\alpha]$ maps to $0$
under $H^1\bigl(G(t,p,\zeta)\bigr)\to
H^1\bigl(\widehat{K}\bigr)\oplus H^1\bigl(\widehat{Z}\bigr)$.

The map $H^1\bigl(G(t,p,\zeta)\bigr)\to
H^1\bigl(\widehat{Z}\bigr)$ is given by $\phi_X^{t,p,\zeta}$
and we have $\phi_X^{t,p,\zeta*}\alpha=\alpha_X$,
which is an exact $1$-form.

Let $\phi_L^{t,p,\zeta}$ be as in Lemma~\ref{Utp},
and let $\alpha_L=\phi_L^{t,p,\zeta*}\alpha$.
Then we have to only prove $[\alpha|_L]=0\in H^1(\widehat{K})$.
Let $u_L$ be as in Lemma~\ref{zeta gluing}.
Then we have
\[t^2\bigl(f_L^*(\alpha|_L+du_L)+\eta_L\bigr)=\beta_C.\]
on $f_L^{-1}(\widehat{K})\cap A_{tR,\delta}$
We have also $t^2[\eta_L]=[\beta_C]$ and so
$f_L^*[\alpha_L]=0\in H^1(\Sigma;\mathbb{R})$.
As in the proof of Lemma~\ref{R^2S^1}
we can construct a diffeomorphism of $L$ onto $S^1\times\mathbb{R}^2$
and we may suppose that
$\widehat{K}$ is diffeomorphic to $S^1\times\mathbb{R}^2$.
Hence we get $H^1_c(\widehat{K};\mathbb{R})=0$.
Notice also that we have an exact sequence
$H^1_c(\widehat{K};\mathbb{R})\to H^1(\widehat{K};\mathbb{R})
\to H^1(\Sigma;\mathbb{R})$.
Then we see that
$f_L^*:H^1(\widehat{K};\mathbb{R})\to H^1(\Sigma;\mathbb{R})$
is injective.
On the other hand we have $f_L^*[\alpha|_L]=0$
and so $[\alpha|_L]=0\in H^1(\widehat{K};\mathbb{R})$
as we want.
\end{proof}

We shall now complete the proof of Theorem~\ref{main}.
Let $\mathcal{U}$ be as in Corollary~\ref{vC},
and let $V\in\mathcal{U}\setminus\mathcal{X}$.
Then we have only to prove $\spt\|V\|=G(t,\zeta)$
for some $(t,\zeta)\in(0,\tau)\times\mathcal{Z}$.
Let $(t,p,\zeta)$ be as in Corollary~\ref{vC}.
Then we can find a function $w:G(t,\zeta)\to\mathbb{R}$
with $\Phi_{t,p,\zeta}(dw)=\spt\|V\|$.
As $\spt\|V\|$ is special Lagrangian $w$ satisfies an elliptic equation
to which we can apply a maximum principle of Hopf~\cite{Hopf}.
Hence we see that $w$ is constant, and so $\|V\|=|G(t,\zeta)|$.

%Clifford torus cones of dimension greater than $3$
%are not stable by a result of
%Joyce~\cite[p11]{Joyce2} and Marshall~\cite[\S6.3.4]{Marshall}.
%There seems to be few known examples of stable
%special Lagrangian cones except the Clifford torus cone of dimension $3$.
%For instance Ohnita~\cite{Ohnita} gives an example of stable cone on $SU_3/SO_3$,
%but Haskins and Pacini~\cite[Remark~2.3]{Haskins Pacini}
%point out that $SU_3/SO_3$ does not bound any compact manifold
%because $SU_3/SO_3$ has a non-zero Stiefel--Whitney number;
%this implies that we cannot smooth the singularity modelled on
%the cone of $SU_3/SO_3$, which is not suitable for our purpose.

%We note that in Theorem~\ref{main} we suppose $X\in\partial\mathcal{N}$
%but we have also:
%\begin{corollary*}[to Corollary~\ref{gluing condition}]
%If $X\notin\partial\mathcal{N}$ then there exists a neighbourhood $\mathcal{U}$
%of $X$ in $\mathcal{V}$ with $\mathcal{U}\subset\mathcal{X}$.
%\end{corollary*}
%\begin{proof}
%We give a proof by contradiction,
%and so suppose that for every neighbourhood $\mathcal{U}$
%of $X$ in $\mathcal{V}$ with $\mathcal{U}\setminus\mathcal{X}\neq\emptyset$.
%By Corollary~\ref{gluing condition}, then, we shall have
%$L\in\{L_1,L_2,L_3\}$ with $[\eta_L]\in\im f_X^*$,
%and so we can glue $L$ to $X$ by the method of Joyce.
%This contradicts $X\notin\partial\mathcal{N}$
%and so completes the proof.
%\end{proof}

\begin{thebibliography}{999}
\bibitem{Allard}
{W. K. Allard},
{\em On the first variation of a varifold},
{Ann. of Math. (2) \textbf{95} (1972), 417--491}.
\bibitem{AA}
{W. K. Allard and F. J. Almgren},
{\em On the radial behavior of minimal surfaces and the uniqueness of their tangent cones}, 
{Ann. of Math. (2) \textbf{113} (1981), 215--265}.
\bibitem{AHS}
{M. F. Atiyah, N. J. Hitchin, and I. M. Singer},
{\em Self-duality in four-dimensional Riemannian geometry},
{Proc. Roy. Soc. London Ser. A \textbf{362} (1978), 425--461}.
\bibitem{AS}
{D. Adams and L. Simon},
{\em Rates of asymptotic convergence near isolated singularities of geometric extrema}, 
{Indiana Univ. Math. J. \textbf{37} (1988), 225--254}.

\bibitem{Bott}
{R. Bott},
{\em Nondegenerate critical manifolds},
{Ann. of Math, (2) \textbf{60} (1954), 248--261}.
%
\bibitem{Donaldson}
{S. K. Donaldson},
{\em An application of gauge theory to four-dimensional topology},
{J. Differential Geom. \textbf{18} (1983), 279--315}.
%\bibitem{Douglis and Nirenberg}
%{A. Douglis and L. Nirenberg},
%{\em Interior Estimates for Elliptic Systems of Partial Differential
%Equations},
%{Communications on Pure and Applied Mathematics \textbf{8} (1955),
%503--538}. 
%
\bibitem{Federer}
{H. Federer},
{\em Geometric measure theory},
{Springer, 1969}.
\bibitem{Freed Uhlenbeck}
{D. S. Freed and K. K. Uhlenbeck},
{\em Instantons and four-manifolds},
{Springer, 1984}.
%\bibitem{Goldstein}
%{E. Goldstein},
%{\em Calibrated fibrations on noncompact manifolds via group actions},
%{Duke Math. J. 110 (2001), no. 2, 309--343.}
\bibitem{Harvey and Lawson}
{R. Harvey and  H. B. Lawson},
{\em Calibrated geometries},
{Acta Math. \textbf{148} (1982), 47--157}.
\bibitem{Haskins}
{M. Haskins},
{\em The geometric complexity of special Lagrangian $T^2$-cones},
{Invent. Math. \textbf{157} (2004), 11--70}. 
%\bibitem{Haskins Pacini}

%\bibitem{Hofer}
%{H. Hofer},
%{\em Pseudoholomorphic curves in symplectizations with applications to the
%Weinstein conjecture in dimension
%three},
%{Invent. math.
%\textbf{114} (1993), 515--563.}
%\bibitem{Hopf}
\bibitem{Hopf}{E. Hopf},
{\em Elementare Bemerkungen \"uber die L\"osungen partieller
Differentialgleichungen
zweiter Ordnung vom elliptischen Typus},
{Sitz. Ber. Preuss. Akad. Wiss. \textbf{19} (1927), 147--152}.
%
\bibitem{Imagi}
{Y. Imagi},
{\em A uniqueness theorem for gluing special Lagrangian submanifolds in Calabi--Yau 
manifolds},
{preprint, arXiv:11052628}.
%\bibitem{JoyceSYZ}
%{D. D. Joyce},
%{\em Singularities of special Lagrangian fibrations and the SYZ conjecture},
%{Comm. Anal. Geom. 11 (2003), 859--907.}
\bibitem{Joyce1}
{D. D. Joyce},
{\em Special Lagrangian submanifolds with isolated conical singularities
I. Regularity},
{Ann. Global Anal. Geom. \textbf{25} (2003), 201--258}.
\bibitem{Joyce2}
{D. D. Joyce},
{\em Special Lagrangian submanifolds with isolated conical singularities
II. Moduli spaces},
{Ann. Global Anal. Geom. \textbf{25} (2003), 301--352}.
\bibitem{Joyce3}
{D. D. Joyce},
{\em Special Lagrangian submanifolds with isolated conical singularities
III. Desingularization, the unobstructed case},
{Annal. Global Anal. Geom. \textbf{26} (2004), 1--58}.
\bibitem{Joyce4}
{D. D. Joyce},
{\em Special Lagrangian submanifolds with isolated conical singularities
IV. Desingularization, obstructions and families},
{Annal. Global Anal. Geom. \textbf{26} (2004), 117--174}.
\bibitem{Joyce5}
{D. D. Joyce},
{\em Special Lagrangian submanifolds with isolated conical singularities
V. Survey and applications},
{J. Differential Geom. \textbf{63} (2003), 279--347}.
%\bibitem{Lawlor}
%{G. Lawlor},
%{\em The angle criterion},
%{Invent. math. \textbf{95} (1989), 437--446.}

%\bibitem{D. Lee}
%{D.A. Lee},
%{\em Connected Sums of Special Lagrangian Submanifolds},
%{Communications in Analysis and Geometry, Vol.12, No.3, 553--579, 2004.}

\bibitem{Lojasiewicz}
{S. \L{}ojasiewicz,}
{\em Ensembles semi-analytiques},
{IHES, 1965.}
%

%\bibitem{Marshall}
%{S. P. Marshall},
%{\em Deformations of special Lagrangian submanifolds},
%{Oxford D.Phil. thesis, 2002.}

\bibitem{Mclean}
{R. C. Mclean},
{\em Deformations of calibrated submanifolds},
{Comm. Anal. Geom. \textbf{6} (1998), 705--747}.
\bibitem{Michael Simon}
{J. H. Michael and L. Simon},
{\em Sobolev and mean-value inequalities on generalized submanifolds of $\mathbb{R}^n$},
{Comm. Pure Appl. Math. \textbf{26} (1973), 361--379}.
%\bibitem{Morrey}
%{C.B. Morrey},
%{\em Second Order Elliptic Systems of Differential Equations},
%{in {\em Contributions to the Theory of Partial Differential Equations}
%Edited by
%L. Bers, S. Bochner, and F. John},
%{Annals of Mathematics Studies \textbf{33}, 101--159, Princeton University
%Press, 1954.}
\bibitem{Oh}
{Y.-G. Oh},
{\em Second variation and stabilities of minimal Lagrangian submanifolds in K\"ahler
manifolds},
{Invent. Math. \textbf{101} (1990), 501--519}.
%\bibitem{Ohnita}
%{Y. Ohnita},
%{\em Stability and rigidity of special Lagrangian cones
%over certain minimal Legendrian orbits},
%{Osaka J. Math. 44 (2007), no. 2, 305--334.}
\bibitem{Simon}
{L. Simon},
{\em Asymptotics for a class of nonlinear evolution equations,
with applications to geometric problems},
{Ann. of Math. (2) \textbf{118} (1983), 525--571}.

\bibitem{Simon2}
{L. Simon},
{\em Isolated singularities of extrema of geometric variational problems}
{in}
{\em Harmonic mappings and minimal immersions},
{Springer, 1985}.
\bibitem{Simon3}
{L. Simon},
{\em Lectures on geometric measure theory},
{Proceedings of the Centre for Mathematical Analysis, Australian National University, 1983}.

%Richard Scheon Leon Simon
%Regularity of stable minimal hypersurfaces.
%Comm. Pure Appl. Math. 34 (1981), no. 6, 741--797.
\bibitem{Taubes}
{C. H. Taubes},
{\em Self-dual Yang-Mills connections on non-self-dual $4$-manifolds},
{J. Differential Geom. \textbf{17} (1982), 139--170}.%
\bibitem{Uhlenbeck}
{K. K. Uhlenbeck},
{\em Removable singularities in Yang-Mills fields},
{Commun. Math. Phys. \textbf{83} (1982), 11--29}.

\bibitem{Weinstein}
{A. Weinstein},
{\em Symplectic manifolds and their Lagrangian submanifolds},
{Adv. in Math. \textbf{6} (1971), 329--346}.
\end{thebibliography}
\end{document}
1112.4309.extracted_bib
\begin{thebibliography}{999}
\bibitem{Allard}
{W. K. Allard},
{\em On the first variation of a varifold},
{Ann. of Math. (2) \textbf{95} (1972), 417--491}.
\bibitem{AA}
{W. K. Allard and F. J. Almgren},
{\em On the radial behavior of minimal surfaces and the uniqueness of their tangent cones}, 
{Ann. of Math. (2) \textbf{113} (1981), 215--265}.
\bibitem{AHS}
{M. F. Atiyah, N. J. Hitchin, and I. M. Singer},
{\em Self-duality in four-dimensional Riemannian geometry},
{Proc. Roy. Soc. London Ser. A \textbf{362} (1978), 425--461}.
\bibitem{AS}
{D. Adams and L. Simon},
{\em Rates of asymptotic convergence near isolated singularities of geometric extrema}, 
{Indiana Univ. Math. J. \textbf{37} (1988), 225--254}.

\bibitem{Bott}
{R. Bott},
{\em Nondegenerate critical manifolds},
{Ann. of Math, (2) \textbf{60} (1954), 248--261}.
%
\bibitem{Donaldson}
{S. K. Donaldson},
{\em An application of gauge theory to four-dimensional topology},
{J. Differential Geom. \textbf{18} (1983), 279--315}.
%\bibitem{Douglis and Nirenberg}
%{A. Douglis and L. Nirenberg},
%{\em Interior Estimates for Elliptic Systems of Partial Differential
%Equations},
%{Communications on Pure and Applied Mathematics \textbf{8} (1955),
%503--538}. 
%
\bibitem{Federer}
{H. Federer},
{\em Geometric measure theory},
{Springer, 1969}.
\bibitem{Freed Uhlenbeck}
{D. S. Freed and K. K. Uhlenbeck},
{\em Instantons and four-manifolds},
{Springer, 1984}.
%\bibitem{Goldstein}
%{E. Goldstein},
%{\em Calibrated fibrations on noncompact manifolds via group actions},
%{Duke Math. J. 110 (2001), no. 2, 309--343.}
\bibitem{Harvey and Lawson}
{R. Harvey and  H. B. Lawson},
{\em Calibrated geometries},
{Acta Math. \textbf{148} (1982), 47--157}.
\bibitem{Haskins}
{M. Haskins},
{\em The geometric complexity of special Lagrangian $T^2$-cones},
{Invent. Math. \textbf{157} (2004), 11--70}. 
%\bibitem{Haskins Pacini}

%\bibitem{Hofer}
%{H. Hofer},
%{\em Pseudoholomorphic curves in symplectizations with applications to the
%Weinstein conjecture in dimension
%three},
%{Invent. math.
%\textbf{114} (1993), 515--563.}
%\bibitem{Hopf}
\bibitem{Hopf}{E. Hopf},
{\em Elementare Bemerkungen \"uber die L\"osungen partieller
Differentialgleichungen
zweiter Ordnung vom elliptischen Typus},
{Sitz. Ber. Preuss. Akad. Wiss. \textbf{19} (1927), 147--152}.
%
\bibitem{Imagi}
{Y. Imagi},
{\em A uniqueness theorem for gluing special Lagrangian submanifolds in Calabi--Yau 
manifolds},
{preprint, arXiv:11052628}.
%\bibitem{JoyceSYZ}
%{D. D. Joyce},
%{\em Singularities of special Lagrangian fibrations and the SYZ conjecture},
%{Comm. Anal. Geom. 11 (2003), 859--907.}
\bibitem{Joyce1}
{D. D. Joyce},
{\em Special Lagrangian submanifolds with isolated conical singularities
I. Regularity},
{Ann. Global Anal. Geom. \textbf{25} (2003), 201--258}.
\bibitem{Joyce2}
{D. D. Joyce},
{\em Special Lagrangian submanifolds with isolated conical singularities
II. Moduli spaces},
{Ann. Global Anal. Geom. \textbf{25} (2003), 301--352}.
\bibitem{Joyce3}
{D. D. Joyce},
{\em Special Lagrangian submanifolds with isolated conical singularities
III. Desingularization, the unobstructed case},
{Annal. Global Anal. Geom. \textbf{26} (2004), 1--58}.
\bibitem{Joyce4}
{D. D. Joyce},
{\em Special Lagrangian submanifolds with isolated conical singularities
IV. Desingularization, obstructions and families},
{Annal. Global Anal. Geom. \textbf{26} (2004), 117--174}.
\bibitem{Joyce5}
{D. D. Joyce},
{\em Special Lagrangian submanifolds with isolated conical singularities
V. Survey and applications},
{J. Differential Geom. \textbf{63} (2003), 279--347}.
%\bibitem{Lawlor}
%{G. Lawlor},
%{\em The angle criterion},
%{Invent. math. \textbf{95} (1989), 437--446.}

%\bibitem{D. Lee}
%{D.A. Lee},
%{\em Connected Sums of Special Lagrangian Submanifolds},
%{Communications in Analysis and Geometry, Vol.12, No.3, 553--579, 2004.}

\bibitem{Lojasiewicz}
{S. \L{}ojasiewicz,}
{\em Ensembles semi-analytiques},
{IHES, 1965.}
%

%\bibitem{Marshall}
%{S. P. Marshall},
%{\em Deformations of special Lagrangian submanifolds},
%{Oxford D.Phil. thesis, 2002.}

\bibitem{Mclean}
{R. C. Mclean},
{\em Deformations of calibrated submanifolds},
{Comm. Anal. Geom. \textbf{6} (1998), 705--747}.
\bibitem{Michael Simon}
{J. H. Michael and L. Simon},
{\em Sobolev and mean-value inequalities on generalized submanifolds of $\mathbb{R}^n$},
{Comm. Pure Appl. Math. \textbf{26} (1973), 361--379}.
%\bibitem{Morrey}
%{C.B. Morrey},
%{\em Second Order Elliptic Systems of Differential Equations},
%{in {\em Contributions to the Theory of Partial Differential Equations}
%Edited by
%L. Bers, S. Bochner, and F. John},
%{Annals of Mathematics Studies \textbf{33}, 101--159, Princeton University
%Press, 1954.}
\bibitem{Oh}
{Y.-G. Oh},
{\em Second variation and stabilities of minimal Lagrangian submanifolds in K\"ahler
manifolds},
{Invent. Math. \textbf{101} (1990), 501--519}.
%\bibitem{Ohnita}
%{Y. Ohnita},
%{\em Stability and rigidity of special Lagrangian cones
%over certain minimal Legendrian orbits},
%{Osaka J. Math. 44 (2007), no. 2, 305--334.}
\bibitem{Simon}
{L. Simon},
{\em Asymptotics for a class of nonlinear evolution equations,
with applications to geometric problems},
{Ann. of Math. (2) \textbf{118} (1983), 525--571}.

\bibitem{Simon2}
{L. Simon},
{\em Isolated singularities of extrema of geometric variational problems}
{in}
{\em Harmonic mappings and minimal immersions},
{Springer, 1985}.
\bibitem{Simon3}
{L. Simon},
{\em Lectures on geometric measure theory},
{Proceedings of the Centre for Mathematical Analysis, Australian National University, 1983}.

%Richard Scheon Leon Simon
%Regularity of stable minimal hypersurfaces.
%Comm. Pure Appl. Math. 34 (1981), no. 6, 741--797.
\bibitem{Taubes}
{C. H. Taubes},
{\em Self-dual Yang-Mills connections on non-self-dual $4$-manifolds},
{J. Differential Geom. \textbf{17} (1982), 139--170}.%
\bibitem{Uhlenbeck}
{K. K. Uhlenbeck},
{\em Removable singularities in Yang-Mills fields},
{Commun. Math. Phys. \textbf{83} (1982), 11--29}.

\bibitem{Weinstein}
{A. Weinstein},
{\em Symplectic manifolds and their Lagrangian submanifolds},
{Adv. in Math. \textbf{6} (1971), 329--346}.
\end{thebibliography}