and $\mathfrak{a}_{2}=d\mathfrak{f}_{2}$.

We write the 1-parameter family of diffeomorphisms as

$\frac{d}{ds}\lambda_{1,s}=tv_{1}\circ\lambda_{1,s},\;\mathrm{and}\;\lambda_{1}:=\lambda_{1,s}|_{s=1}.$

In addition, we define $\varphi_{2,t}:=\lambda_{1}$. As $\lim_{t\to
0}\varphi_{2,t}=\mathrm{Id}$, there exists a positive number
$T_{2}^{\prime\prime}$ such that for any $t<T_{2}^{\prime\prime}$,
$\varphi_{2,t}^{-1}(\Sigma)$ is an embedded submanifold.

$\bullet\;\mathbf{Step\;2.2}:$ We will prove for
$\|r\mathrm{SL}_{\lambda_{1}^{\star}g}(\mathfrak{A}_{2})\|_{\mathcal{C}^{,\gamma}}\lesssim
t^{3}$.

We define $\mathfrak{A}_{2}:=\mathfrak{A}_{1}+t^{2}\mathfrak{a}_{2}$ and we
will compute the formal power series expansions of the special Lagrangian
equation in terms of $t$. By Proposition 6.9, we could write

(54)
$\begin{split}\mathrm{SL}_{\lambda_{1}^{\star}g}(\mathfrak{A}_{2})=&\mathrm{SL}_{g}(\mathfrak{A}_{1})-t^{2}(\Delta_{g}\mathfrak{f}_{2}-\Delta_{g}\nabla_{v_{1}}\mathfrak{f}_{1})-t\nabla_{v_{1}}\Delta_{g}\mathfrak{f}_{1}+E_{\delta
Q}+E_{Q_{\delta g}}+E_{\Delta},\\\
=&t^{2}(\rho_{2}-\Delta_{g}(\mathfrak{f}_{2}-\nabla_{v_{1}}\mathfrak{f}_{1}))-t\nabla_{v_{1}}\Delta_{g}\mathfrak{f}_{1}+E_{\mathrm{SL}}+E_{\delta
Q}+E_{Q_{\delta g}}+E_{\Delta}.\end{split}$

where

(55)
$\begin{split}&E_{\Delta}=e_{\Delta}(d\mathfrak{F}_{1},\lambda_{1},g)+t^{3}(\nabla_{v_{1}}\Delta\mathfrak{f}_{2}-\Delta\nabla_{v_{1}}\mathfrak{f}_{2}),\;E_{Q_{\delta
g}}:=Q_{\lambda_{1}^{\star}g}(\mathfrak{A}_{1})-Q_{g}(\mathfrak{A}_{1}),\\\
&E_{\delta
Q}:=Q_{g}(\mathfrak{A}_{1})-Q_{g}(\mathfrak{A}_{0}),\;E_{\mathrm{SL}}=\mathrm{SL}_{g}(\mathfrak{A}_{1})-\rho_{2}t^{2}.\end{split}$

We define $E:=E_{\Delta}+E_{\delta Q}+E_{Q_{\delta
g}}+E_{\mathrm{SL}}+t\nabla_{v_{1}}\Delta_{g}\mathfrak{F}_{1}$, then (54)
could be rewritten as

(56)
$\begin{split}\mathrm{SL}_{\lambda_{1}^{\star}g}(\tilde{\mathfrak{A}}_{2})=t^{2}(\rho_{2}-\Delta_{g}\nabla_{v_{1}}\mathfrak{f}_{1}+\Delta_{g}\mathfrak{f}_{2})+E.\end{split}$

By Lemma 6.3, 6.7, 6.8 and Proposition 6.4, we could write $E=t^{3}E^{\prime}$
with $\|rE^{\prime}\|_{\mathcal{C}^{,\gamma}}\lesssim 1$, which implies

$\|r\mathrm{SL}_{g_{2,t}}(\mathfrak{A}_{2})\|_{\mathcal{C}^{,\gamma}}\lesssim
t^{3}.$

$\bullet\;\mathbf{Step\;3}:$ We will finish the inductive step. Suppose we
have construct the nondegenerate pair $\mathfrak{A}_{k}=d\mathfrak{F}_{k}$,
metric $g_{k}$ and $T_{k}$ such that for any $t<T_{k}$, $\mathfrak{A}_{k}$ is
nondegenerate and
$\|r\mathrm{SL}_{g_{k}}(\mathfrak{A}_{k})\|_{\mathcal{C}^{,\gamma}}\lesssim
t^{k+1}.$ We will construct $\mathfrak{A}_{k+1}$, $g_{k+1}$ and $T_{k+1}$ with
the desire property.

We could write

$\mathrm{SL}_{g_{k}}(\mathfrak{A}_{k})=\rho_{k+1}t^{k+1}+E_{\mathrm{SL}}.$

We will find
$\mathfrak{a}_{k+1}=d\mathfrak{f}_{k+1}=(df_{k+1}^{+},df_{k+1}^{-})$, a vector
field $v_{k+1}$ such that $f_{k+1}^{-}$ has leading asymptotic
$f_{k+1}^{-}\sim\Re(B_{f_{k+1}^{-}}z^{\frac{3}{2}})$ and $\mathfrak{a}_{k+1}$
solves the following equations:

(57)
$\begin{split}\rho_{k+1}^{-}-\Delta_{g_{k}}(f_{k+1}^{-}-\nabla_{v_{k}}f_{1}^{-})=0,\;\rho_{k+1}^{+}-\Delta_{g_{k}}(f_{k+1}^{+}-\nabla_{v_{k}}f_{1}^{+})=0.\end{split}$

Similarly to $\mathbf{Step\;2.1}$, we first solve
$-\Delta_{g_{k}}P_{k+1}+\rho_{k+1}^{-}=0$ with

$P_{k+1}=\Re(a_{k+1}z^{\frac{1}{2}}+b_{k+1}z^{\frac{3}{2}})+E_{P_{k+1}}.$

We choose $v_{k}$ be an extension of $-\frac{2a_{k+1}}{3B_{1}}$ over $L$, then
$\nabla_{v_{k}}f_{1}^{-}$ has leading expansion
$-\Re(a_{k+1}z^{\frac{1}{2}})$. We define
$f_{k+1}^{-}:=P_{k+1}+\nabla_{v_{k}}f_{1}^{-}$. By Proposition 5.5, 6.10, we
could find $f_{k+1}^{+}$ solves the second equation.

Let $\frac{d}{ds}\lambda_{k,s}=t^{k}v_{k}\circ\lambda_{k,s}$ be the
1-parameter family and define

(58) $\varphi_{k+1,t}=\varphi_{k,t}\circ\lambda_{k,1},$

then $g_{k+1,t}:=\varphi_{k+1,t}^{\star}g=\lambda_{k,1}^{\star}g_{k,t}.$ We
define $\mathfrak{A}_{k+1}:=\mathfrak{A}_{k}+t^{k+1}\mathfrak{a}_{k+1}$, then
by Lemma 6.9, we have

(59)
$\begin{split}\mathrm{SL}_{g_{k+1}}(\mathfrak{A}_{k+1})=&t^{k+1}(\rho_{k+1}-\Delta_{g}(\mathfrak{f}_{k+1}-\nabla_{v_{k}}\mathfrak{f}_{1}))+E.\end{split}$

By Lemma 6.3, 6.7, 6.8, we could write $E=t^{k+2}E^{\prime}$ and by
Proposition 6.4, $\|rE^{\prime}\|_{\mathcal{C}^{,\gamma}}\lesssim 1$.
Therefore, we obtain

$\|r\mathrm{SL}_{g_{k+1}}(\mathfrak{A}_{k+1})\|_{\mathcal{C}^{,\gamma}}\lesssim
t^{k+2}.$

As $\lim_{t\to 0}\lambda_{k,1}=\mathrm{Id}$, there exists $T_{k+1}^{\prime}$
such that $t<T_{k+1}^{\prime}$, $\varphi_{k+1,t}^{-1}(\Sigma)$ is a embedded
submanifold. We write the leading expansion as

$\mathfrak{A}_{k+1}\sim\Re((tB_{\mathfrak{A}_{k}}+t^{k+1}B_{f_{k+1}^{-}})z^{\frac{3}{2}}),$

where $B_{f_{k+1}^{-}}$ is the $r^{\frac{3}{2}}$ expansion for $f_{k+1}^{-}$.
As $\mathfrak{A}_{k}$ is nondegenerate, $\min|B_{A_{k}^{-}}|>0$. Thus we could
define a positive number

$T_{k+1}:=\min\\{T_{k},\frac{1}{4}\frac{\min|B_{\mathfrak{A}_{k}}|}{\max|B_{f_{k+1}^{-}}|},T_{k+1}^{\prime}\\},$

such that for any $t<T_{k+1}$, $\mathfrak{A}_{k+1}$ will be nondegenerate.
This completes $\mathbf{Step\;3}$.

Now, we will give a proof of Theorem 6.1.

Proof of Theorem 6.1: Given positive integer $N$, we define
$\phi_{t}:=\varphi_{N,t}^{-1}$ and $\mathfrak{A}_{t}:=\mathfrak{A}_{t,N}$.
Recall that $g_{N,t}:=\varphi_{N,t}^{\star}g$, then by Proposition 6.5, we
have

$\phi_{t}^{\star}\mathrm{SL}_{g_{N,t}}(\mathfrak{A}_{t})=\mathrm{SL}_{g}(\phi_{t}^{\star}\mathfrak{A}_{t}).$

In addition, as $\phi_{t}$ is a smooth diffeomorphism with bounded derivative,
we have

$\|r_{t}\mathrm{SL}_{g}(\phi_{t}^{\star}\mathfrak{A}_{t})\|_{\mathcal{C}^{,\gamma}}=\|\phi_{t}^{\star}(r\mathrm{SL}_{g_{N,t}}(\mathfrak{A}_{t}))\|_{\mathcal{C}^{,\gamma}}\lesssim\|r\mathrm{SL}_{\varphi^{\star}g}(\mathfrak{A}_{t})\|_{\mathcal{C}^{,\gamma}},$

where $r_{t}:=\phi_{t}^{\star}r.$ Therefore, we obtain (i):
$\|r_{t}\mathrm{SL}_{g}(\phi_{t}^{\star}\mathfrak{A}_{t})\|_{\mathcal{C}^{,\gamma}}\lesssim
t^{N+1}.$

(ii), (iii) and (iv) follows straight forward by the construction. For (v), as
$t\to 0$, $\lambda_{i}$ convergence smoothly to the identity map, when $t\to
0$, we have $\phi_{t}$ convergence smoothly to the identity map.

We still need to check (v). Near $p\in\Sigma$, let
$(x_{1},x_{2},\cdots,x_{n})$ be the coordinate system on a neighborhood $U$ of
$p\in\Sigma$, which we used in 6.2 with $z=x_{1}+\sqrt{-1}x_{2}$ and
$\Sigma\cap U=\\{z=0\\}$. Over $T^{\star}L|_{U}$, we write the corresponding
fiber coordinate as $(y_{1},\cdots,y_{n})$, then the Riemannian metric could
be written as

$g_{U_{L}}|_{U}=\sum_{i,j=1}^{n}A_{ij}dx_{i}\otimes dx_{j}+B_{ij}dx_{i}\otimes
dy_{j}+C_{ij}dy_{i}\otimes dy_{j}.$

As $[\tilde{L}_{t}]$ is the graph manifold of $\phi_{t}^{\star}\mathfrak{A}$
and WLOG, we could assume $\phi_{t}=\mathrm{Id}$ and write
$\phi_{t}^{\star}\mathfrak{A}=d\mathfrak{F}$,
$\mathfrak{F}_{ij}=\partial_{x_{i}}\partial_{x_{k}}\mathfrak{F}$. Suppose we
write $\tilde{g}^{t}$ be the induced metric on $\tilde{L}_{t}$ and
$\tilde{g}_{ij}^{t}=\tilde{g}^{t}(\partial_{x_{i}},\partial_{x_{j}})$, then

$\tilde{g}_{ij}^{t}=A_{ij}+\sum_{p=1}^{n}B_{ip}\mathfrak{F}_{pj}+\sum_{p,q=1}^{n}\mathfrak{F}_{ip}\mathfrak{F}_{jq}C_{pq}.$

As $\mathfrak{F}_{ij}\sim r^{-\frac{1}{2}}$ when $i,j\in\\{1,2\\}$,
$\mathfrak{F}_{ij}\sim\mathcal{O}(1)$ in other situation, the worst
singularity comes from the third terms, which will be order $r^{-1}$. We are
in particular interested in the behavior of $\lim_{r\to 0}r\tilde{g}_{ij}^{t}$
and it is straight forward to check that suppose one of $i$ or
$j\notin\\{1,2\\}$, we have $\lim_{r\to 0}r\tilde{g}_{ij}^{t}=0$.

Let $\mathfrak{f}$ be the leading term of $\mathfrak{F}$, then we could write
$\mathfrak{f}=\Re(Bz^{\frac{3}{2}})$ with $B$ is no where vanishing along
$\Sigma$. We write
$\mathfrak{f}_{ij}=\partial_{x_{i}}\partial_{x_{j}}\mathfrak{f}$, then we
compute

$\mathfrak{f}_{11}=\Re(Bz^{-\frac{1}{2}}),\;\mathfrak{f}_{22}=-\Re(Bz^{-\frac{1}{2}}),\;\mathfrak{f}_{12}=-\Im(Bz^{-\frac{1}{2}}).$

Moreover, for $i,j\in\\{0,1\\}$, we compute

$\lim_{r\to 0}r\tilde{g}_{ij}^{t}=\lim_{r\to
0}t^{2}\sum_{p,q=1}^{2}C_{pq}r\mathfrak{f}_{pi}\mathfrak{f}_{qj}.$

We define the vector fields
$v_{i}=r^{\frac{1}{2}}(\mathfrak{f}_{1i}\partial_{y_{1}}+\mathfrak{f}_{2i}\partial_{y_{2}}),$
then by the previous computations, for $i\in\\{1,2\\}$, $v_{i}$ are nowhere
vanishing in this neighborhood. In addition, for $i,j\in\\{1,2\\}$, we have

$\lim_{r\to 0}r\tilde{g}_{ij}^{t}=t^{2}g(v_{i},v_{j}),$

which is also nowhere vanishing near $\Sigma$.

Therefore, we could find a positive function $\mu$ such that

$r\det(\tilde{g}_{ij}^{t})dx_{1}\wedge\cdots\wedge dx_{n}=\mu
dx_{1}\wedge\cdots\wedge dx_{n},$

with a lower bound $\mu\geq t^{2}c_{0}$. Let $g_{0}$ be the Riemannian metric
on the zero section, then

$t^{2}d\mathrm{Vol}_{g_{0}}=t^{2}\det(g_{0})dx_{1}\wedge\cdots\wedge
dx_{n}\lesssim r\det(\tilde{g}_{ij}^{t})dx_{1}\wedge\cdots\wedge
dx_{n}=rd\mathrm{Vol}_{\tilde{g}^{t}}.$

In addition, we have
$\tilde{\iota}_{t}^{\star}\Im\Omega=\sin\theta_{t}d\mathrm{Vol}_{\tilde{g}^{t}}$
and
$\tilde{\iota}_{t}^{\star}\Im\Omega=\mathrm{SL}(\phi_{t}^{\star}\mathfrak{A}_{t})d\mathrm{Vol}_{g_{0}}$.
Therefore, we obtain

$\|\theta_{t}\|_{\mathcal{C}^{,\gamma}}\lesssim
t^{-2}\|r_{t}\mathrm{SL}(\phi_{t}^{\star}\mathfrak{A}_{t})\|_{\mathcal{C}^{,\gamma}}\lesssim
t^{N-1}$

For (vi), over a neighborhood $U$ of the zero section $T^{\star}\tilde{L}$, as
currents, $[\tilde{L}_{t}]$ will be the graph of $p^{\star}\mathfrak{A}_{t}$
and $2[L]$ is the zero section in $T^{\star}\tilde{L}$. By (iii),
$p^{\star}\mathfrak{A}_{t}$ have the following asymptotic expansions

$p^{\star}\mathfrak{A}_{t}\sim\tilde{a}_{0}+\tilde{a}_{1}\tilde{r}+\tilde{a}_{2}\tilde{r}^{2}+\sum_{k=1}^{p}\tilde{a}_{3,k}\tilde{r}^{3}(\log\tilde{r})^{k}+\cdots,$

where $\tilde{r}^{2}=p^{\star}r.$ Therefore, for a smooth Riemannian metric
$g^{\prime}$ over $\tilde{L}$, $p^{\star}\mathfrak{A}_{t}$ is a
$\mathcal{C}^{2,\gamma}_{g^{\prime}}$ 1-form with $\lim_{t\to
0}\|p^{\star}\mathfrak{A}_{t}\|_{\mathcal{C}_{g^{\prime}}^{2,\gamma}}=0.$
Therefore, for any differential form $\omega$ on $U$, we have $\lim_{t\to
0}\int_{\tilde{L}}\iota_{t}^{\star}\omega=\int_{\tilde{L}}(\iota_{0}\circ
p)^{\star}\omega$, which implies the current convergence
$\tilde{\iota}_{t}(\tilde{L})\to 2\iota_{0}(L)$. ∎

######  Remark.

As the construction on Theorem 6.1 depends on a positive integer $N$, we could
label the approximate nondegenerate pair as $\mathfrak{A}_{t,N}$ and
$\Sigma_{t,N}$. As $N$ become large, the distance between $\Sigma_{t,N}$ and
$\Sigma_{t,N+1}$ will be size with order $t^{N+1}$. Suppose we obtain a
uniform bound on $v_{k}$ during the construction process, we might prove that
$\lim_{N\to\infty}\Sigma_{t,N}$ exists and this will be the branch locus of
the real solution. However, in terms of proving the existence of solutions, we
don’t need infinite order correction.

###  6.5. The construction over fixed real locus

Suppose our initial special Lagrangian submanifold is locally a fixed real
locus as in Definition 5.6, we could also obtain a symmetry for the
approximate solutions we construction in Theorem 6.1. Let $R_{0}:T^{\star}L\to
T^{\star}L$ be the canonical involution and $(U_{L},J,\omega,\Omega)$ be a
$R_{0}$-invariant Calabi-Yau structure. The approximate solutions will also
obtain a symmetry.

######  Theorem 6.11.

Suppose $[L]$ is locally a fixed real locus and $\mathfrak{a}=\alpha^{-}\in
T^{\star}L\otimes\mathcal{I}$ be a nondegenerate $\mathbb{Z}_{2}$ harmonic
1-form, then in Theorem 6.1, we could choose the construction in Theorem 6.1
such that $\mathfrak{A}_{t}=\mathfrak{A}^{-}_{t}\in
T^{\star}L\otimes\mathcal{I}$. In addition, let
$[\tilde{L}_{t}]=(\tilde{L},\tilde{\iota}_{t})$ be the graph of
$\mathfrak{A}_{t}$, then
$R_{0}\circ\tilde{\iota}_{t}=\tilde{\iota}_{t}\circ\sigma$.

###### Proof.

By Proposition 5.8, we have $Q(\alpha^{-})^{+}=0$. Suppose we start with
$\mathfrak{a}=\alpha^{-}\in T^{\star}L\otimes\mathcal{I}$, then in each step
of the construction in Subsection 6.4, we could always choose the correction
to be $\mathfrak{A}_{i}=\alpha_{i}^{-}\in T^{\star}L\otimes\mathcal{I}.$ The
rest follows straight forward. ∎

If we write $\theta_{t}$ be the Lagrangian angle of $[\tilde{L}_{t}]$ and
$\tilde{g}_{t}:=\tilde{\iota}_{t}^{\star}g$ be the induced metric, then as
$R_{0}\circ\tilde{\iota}_{t}=\tilde{\iota}_{t}\circ\sigma$, we obtain

$\sigma^{\star}\tilde{g}_{t}=\tilde{g}_{t},\;\sigma^{\star}e^{i\theta_{t}}=e^{-i\theta_{t}}.$

In particular, over the branch locus $\Sigma$, we have
$\sin\theta_{t}|_{\Sigma}=0$.

By Proposition 7.3, $[\tilde{L}_{t}]$ is an immersed submanifold and we could
choose a Weinstein neighborhood of $[\tilde{L}_{t}]$ and consider the special
Lagrangian equation $\mathrm{SL}_{\tilde{g}_{t}}(\alpha)$ for the graph of
1-form $\alpha$ on $\tilde{L}$.

######  Corollary 6.12.

$\sigma^{\star}\mathrm{SL}_{\tilde{g}_{t}}(\alpha)=-\mathrm{SL}_{\tilde{g}_{t}}(-\sigma^{\star}\alpha)$.

###### Proof.

We define $\tilde{\iota}_{\alpha}:\tilde{L}\to U_{L}$ as
$\tilde{\iota}_{\alpha}(x):=\exp_{\tilde{\iota}_{t}(x)}(J(\tilde{\iota}_{t})_{\star}V_{\alpha})$,
where $V_{\alpha}$ is the vector field on $\tilde{L}$ dual to $\alpha$ given
by the Riemannian metric $\tilde{g}_{t}$. We will first prove that
$R_{0}\circ\tilde{\iota}_{\alpha}(\tilde{x})=\tilde{\iota}_{-(\sigma^{-1})^{\star}\alpha}\circ\sigma.$
As $R_{0}$ is an isomorphism and
$R_{0}\circ\tilde{\iota}_{t}=\tilde{\iota}_{t}\circ\sigma$, we compute

(60)
$\begin{split}R_{0}\circ\tilde{\iota}_{\alpha}(x)=&R_{0}\exp_{\tilde{\iota}_{t}(x)}(J(\tilde{\iota}_{t})_{\star}V_{\alpha})\\\
=&\exp_{R_{0}\tilde{\iota}_{t}(x)}((R_{0})_{\star}J(\tilde{\iota}_{t})_{\star}V_{\alpha})\\\
=&\exp_{\tilde{\iota}_{t}\sigma(x)}(-J(R_{0})_{\star}(\tilde{\iota}_{t})_{\star}V_{\alpha})\\\
=&\exp_{\tilde{\iota}_{t}\sigma(x)}(-J(\tilde{\iota}_{t})_{\star}\sigma_{\star}V_{\alpha})\\\
\end{split}$

As $\sigma^{\star}g_{t}=g_{t}$, we obtain
$-\sigma_{\star}V_{\alpha}=V_{(-\sigma^{-1})^{\star}\alpha}$. Therefore, we
obtain

(61)
$R_{0}\circ\tilde{\iota}_{\alpha}(x)=\exp_{\tilde{\iota}_{t}\sigma(x)}(J(\tilde{\iota}_{t})_{\star}V_{(-\sigma^{-1})^{\star}\alpha})=\tilde{\iota}_{-(\sigma^{-1})^{\star}\alpha}\circ\sigma.$

As
$\tilde{\iota}_{\alpha}^{\star}\Im\Omega=\mathrm{SL}_{\tilde{g}_{t}}(\alpha)d\mathrm{Vol}_{\tilde{g}_{t}}$,
we compute

$-\mathrm{SL}_{\tilde{g}_{t}}(\alpha)d\mathrm{Vol}_{\tilde{g}_{t}}=-\tilde{\iota}_{\alpha}^{\star}\Im\Omega=\tilde{\iota}_{\alpha}^{\star}R_{0}^{\star}\Im\Omega=\sigma^{\star}\tilde{\iota}_{-(\sigma^{-1})^{\star}\alpha}^{\star}\Im\Omega=\sigma^{\star}\mathrm{SL}_{\tilde{g}_{t}}(-(\sigma^{-1})^{\star}\alpha)d\mathrm{Vol}_{\tilde{g}_{t}},$

which implies the claim. ∎

##  7\. The Geometry of the approximate Special Lagrangian Submanifolds

In this section, we will study the geometry for the family of graphic special
Lagrangian submanifolds constructed in Theorem 6.1. Let
$(U_{L},J,\omega,\Omega)$ be the pull-back Calabi-Yau structure on the
Weinstein neighborhood with Calabi-Yau metric $g$, and
$[\tilde{L}_{t}]=(\tilde{L},\tilde{\iota}_{t})$ be the family of submanifolds
constructed in Theorem 6.1, such that $\tilde{\iota}_{t}$ is given by the
graph of a nondegenerate pair
$\mathfrak{A}_{t}=\mathcal{A}^{+}_{t}+\mathcal{A}^{-}_{t}$. We will study the
induced Riemannian metric, the second fundamental form and the Schauder
estimates on $[\tilde{L}_{t}]$.

###  7.1. The induced metric on the approximate family

In this subsection, we will prove the induced metric over $\tilde{L}$ under
the pullback of the branched covering map is a cone metric will angle $4\pi$.

####  7.1.1. Cone Metric

Let $\mathbb{R}^{n}=\mathbb{C}\times\mathbb{R}^{n-2}$ with coordinates
$z=re^{i\theta},x_{3},\cdots,x_{n}$, consider the model metric

(62) $g_{4\pi}=4r^{2}(dr^{2}+r^{2}d\theta^{2})+\sum_{i=3}^{n}dx_{i}^{2},$

which has cone singularity of angle $4\pi$ along the submanifold $\\{z=0\\}$.
Let $g_{1},g_{2}$ be two metric, we write $g_{1}\leq g_{2}$ if for any $v$, we
have $g_{1}(v,v)\leq g_{2}(v,v)$.

######  Definition 7.1.

Let $\Sigma\subset L$ be a codimension two embedded submanifold, we call $g$
is a metric with cone angle $4\pi$ along $\Sigma$ if for every $p\in\Sigma$,
we could find coordinates $(z,x_{3},\cdots,x_{n})$ centered at $p$ with
$\Sigma=\\{z=0\\}$ and constant $C$ such that $C^{-1}g_{4\pi}\leq g\leq
Cg_{4\pi}$.

For (62), passing to the branched covering, we write $r=\sqrt{\tilde{r}}$,
then the model metric become
$d\tilde{r}^{2}+4\tilde{r}^{2}d\theta^{2}+\sum_{i=3}^{n}dx_{i}^{2},$ which is
quasi-isometric to the Euclidean metric. Therefore, when we define the Hölder
norm or $W^{1,2}$ norms using the cone metric, it will be the same as defining
using the smooth Riemannian metric.

######  Proposition 7.2.

$p^{\star}g_{L}$ is a cone metric over $\tilde{L}$ with cone angle $4\pi$.

###### Proof.

Let $N_{\Sigma}$ be the normal bundle of $\Sigma\subset L$, we identified a
neighborhood $U$ of $\Sigma$ in $L$ with a neighborhood of zero section of
$N_{\Sigma}$ and write $z$ be the fibre coordinates of $N_{\Sigma}$. We could
define another Riemannian metric $g_{0}$ on $U$ as

$g_{0}=|dz|^{2}+g_{\Sigma},$

then $g_{0}$ is also smooth Riemannian metric which satisfies $C^{-1}g_{0}\leq
g_{L}\leq Cg_{0}$.

We could write the branched covering map $p:\tilde{L}\to L$ in this local
coordinate as

$p(\tilde{z},x_{3},\cdots,x_{n})=(\tilde{z}^{2},x_{3},\cdots,x_{n}),$

then

(63)
$p^{\star}g_{0}=4\tilde{r}^{2}(d\tilde{r}^{2}+\tilde{r}^{2}d\tilde{\theta}^{2})+p^{\star}g_{\Sigma},$

which is a cone metric along $\tilde{\Sigma}$ with cone angle $4\pi$. As
$C^{-1}p^{\star}g_{0}\leq p^{\star}g_{L}\leq Cp^{\star}g_{0}$,
$p^{\star}g_{L}$ is also a cone metric with cone angle $4\pi$. ∎

####  7.1.2. Induced metric and convergence

Recall that $\tilde{\iota}_{t}:\tilde{L}\to U_{L}$ is given by the graph of
the nondegenerate pair $\mathfrak{A}=\mathcal{A}_{t}^{+}+\mathcal{A}_{t}^{-}$
and we would like to understand the map. By the toy model in Section 5.1.1, we
need to prove $\tilde{\iota}_{t}$ is well-behaved near $\Sigma$ using the
nondegenerate condition. Moreover, we define

$S_{t}:=\\{x\in\tilde{\iota}_{t}(\tilde{L})|\mathrm{there\;exists\;}x_{1}\neq
x_{2}\in\tilde{L}\;\mathrm{such\;that\;}\tilde{\iota}_{t}(x_{1})=\tilde{\iota}_{t}(x_{2})\\},$

which is the self-intersection points of $\tilde{\iota}_{t}(\tilde{L})$.

######  Proposition 7.3.

For the map $\tilde{\iota}_{t}:\tilde{L}\to U_{L}$, the following holds:

  * (i)

$\tilde{\iota}_{t}$ is a $\mathcal{C}^{2,\gamma}$ immersion and
$\tilde{\iota}_{t}^{\star}g$ is a $\mathcal{C}^{1,\gamma}$ Riemannian metric.

  * (ii)

for each $t$, there exists a neighborhood $\tilde{U}_{\Sigma}$ of
$\Sigma\subset\tilde{L}$ such that
$\tilde{\iota}_{t}|_{\tilde{U}_{\Sigma}}:\tilde{U}_{\Sigma}\to U_{L}$ is an
embedding.

  * (iii)

$S_{t}=\\{x\in L\setminus\Sigma||\mathcal{A}_{t}^{-}|(x)=0\\}$.

###### Proof.

For (i), outside of the branch locus, $\tilde{\iota}_{t}$ will locally given
by graph of different 1-forms which is an immersion. Therefore, we only need
to consider the map near the branch locus. Near the branch locus, we choose
coordinates $(\tilde{z},\tilde{x}_{3},\cdots,\tilde{x}_{n})$ on $\tilde{L}$
and $(z,x_{3},\cdots,x_{n})$ on $L$ such that the branched covering map could
be written as
$p(\tilde{z},\tilde{x}_{3},\cdots,\tilde{x}_{n})=(\tilde{z}^{2},x_{3},\cdots,x_{n})$.

Then under the identification of $\Gamma:T(T^{\star}L)\cong TL\oplus
T^{\star}L$, we have

(64)
$\Gamma\circ\tilde{\iota}_{\star}(\partial_{\tilde{x}_{i}})=(p_{\star}\partial_{\tilde{x}_{i}},\nabla_{p_{\star}\partial_{\tilde{x}_{i}}}\mathcal{A}_{t}^{-}+\nabla_{p_{\star}\partial_{\tilde{x}_{i}}}\mathcal{A}_{t}^{+}).$

As $p_{\star}\partial_{\tilde{z}}=2\tilde{z}\partial_{z}$ and
$\mathcal{A}_{t}^{-}$ is nondegenerate with expansion

$\mathcal{A}_{t}^{-}\sim\Re(Bz^{\frac{1}{2}}dz)+\cdots,$

where $B$ is nowhere vanishing along $\Sigma$. We compute

$\nabla_{p_{\star}\partial_{\tilde{z}}}\mathcal{A}_{t}^{-}|_{\Sigma}=2B\tilde{z}\nabla_{\partial_{z}}(z^{\frac{1}{2}}dz)|_{\Sigma}=\frac{1}{2}Bdz\neq
0.$

In addition, we compute
$\nabla_{p_{\star}\partial_{\tilde{z}}}(\mathcal{A}_{t}^{+})|_{\Sigma}=\tilde{z}\nabla_{\partial_{z}}\mathcal{A}_{t}^{+}|_{\Sigma}=0.$
Therefore, $(\tilde{\iota}_{t})_{\star}$ is injective thus an immersion.

(ii) follows by the definition of the double branched cover.

For (iii), for $x\in L\setminus\Sigma$, we write $x_{1},x_{2}\in
p^{-1}(x)\in\tilde{L}$, then
$\tilde{\iota}_{t}(x_{1})=\tilde{\iota}_{t}(x_{2})$ if and only if
$|\mathcal{A}_{t}^{-}|(x)=0$ and this gives all the self-intersection points.
∎

######  Example 7.4.

Let $\tilde{L}_{t}:=\\{w^{2}=t^{2}z\\}\subset\mathbb{C}^{2}$ be the graph
manifold of $td\Re(z^{\frac{3}{2}})$ over the complex $z$ plane $L=\\{w=0\\}$,
we write $z=re^{i\theta}$, then use the coordinate over $L$, the induced
metric $\tilde{g}_{t}$ over $L_{t}$ using the $z$ plane coordinate is
$\tilde{g}_{t}=(1+\frac{t^{2}}{4r})(dr^{2}+r^{2}d\theta^{2}).$ Let
$\tilde{z}=\tilde{r}e^{i\tilde{\theta}}$ be the coordinate on $\tilde{L}$ with
$z=\tilde{z}^{2}$, then in the $\tilde{z}$ coordinates, the metric would be
$\tilde{g}_{t}=(4\tilde{r}^{2}+t^{2})(d\tilde{r}^{2}+\tilde{r}^{2}d\tilde{\theta}^{2}).$
When $t\to 0$, $\lim_{t\to
0}\tilde{g}_{t}=4\tilde{r}^{2}(d\tilde{r}^{2}+\tilde{r}^{2}d\tilde{\theta}^{2}),$
which is the cone metric with angle $4\pi$.

######  Theorem 7.5.

Let $g^{\prime}$ be any smooth Riemannian metric on $\tilde{L}$ and $g_{L}$ be
the Riemannian metric on the zero section of $U_{L}\subset T^{\star}L$, then
$\lim_{t\to
0}\|\tilde{g}_{t}-p^{\star}g_{L}\|_{\mathcal{C}_{g^{\prime}}^{1,\gamma}}=0$,
where $\mathcal{C}_{g^{\prime}}^{1,\gamma}$ is defined using $g^{\prime}$.

###### Proof.

We prove the above theorem using local charts. For $U_{L}\subset T^{\star}L$,
let $(x_{1},\cdots,x_{n})$ be a local coordinate on $L$ and
$(y_{1},\cdots,y_{n})$ be coordinates on fiber. The Riemannian metric on
$U_{L}$ in these coordinates could be written as

$g_{U_{L}}=g_{L}+B_{ij}dx_{i}\otimes dy_{j}+C_{ij}dy^{i}\otimes dy^{j}.$

Then the pullback metric $\tilde{g}_{t}:=\tilde{\iota}_{t}^{\star}g$ could be
written as

(65)
$\begin{split}\tilde{g}_{t}=p^{\star}g_{L}+t(B_{ij}^{\prime}d\tilde{x}_{i}\otimes
d\Theta_{j})+t^{2}C_{ij}d\Theta_{i}\otimes d\Theta_{j},\end{split}$

where
$\Theta_{i}:=p^{\star}\langle\partial_{x_{i}},\frac{1}{t}\mathfrak{A}_{t}\rangle.$
Therefore, we could write $\tilde{g}_{t}=p^{\star}g_{L}+t\hat{g}_{t}$ with

$\|\hat{g}_{t}\|_{\mathcal{C}^{1,\gamma}}\lesssim\sum_{i,j=1}^{n}(\|B_{ij}^{\prime}\|_{\mathcal{C}_{g^{\prime}}^{1,\gamma}}+\|C_{ij}\|_{\mathcal{C}_{g^{\prime}}^{1,\gamma}})+\|\frac{1}{t}p^{\star}\mathcal{A}_{t}\|_{\mathcal{C}_{g^{\prime}}^{2,\gamma}}\lesssim
1.$

In addition, by Theorem 6.1, $p^{\star}\mathfrak{A}_{t}$ is a
$\mathcal{C}_{g^{\prime}}^{2,\gamma}$ form over $\tilde{L}$. Therefore,
$\lim_{t\to 0}\|\tilde{g}_{t}-p^{\star}g\|_{\mathcal{C}^{1,\gamma}}=0$, which
implies the statement. ∎

###  7.2. First eigenvalue estimate

Let $[\tilde{L}_{t}]=(\tilde{L},\tilde{\iota}_{t})$ be the immersed
submanifold constructed in Theorem 6.1 with $\theta_{t}$ be the Lagrangian
angle and let $\tilde{g}_{t}:=\iota_{t}g$ be the pull-back metric, which by
Proposition 7.3 is $\mathcal{C}^{1,\gamma}$. Let $f$ be a function on
$\tilde{L}$, we write $\mathcal{D}_{t}f:=d^{\star_{t}}(\cos\theta_{t}df)$ be
the linear term of the special Lagrangian equation, where $d^{\star_{t}}$ is
taken for the Riemannian metric $\tilde{g}_{t}$. It is straight forward to
check that $\mathcal{D}_{\theta_{t}}$ is a self-adjoint operator. If we write
$\hat{g}_{t}={(\cos\theta_{t})}^{\frac{2}{n-2}}g_{t}$, then
$\mathcal{D}_{\theta_{t}}f=(\cos\theta_{t})^{\frac{2}{n-2}}\Delta_{\hat{g}_{t}}f$,
where $\Delta_{\tilde{g}_{t}}$ is the Laplace operator for $\hat{g}_{t}$.

To begin with, we will understand the eigenvalue for the cone metric. Let
$g_{0}:=p^{\star}g_{L}$ be the pullback cone metric and $\Delta_{g_{0}}$ be
the corresponding cone Laplace operator. We define

$\lambda_{0}:=\inf_{f}\\{\frac{\int|df|_{g_{0}}^{2}d\mathrm{Vol}_{0}}{\int|f|^{2}d\mathrm{Vol}_{0}}|f\in
W^{1,2},\int fd\mathrm{Vol}_{0}=0\\},$

where $d\mathrm{Vol}_{0}$ is the volume form w.r.t. the metric $g_{0}$. As the
cone metric is quasi-isometric to the smooth metric, the
$\mathcal{C}^{,\gamma}$ norms and $W^{1,2}$ norms using the cone metric will
be same as using any smooth metric.

######  Theorem 7.6.

$\lambda_{0}$ is the first positive eigenvalue for the cone metric Laplace
operator $\Delta_{g_{0}}$.

###### Proof.

We first prove that $\lambda_{0}$ is positive and could be achieved. Suppose
$\lambda_{0}=0$, then there exists a sequence of functions $f_{k}$ such that
$\int|df_{k}|^{2}\leq\frac{1}{k}$ and $\int|f_{k}|^{2}=1$, which is a bounded
sequence in $W^{1,2}$. Thus after passing to a subsequence, there exists
$f_{\infty}$ such that $f_{k}$ convergence to $f_{\infty}$ weakly in $W^{1,2}$
and strongly in $L^{2}$. Therefore, $\int|f_{\infty}|^{2}=1,\int
f_{\infty}d\mathrm{Vol}_{0}=0$. In addition,
$\|df_{\infty}\|_{L^{2}}\leq\liminf_{k\to\infty}\|df_{k}\|_{L^{2}}=0,$ thus
$f_{\infty}$ is a constant, which contradicts to $\int
f_{\infty}d\mathrm{Vol}_{0}=0$. Therefore, $\lambda_{0}>0$ and we still denote
$f_{\infty}$ be the weak limit for a sequence of minimizers.

By Lagrange multiplier theorem, we have

$\int\langle df_{\infty},df^{\prime}\rangle d\mathrm{Vol}_{0}=\lambda_{0}\int
f_{\infty}f^{\prime}d\mathrm{Vol}_{0},$

for any $f^{\prime}\in W^{1,2}$. Therefore, $f_{\infty}$ is a weak solution to
$\Delta_{g_{0}}-\lambda_{0}$. In addition, as the cone metric is quasi-
isomorphism to the Euclidean metric, the coefficients of the Laplace operator
will be $L^{\infty}$, then by the elliptic theory for $L^{\infty}$
coefficients elliptic operator, $f_{\infty}$ is a eigenfunction with
eigenvalue $\lambda_{0}$. ∎

We write $\lambda_{t}$ be the first eigenvalue of $\mathcal{D}_{\theta_{t}}$,
then $\lambda_{t}$ could be realized by the following variation

$\lambda_{t}=\inf_{f}\\{\frac{\int\cos\theta_{t}|df|_{\tilde{g}_{t}}^{2}d\mathrm{Vol}_{t}}{\int|f|_{\tilde{g}_{t}}^{2}d\mathrm{Vol}_{t}}|f\in
W^{1,2},\int fd\mathrm{Vol}_{t}=0\\},$

where $d\mathrm{Vol}_{t}$ is the volume form for the metric $\tilde{g}_{t}$.

######  Theorem 7.7.

Let $\lambda_{t}$ be the first eigenvalue of $\mathcal{D}_{\theta_{t}}$, then
$\lim_{t\to 0}\lambda_{t}=\lambda_{0}>0$. In particular, there exists a
constant $c_{0}$ independent of $t$ such that $\lambda_{t}\geq c_{0}>0$.

###### Proof.

Let $\mathrm{Vol}_{t}$ be the volume form of $\tilde{g}_{t}$ over $\tilde{L}$
and $\mathrm{Vol}_{0}$ be the volume form of $g_{0}$ over $\tilde{L}$. In
local coordinates $(x_{1},\cdots,x_{n})$, we could write

$\cos\theta_{t}|df|_{\tilde{g}_{t}}^{2}d\mathrm{Vol}_{t}=\sum_{i,j=1}^{n}\cos\theta_{t}(\tilde{g}_{t})^{ij}\partial_{x_{i}}f\partial_{x_{j}}f\det(\tilde{g}_{t})dx_{1}\wedge\cdots\wedge
dx_{n}.$

As $\lim_{t\to 0}\tilde{g}_{t}=g_{0}$ and $\lim_{t\to 0}\cos\theta_{t}=1$, we
obtain

(66)
$(1-\epsilon)\int|df|_{g_{0}}^{2}d\mathrm{Vol}_{0}\leq\int\cos\theta_{t}|df|^{2}_{\tilde{g}_{t}}d\mathrm{Vol}_{t}\leq(1+\epsilon)\int|df|^{2}_{g_{0}}d\mathrm{Vol}_{0}.$

Let $f_{t}$ be the eigenfunction
$\mathcal{D}_{\theta_{t}}f_{t}=\lambda_{t}f_{t}$, then we compute

$\begin{split}&\int\cos\theta_{t}|df_{t}|_{\tilde{g}_{t}}^{2}d\mathrm{Vol}_{t}=\lambda_{t}\int|f_{t}|_{\tilde{g}_{t}}^{2}\mathrm{Vol}_{t}\leq\lambda_{t}(1+\epsilon)\int|f_{t}|_{g_{0}}^{2}d\mathrm{Vol}_{0},\\\
&\int\cos\theta_{t}|df_{t}|_{\tilde{g}_{t}}^{2}d\mathrm{Vol}_{t}\geq(1-\epsilon)\int|df_{t}|^{2}_{g_{0}}d\mathrm{Vol}_{0}\geq(1-\epsilon)\lambda_{0}\int|f_{t}|_{g_{0}}^{2}d\mathrm{Vol}_{0}.\end{split}$

Therefore, $\lambda_{t}\geq\frac{1-\epsilon}{1+\epsilon}\lambda_{0}$.

On the other hand, by the definition of $\lambda_{0}$, for every $\epsilon$,
we can find a function $f$ such that

$\int|df|_{g_{0}}^{2}d\mathrm{Vol}_{0}\leq(\lambda_{0}+\epsilon)\int|f|^{2}_{g_{0}}d\mathrm{Vol}_{0}.$

In addition, we have

$\begin{split}\int|df|^{2}_{g_{0}}d\mathrm{Vol}_{g_{0}}\geq(1-\epsilon)\int|df|^{2}_{g_{t}}d\mathrm{Vol}_{t}\geq(1-\epsilon)\lambda_{t}\int|f|^{2}d\mathrm{Vol}_{t}\geq(1-\epsilon)^{2}\lambda_{t}\int|f|^{2}d\mathrm{Vol}_{0}\end{split}$

Therefore, we conclude that
$\lambda_{t}\leq\frac{\lambda_{0}+\epsilon}{(1-\epsilon)^{2}}$, which implies
the claim. ∎

###  7.3. Curvature and injective radius estimate for the graph family

In this subsection, we will compute the second fundamental form and the
injective radius for $[\tilde{L}_{t}]$.

######  Definition 7.8.

Let $(X,g)$ be a Riemannian manifold with Levi-Civita connection $\nabla$, let
$\iota:L\to X$ be an immersion, the second fundamental form $K_{m}$ of
$(L,\iota)$ at $m\in L$ is defined as

$\begin{split}K_{m}&:T_{m}L\times T_{m}L\times N_{m}\to\mathbb{R},\\\
K_{m}&(X,Y,n):=\langle\nabla_{\iota_{\star}X}(\iota_{\star}Y),n\rangle,\end{split}$

where $N_{m}$ is the normal bundle of the immersion at $\iota(m)$ with section
$n$ of $N_{m}$, $X,Y$ are vector fields on $L$.

Let $[\tilde{L}_{t}]$ be the family of immersed Lagrangians constructed in
Theorem 6.1 on $(U_{L},J,\omega,\Omega)$ and $\iota_{0}:L\to U_{L}$ be the
inclusion of the zero section.

######  Proposition 7.9.

For $[\tilde{L}_{t}]$, let $\tilde{m}\in\tilde{L}_{t}$, let
$K^{t}_{\tilde{m}}$ be the second fundamental form of $X$ at $\tilde{m}$, then

  * (i)

For $\tilde{m}\notin\tilde{\Sigma}$, we write $m=p(\tilde{m})\in L$ and
$K^{0}_{m}$ be the second fundamental form of $\iota_{0}$. Let $n_{t}$ be a
sequence of smooth section of $N_{\tilde{\iota}_{t}(\tilde{m})}$ which
convergence continuously to $n_{0}\in N_{\iota_{0}(m)}$ on $T^{\star}U_{L}$,
then for any $X,Y\in T_{\tilde{m}}\tilde{L}$, we have

$\lim_{t\to
0}K^{t}_{\tilde{m}}(X,Y,n_{t}):=K^{0}_{m}(p_{\star}X,p_{\star}Y,n_{0}),$

  * (ii)

in a suitable neighborhood of $\Sigma$, we write
$K^{t}:=\max_{|X|=|Y|=|n|=1,m\in\Sigma}|K_{m}^{t}(X,Y,n)|$, then we have
$|K^{t}|\leq\frac{C}{r+t^{2}}\leq Ct^{-2}.$

###### Proof.

For (i), let$(\tilde{x}_{1},\cdots,\tilde{x}_{n})$ be a coordinate on a
neighborhood of $\tilde{m}$ and $(x_{1},\cdots,x_{n})$ be a coordinate on a
neighborhood of $m$ such that
$p_{\star}\partial_{\tilde{x}_{i}}=\partial_{x_{i}}$. We also write
$(y_{1},\cdots,y_{n})$ be the fiber coordinates. We define
$E_{i}^{t}:=(\tilde{\iota}_{t})_{\star}\partial_{\tilde{x}_{i}}$, then under
the identification $\Gamma:T(T^{\star}L)\to TL\oplus T^{\star}L$, we could
write

$\Gamma(E_{i}^{t})=(p_{\star}\partial_{\tilde{x}_{i}},\nabla_{p_{\star}\partial_{\tilde{x}_{i}}}\mathcal{A}_{t}^{+}+\nabla_{p_{\star}\partial_{\tilde{x}_{i}}}\mathcal{A}_{t}^{-})$

As $m\in L\setminus\Sigma$, $\lim_{t\to
0}\|\nabla_{p_{\star}\partial_{\tilde{x}_{i}}}\mathcal{A}_{t}^{+}\|_{\mathcal{C}^{k,\gamma}}+\|\nabla_{p_{\star}\partial_{\tilde{x}_{i}}}\mathcal{A}_{t}^{-}\|_{\mathcal{C}^{k,\gamma}}=0$,
we have $\lim_{t\to 0}E_{i}^{t}=\partial_{x_{i}}.$ If we write $\bar{\nabla}$
be the Levi-Civita connection of $(U_{L},g_{U_{L}})$, then $\lim_{t\to
0}\bar{\nabla}_{E_{i}^{t}}E_{j}^{t}=\bar{\nabla}_{p_{\star}\partial_{\tilde{x}_{i}}}p_{\star}(\partial_{\tilde{x}_{j}}).$

As $\lim_{t\to 0}n_{t}=n_{0}$, we compute

$\lim_{t\to
0}K^{t}_{\tilde{m}}(\partial_{\tilde{x}_{i}},\partial_{\tilde{x}_{j}},n_{t})=\lim_{t\to
0}\langle\bar{\nabla}_{E_{i}^{t}}E_{j}^{t},n_{t}\rangle=\langle\bar{\nabla}_{p_{\star}\partial_{\tilde{x}_{i}}}p_{\star}(\partial_{\tilde{x}_{j}}),n_{0}\rangle=K^{0}_{m}(p_{\star}\partial_{\tilde{x}_{i}},p_{\star}\partial_{\tilde{x}_{j}},n_{0}),$

which implies (i).

For (ii), near $\Sigma$, under the identification $\Gamma:T(T^{\star}L)\to
TL\oplus T^{\star}L$, we have

(67)
$\Gamma\circ\tilde{\iota}_{\star}(\partial_{\tilde{x}_{i}})=(p_{\star}\partial_{\tilde{x}_{i}},\nabla_{p_{\star}\partial_{\tilde{x}_{i}}}\mathcal{A}_{t}^{+}+\nabla_{p_{\star}\partial_{\tilde{x}_{i}}}\mathcal{A}_{t}^{-}).$

In addition, we could write the expansion of $\mathcal{A}_{t}^{-}\sim
t\Re(B_{t}z^{\frac{1}{2}}dz)$ with $|B_{t}|\geq C_{0}>0$ on $\Sigma$. Then we
compute

$p_{\star}\partial_{\tilde{z}}=2\tilde{z}\partial_{z},\;\nabla_{p_{\star}\partial_{\tilde{z}}}\mathcal{A}^{-}|_{\Sigma}=tB_{t}dz,\;\nabla_{p_{\star}\partial_{\tilde{z}}}(\mathcal{A}^{+})=0.$

We denote $E_{i}^{t}:=(\tilde{\iota}_{t})_{\star}(\partial_{\tilde{x}_{i}})$,
then by previous computation, we obtain

$|E_{i}|^{2}\leq
r+t^{2},\;\mathrm{for\;i=1\;or\;2\;and\;}|E_{i}|^{2}\sim\mathcal{O}(1)\;\mathrm{for\;i}\neq
1,2,$

which implies the claim. ∎

######  Proposition 7.10.

Let $(X,g)$ be a Riemannian manifold, $\iota:Y\to X$ be an immersion and let
$K_{Y}$ be the second fundamental form of $Y$, suppose $|K_{Y}|\leq\tau_{Y}$,
then the injective radius $\mathrm{inj}_{p}$ of $Y$ at any point $p\in Y$ have
$\mathrm{inj}_{p}\geq\frac{\pi}{C+\tau_{Y}}$, where $C$ is a constant depends
on the Riemannian metric on $X$ but independent of $\tau_{Y}.$

###### Proof.

By the Gauss equation, let $\mathrm{Riem}$ be the Riemannian curvature tensor
of $(Y,\iota^{\star}g)$, then $\max|\mathrm{Riem}|\leq\tau^{2}$. As
$(Y,\iota^{\star}g)$ is a complete Riemannian manifold, for any point $p\in
Y$, the injective radius of $Y$ satisfies

$\mathrm{inj}(Y)\geq\min\\{\frac{\pi}{\tau},\frac{1}{2}l(Y)\\},$

where $l(Y)$ is the infimum of lengths of closed geodesics.

Now, we will control $l(Y)$. We take a isometric embedding
$(X,g)\to\mathbb{R}^{N}$, if we write $\nabla^{\mathbb{R}^{N}}$ be the Levi-
Civita connection on $\mathbb{R}^{N}$, then for $\alpha,\beta\in TY$, we could
write

$\nabla_{\alpha}^{\mathbb{R}^{N}}\beta=\nabla^{X}_{\alpha}\beta+K_{X},$

where $\nabla^{X}$ is the Levi-Civita connection on $X$ and $K_{X}$ is a
section of the normal bundle of $X$ in $\mathbb{R}^{N}$. Similarly, we have
$\nabla^{X}_{\alpha}\beta=\nabla^{Y}_{\alpha}\beta+K_{Y}$, where $\nabla^{Y}$
is the Levi-Civita connection on $Y$ and $K_{Y}$ is a section of the normal
bundle of $Y$ in $X$. Therefore, we obtain

(68) $\nabla_{\alpha}^{\mathbb{R}^{N}}\beta=\nabla_{\alpha}\beta+K_{Y}+K_{X},$

where $\nabla^{\mathbb{R}^{N}}$ is the covariant derivative on
$\mathbb{R}^{N}$ and $\nabla_{\alpha}\beta$ is the covariant derivative on
$Y$. Let $\gamma$ be a geodesic on $Y$ and we could consider $\gamma$ is a
curve on $\mathbb{R}^{N}$, then the extrinsic curvature $k_{\gamma}$ of
$\gamma$ is defined as
$\kappa_{\gamma}=|\nabla_{\gamma^{\prime}}^{\mathbb{R}^{n}}\gamma^{\prime}|$.
In addition, we compute

(69)
$\begin{split}|\nabla_{\gamma^{\prime}}^{\mathbb{R}^{n}}\gamma^{\prime}|=&|\nabla_{\gamma^{\prime}}\gamma^{\prime}+K_{Y}(\gamma^{\prime},\gamma^{\prime})+K_{X}(\gamma^{\prime},\gamma^{\prime})|\\\
\leq&|K_{Y}(\gamma^{\prime},\gamma^{\prime})|+|K_{X}(\gamma^{\prime},\gamma^{\prime})|\\\
\leq&\tau_{X}+\tau_{Y},\end{split}$

where $\tau_{X}$ is the maximal of the second fundamental form of $X$.

By Fenchel’s theorem on $\mathbb{R}^{N}$ [2], we have
$\int_{\gamma}\kappa_{\gamma}ds\geq 2\pi$. Let $l(\gamma)$ be the arc length
of $\gamma$, then we have

$2\pi\leq\int_{\gamma}\kappa_{\gamma}ds\leq l(\gamma)(\tau_{X}+\tau_{Y}),$

which implies $l(Y)\geq\frac{2\pi}{\tau_{X}+\tau_{Y}}.$ Therefore,
$\mathrm{inj}(Y)\geq\frac{\pi}{\tau_{X}+\tau_{Y}}.$ ∎

Over $(\tilde{L},g_{t})$, let $\mathrm{inj}_{t}$ be minimal of injective
radius along different points of $\tilde{L}$, then we obtain the following
estimate.

######  Corollary 7.11.

There exist constants $t_{0},C$ independent of $t$ such that for any $t\leq
t_{0}$, we have $\mathrm{inj}_{t}\geq Ct^{2}.$

###### Proof.

By Proposition 7.9, for the second fundamental form $K_{t}$, we have
$\max_{p\in\tilde{L}}|K_{t}|\leq Ct^{-2}$. By Proposition 7.10, we have
$\mathrm{inj}_{t}\geq\frac{t^{2}\pi}{C+\tau_{X}t^{2}}$, where $\tau_{X}$ is
the maximal of the second fundamental form for an isomorphic embedding into
$\mathbb{R}^{n}$. As the ambient manifold $(X,g)$ is fixed, we take
$t_{0}<\frac{1}{\sqrt{\tau_{X}}}$, then $\mathrm{inj}_{t}\geq
C^{\prime}t^{2}$. ∎

###  7.4. Size estimate for the Weinstein neighborhood.

Let $[L]=(L,\iota)$ be an immersed Lagrangian submanifold in a Calabi-Yau
manifold $(X,J,\omega,\Omega)$ with Calabi-Yau metric $g_{X}$. By Theorem 2.9,
we could find a Weinstein neighborhood of $[L]$ and in this subsection, we
will prove a Weinstein neighborhood theorem with size controlled by $\tau$,
where $\tau$ is maximum of the second fundamental form of $[L]$. Moreover, we
assume $\tau\geq 1$.

We define $g_{L}:=\iota^{\star}g_{X}$ and

(70) $U(s):=\\{\alpha\in T^{\star}L|\|\alpha\|_{\mathcal{C}^{0}}\leq s\\},$

where the $\mathcal{C}^{0}$ norm is taken using $g_{L}$.

To begin with, we introduce several lemmas to prove the Weinstein tubular
neighborhood theorem.

######  Lemma 7.12.

[6, Proposition 8.3] Let $V$ be a $2n$-dimensional vector space, let
$\Omega_{0}$ and $\Omega_{1}$ be symplectic forms in $V$. Let $S$ be a
subspace of $V$ and is Lagrangian for $\Omega_{0}$ and $\Omega_{1}$, and let
$W$ be any complement to $S$ in $V$, then from $W$, we can canonically
construct a linear isomorphism $A:V\to V$ such that $A|_{S}=\mathrm{Id}_{S}$
and $A^{\star}\Omega_{1}=\Omega_{0}$.

######  Lemma 7.13.

[6, Proposition 6.8] Let $U$ be a tubular neighborhood of $L$ in $X$ and
$\beta$ is a closed $k$-form. Suppose the homology class of $\beta$ restricts
to $L$ vanishes, then we could find a $(k-1)$-form $\mu$ on $U$ such that
$\beta=d\mu$ and $\mu|_{x}=0$, where $x\in L$.

######  Theorem 7.14.

Under the previous assumptions, for the immersed Lagrangian submanifold
$[L]=(L,\iota:L\to X)$, there exists a neighborhood $V$ of $[L]$, a
neighborhood $U$ of the zero section of $T^{\star}L$, a diffeomorphism
$\varphi:U\to V$ and a constant $C$ independent of $[L]$ such that

  * •

$\varphi^{\star}(\omega)=\omega_{\mathrm{can}}$,

  * •

we could take $U=U(C\tau^{-1})$, to be a size $C\tau^{-1}$ neighborhood of
$T^{\star}L$ defined in (70).

###### Proof.

Let $U(s)$ be an size $s$ neighborhood on $T^{\star}L$ defined in (70), let
$\Phi:U(s)\to X$ be decomposition of the complex structure and the normal
exponential map defined in (4). Let $\mathrm{inj}_{X}$ be the injective radius
of $X$, then for any $s<\mathrm{inj}_{X}$, $\Phi$ will be an immersion.
Therefore, we could obtain a pullback Calabi-Yau structure
$(U(s),\Phi^{\star}J,\Phi^{\star}\omega,\Phi^{\star}\Omega)$ with pull-back
metric $g_{U}=\Phi^{\star}g$.

Let $x\in L$ and we write $(x,0)$ be a point on the zero section of $U(s)$, we
write $S=T_{x}L$ and $W=(T_{p}L)^{\perp}$, where the $\perp$ is taken using
the metric $g$ over $T_{(x,0)}U(s)$. By Lemma 7.12, there exists a linear
isomorphism $A_{x}:T_{(x,0)}U(s)\to T_{(x,0)}U(s)$ such that
$A_{x}|_{S}=\mathrm{Id}$ and
$A_{x}^{\star}\omega_{t}|_{(x,0)}=\omega_{\mathrm{can}}$. In addition, as the
choice of $A_{x}$ is canonical, $A_{x}$ varies smoothly with respect to $x$.
Using $A_{x}$, we define a diffeomorphism $h:U(s)\to U(s)$ as

$h(p):=\exp_{x}(A_{x}v),\;\mathrm{where}\;p=(x,v)\in U(s)\subset T^{\star}L.$

Then we have $h|_{(x,0)}=(x,0)$ and $dh|_{(x,0)}=A_{x}$ for $p\in U$.
Therefore, for $x\in L$, we have
$h^{\star}\omega|_{(x,0)}=\omega_{\mathrm{can}}|_{(x,0)}$.

Over the neighborhood of $L$, we write
$\omega_{s}=sh^{\star}\omega+(1-s)\omega_{\mathrm{can}}$ and we wish to find
the size of the tubular neighborhood when $\omega_{s}$ is symplectic. As
$h^{\star}\omega|_{(x,0)}=\omega_{\mathrm{can}}|_{(x,0)}$ for $x\in L$,
$\omega_{s}$ is also a symplectic form in a small neighborhood of $L$.

Now, we will try to understand the size the neighborhood that $\omega_{s}$ is
symplectic. As $h^{\star}\omega$ is the Kähler form for the metric
$h^{\star}g_{U}$, we have $\|h^{\star}\omega\|_{\mathcal{C}^{0}}\lesssim 1$,
where $\mathcal{C}^{0}$ is taken using $h^{\star}g_{U}$. As
$h^{\star}\omega|_{x}=\omega_{\mathrm{can}}|_{x}$ for $x\in L$, we also have
$\|\omega_{\mathrm{can}}\|_{\mathcal{C}^{0}}\lesssim 1$, thus
$\|\omega_{s}\|_{\mathcal{C}^{0}}\lesssim 1$.

Let $R=\tau^{-1}$, we cover $L$ by radius $R$ ball and over each ball $B_{R}$,
we rescale the size of ball by $R^{-1}$, then the Riemannian metric is in
Euclidean size. After rescaling back, we obtain the following estimate
$\|\det(\omega_{s})\|_{\mathcal{C}^{,\gamma}}\lesssim\tau^{\gamma}.$

Let $C_{0}=\min_{p\in L}\|\det(\omega_{s})\|_{\mathcal{C}^{0}}$, over a
neighborhood $U_{L}(C\tau^{-1})\subset T^{\star}L$, for $p=(x,v)\in
U_{L}(C\tau^{-1})$, we compute

(71)
$\begin{split}|\det(\omega_{s})|(p)&\geq|\det(\omega_{s})||(x)-\|\det(\omega_{s})\|_{\mathcal{C}^{,\gamma}}\|v\|_{\mathcal{C}^{0}}\\\
&\geq C_{0}-C\tau^{-\gamma}\|\det(\omega_{s})\|_{\mathcal{C}^{,\gamma}}\\\
&\geq C_{0}-CC^{\prime},\end{split}$

where the last inequality, we use
$\|\det(\omega_{s})\|_{\mathcal{C}^{,\gamma}}\leq C^{\prime}\tau^{\gamma}$.
Therefore, for $C\leq\frac{C_{0}}{2C^{\prime}}$, over $U_{L}(C\tau^{-1})$, we
have $|\det(\omega_{s})|>0$ which implies $\det(\omega_{s})$ is symplectic.

As the 2-form $h^{\star}\omega-\omega_{\mathrm{can}}$ is closed and
$(h^{\star}\omega-\omega_{\mathrm{can}})_{p}=0$ for all $p\in L$, then by
Lemma 7.13, there exists a 1-form $\beta$ on $U_{L}(C_{1}\tau^{-1})$ such that
$h^{\star}\omega-\omega_{\mathrm{can}}=d\beta$ and $\beta|_{(x,0)}=0$ for
$x\in L$. We solve the Moser equation for vector field $v_{s}$:
$\iota_{v_{s}}\omega_{s}=-\beta$ with $v_{s}=0$ on $L$. Let
$\rho_{s}:U_{L}(C\tau^{-1})\to U_{L}(C\tau^{-1})$ be family of diffeomorphisms
generates by $v_{s}$ then $\rho_{s}^{\star}\omega_{s}=\omega_{\mathrm{can}}$
over $U_{L}(C\tau^{-1})$. Let $\varphi:=\Phi\circ h\circ\rho_{1}$, then
$\varphi$ satisfies the claim. ∎

Let $\tau_{t}$ be the maximal of the second fundamental form for the family of
approximate Lagrangian submanifolds $[\tilde{L}_{t}]$ constructed in Theorem
6.1 and we assume $\tau_{t}\geq 1$. By Proposition 7.9, we have $\tau_{t}\leq
Ct^{-2}$, then we obtain the following corollary.

######  Corollary 7.15.

There exists constant $C$ independent of $t$ such that for each $t$, we could
choose

$U_{\tilde{L}}(t):=\\{\alpha\in
T^{\star}\tilde{L}|\|\alpha\|_{\mathcal{C}^{0}}\leq Ct^{-2}\\}$

to be a Weinstein neighborhood of $[\tilde{L}_{t}]$ and $\mathcal{C}^{0}$ is
taken by the induced metric $\tilde{g}_{t}$ on $[\tilde{L}_{t}]$.

Note that by Theorem 7.5, the induced metric $\tilde{g}_{t}$ on
$[\tilde{L}_{t}]$ will convergence to a cone metric, thus the
$\mathcal{C}^{0}$ norms define by different metric will be equivalent.

###  7.5. Schauder estimate and quadratic term estimate on submanifolds

In this subsection, we will study the Schauder estimate and quadratic term
estimate for the immersed Lagrangian submanifolds. We will first state the
Schauder estimate for a general immersed Lagrangian in a Calabi-Yau and later
we will consider the family that constructed in Theorem 6.1.

####  7.5.1. Schauder estimate for submanifold

Let $(X,J,\omega,\Omega)$ be a Calabi-Yau manifold with Calabi-Yau metric $g$,
and let $[L]=(L,\iota:L\to X)$ be a Lagrangian submanifold, where $\iota$ is a
$\mathcal{C}^{2,\gamma}$ immersion with $\iota^{\star}g$ a
$\mathcal{C}^{1,\gamma}$ metric. Let $\tau$ be the maximal of the second
fundamental form of $[L]$ and we assume that $\tau\geq 1$. Let $\theta$ be the
Lagrangian angle for $[L]$ and $\mathcal{D}_{\theta}f:=-d^{\star}(\cos\theta
df)$, where $\star$ is taken using $\iota^{\star}g$. In addition, as
$\iota^{\star}g$ is only a $\mathcal{C}^{1,\gamma}$ metric,
$\mathcal{D}_{\theta}$ is an elliptic operator with $\mathcal{C}^{,\gamma}$
coefficients. We choose a smooth metric $\hat{g}$ such that the second
fundamental form of $\hat{g}$ and $\iota^{\star}g$ are bounded by each other.
The different choices of $\hat{g}$ will not influence the following estimates
as we only would like to obtain a bound in terms of the second fundamental
form.

######  Proposition 7.16.

For any function $f$ over the Riemannian manifold $(L,\iota^{\star}g)$, we
have

(72) $\tau\|df\|_{\mathcal{C}^{,\gamma}}+\|\nabla
df\|_{\mathcal{C}^{,\gamma}}\lesssim\|\mathcal{D}_{\theta}f\|_{\mathcal{C}^{,\gamma}}+\tau^{(2+\alpha+\frac{n}{2})}\|f\|_{L^{2}},$

where $\nabla,\mathcal{D}_{\theta}$ are taking for the metric
$\iota^{\star}g$.

Let $U\subset T^{\star}L$ be a Weinstein neighborhood of $[L]$ and we still
write $\Omega$ to denote the holomorphic volume form on $U$. Let
$\hat{\nabla}$ be the Levi-Civita connection induced by the Sasaki metric of
$\hat{g}$ to $U$, we have

(73)
$\|\hat{\nabla}^{k}\Omega\|_{\mathcal{C}^{,\gamma}}\lesssim\tau^{(k+\gamma)}.$

###### Proof.

Let $B_{R}$ be a radius $R$ ball centered at point $p$, then set $R=\tau^{-1}$
and scale by $\tau$, we obtain a radius $1$ ball with Euclidean geometry.
Applying the classical Schauder estimate for elliptic operator with
$\mathcal{C}^{,\gamma}$ coefficients and rescale back, we obtain the following
with $R=\tau^{-1}$:

$R^{1+\gamma}\|\nabla
f\|_{\mathcal{C}^{,\gamma}(B_{\frac{R}{2}})}+R^{2+\gamma}\|\nabla^{2}f\|_{\mathcal{C}^{,\gamma}(B_{\frac{R}{2}})}\lesssim
R^{2+\gamma}\|\mathcal{D}_{\theta}f\|_{\mathcal{C}^{,\gamma}(B_{R})}+R^{\frac{n}{2}}\|f\|_{L^{2}(B_{R})},$

which implies the desire estimate.

Moreover, let $\pi:T^{\star}L\to L$ be the projection map, we define
$D_{R}:={\pi}^{-1}(B_{R})\cap U$. Then over $D_{R}$, by a rescale argument, we
have
$R^{k+\gamma}\|\hat{\nabla}^{k}\Omega\|_{\mathcal{C}^{,\gamma}(B_{R})}\lesssim
1$ for any integer $k\geq 0$.

We cover $L$ by size $\tau^{-1}$ balls, which implies the claim. ∎

######  Corollary 7.17.

Under the previous assumption, let $f$ be a function on $L$ with $\int
fd\mathrm{Vol}=0$, $\lambda$ be the first eigenvalue of $\mathcal{D}_{\theta}$
and $\mathrm{Vol}(L)$ be the volume of $L$, then we have

(74) $\tau\|\nabla
f\|_{\mathcal{C}^{,\gamma}}+\|\nabla^{2}f\|_{\mathcal{C}^{,\gamma}}\leq
C(1+\lambda^{-1}\mathrm{Vol}(L)\tau^{2+\gamma+\frac{n}{2}})\|\mathcal{D}_{\theta}f\|_{\mathcal{C}^{,\gamma}}.$

###### Proof.

We compute
$\|f\|_{L^{2}(L)}\leq\lambda^{-1}\|\mathcal{D}_{\theta}f\|_{L^{2}(L)}\leq\lambda^{-1}\mathrm{Vol}(L)\|\mathcal{D}_{\theta}f\|_{\mathcal{C}^{,\gamma}}$.
The corollary follows by the previous computation and Proposition 7.16. ∎

####  7.5.2. Quadratic term estimate

Let $\alpha$ be a 1-form on $L$, for the immersed submanifold $[L]$ discussed
above, we will estimate the quadratic terms $Q(\alpha)$ of the special
Lagrangian equation in a Weinstein tubular neighborhood of $[L]$, which
follows from Joyce [26].

######  Proposition 7.18.

[26, Proposition 5.8] Let $\alpha_{1},\alpha_{2}$ be 1-forms on $L$ with
$\|\alpha_{1}\|_{\mathcal{C}^{0}}+\|\alpha_{1}\|_{\mathcal{C}^{0}}\leq
C\tau^{-1}$ and
$\|\nabla\alpha_{1}\|_{\mathcal{C}^{0}}+\|\nabla\alpha_{2}\|_{\mathcal{C}^{0}}\leq
C$, we have

(75)
$\begin{split}\|Q(\alpha_{1})-Q(\alpha_{2})\|_{\mathcal{C}^{,\gamma}}\lesssim&\tau^{\gamma}(\tau\|\alpha_{1}-\alpha_{2}\|_{\mathcal{C}^{,\gamma}}+\|\nabla\alpha_{1}-\nabla\alpha_{2}\|_{\mathcal{C}^{,\gamma}})\cdot\\\
&(\tau\|\alpha_{1}\|_{\mathcal{C}^{,\gamma}}+\tau\|\alpha_{2}\|_{\mathcal{C}^{,\gamma}}+\|\nabla\alpha_{1}\|_{\mathcal{C}^{,\gamma}}+\|\nabla\alpha_{2}\|_{\mathcal{C}^{,\gamma}}).\end{split}$

###### Proof.

Let $\alpha_{1},\alpha_{2}$ be two 1-forms, for each $x\in L$, we write
$P(r,s)=Q(r(\alpha_{1}-\alpha_{2})+s\alpha_{2})$.

Then we compute

$Q(\alpha_{1})-Q(\alpha_{2})=P(1,1)-P(0,1)=\int_{0}^{1}\frac{\partial
P}{\partial r}(r,1)dr.$

As $dQ(0)=0$, we obtain $\frac{\partial P}{\partial r}(0,0)=0$. Therefore, we
obtain

$\frac{\partial P}{\partial r}(u,1)=\int_{0}^{1}(\frac{d}{ds}(\frac{\partial
P}{\partial r}(us,s)))ds=\int_{0}^{1}u\frac{\partial^{2}P}{\partial
r^{2}}(us,s)+\frac{\partial^{2}P}{\partial r\partial s}(us,s)ds.$

By a change of variable, we obtain

(76)
$\begin{split}Q(\alpha_{1})-Q(\alpha_{2})=\int_{0}^{1}\int_{0}^{s}[\frac{r}{s^{2}}\frac{\partial^{2}P}{\partial
r^{2}}(r,s)+\frac{1}{s}\frac{\partial^{2}P}{\partial r\partial
s}(r,s)]drds.\end{split}$

By (LABEL:eq_speiclalLagrangisntitlde), we have
$\tilde{\mathrm{SL}}:\\{(x,y,z):x\in L,\;y\in
T_{x}^{\star}L,\;z\in\otimes^{2}T_{x}^{\star}L\\}\to\mathbb{R}$ such that
$\mathrm{SL}(\alpha)_{x}=\tilde{\mathrm{SL}}(x,\alpha,\nabla\alpha)$. Let
$\partial_{1}\tilde{\mathrm{SL}}$ be taking the derivative for $y$ variable
and $\partial_{2}\tilde{\mathrm{SL}}$ be taking the derivative for $z$
variable, then

$\begin{split}\frac{\partial^{2}P}{\partial
r^{2}}(r,s)=&\otimes^{2}(\alpha_{1}-\alpha_{2})\cdot\partial_{1}^{2}\tilde{\mathrm{SL}}+2(\alpha_{1}-\alpha_{2})\otimes(\nabla\alpha_{1}-\nabla\alpha_{2})\cdot\partial_{1}\partial_{2}\tilde{\mathrm{SL}}\\\
&+\otimes^{2}(\nabla\alpha_{1}-\nabla\alpha_{2})\cdot\partial_{2}^{2}\tilde{\mathrm{SL}},\;\mathrm{and}\\\
\frac{\partial^{2}P}{\partial r\partial
s}(r,s)=&(\alpha_{1}-\alpha_{2})\otimes\alpha_{2}\cdot\partial_{1}^{2}\tilde{\mathrm{SL}}+((\alpha_{1}-\alpha_{2})\otimes\nabla\alpha_{2}+\alpha_{1}\otimes(\nabla\alpha_{1}-\nabla\alpha_{2}))\cdot\partial_{1}\partial_{2}\tilde{\mathrm{SL}}\\\
&+(\nabla\alpha_{1}-\nabla\alpha_{2})\otimes\nabla\alpha_{2}\cdot\partial_{2}^{2}\tilde{\mathrm{SL}},\end{split}$

where we are evaluating $\partial_{i}\partial_{j}\tilde{\mathrm{SL}}$ at
$(x,r(\alpha_{1}-\alpha_{2})+s\alpha_{2},r(\nabla\alpha_{1}-\nabla\alpha_{2})+s\nabla\alpha_{2})$.

We define

(77)
$\begin{split}E_{1}:=&rs^{-2}\|\alpha_{1}-\alpha_{2}\|_{\mathcal{C}^{,\gamma}}^{2}+s^{-1}\|\alpha_{1}-\alpha_{2}\|_{\mathcal{C}^{,\gamma}}\|\alpha_{2}\|_{\mathcal{C}^{,\gamma}},\\\
E_{2}:=&rs^{-2}\|\nabla\alpha_{1}-\nabla\alpha_{2}\|^{2}_{\mathcal{C}^{,\gamma}}+s^{-1}\|\nabla(\alpha_{1}-\alpha_{2})\|_{\mathcal{C}^{,\gamma}}\|\nabla\alpha_{2}\|_{\mathcal{C}^{,\gamma}},\\\
E_{3}:=&2rs^{-2}\|\alpha_{1}-\alpha_{2}\|_{\mathcal{C}^{,\gamma}}\|\nabla(\alpha_{1}-\alpha_{2})\|_{\mathcal{C}^{,\gamma}}+s^{-1}\|\alpha_{1}-\alpha_{2}\|_{\mathcal{C}^{,\gamma}}\|\nabla\alpha_{2}\|_{\mathcal{C}^{,\gamma}}\\\
&+s^{-1}\|\alpha_{2}\|_{\mathcal{C}^{,\gamma}}\|\nabla(\alpha_{1}-\alpha_{2})\|_{\mathcal{C}^{,\gamma}}.\end{split}$

Then (76) implies

(78) $\begin{split}&\|Q(\alpha_{1})-Q(\alpha_{2})\|_{\mathcal{C}^{,\gamma}}\\\
\leq&\int_{0}^{1}\int_{0}^{s}(E_{1}\|\partial_{1}^{2}\tilde{\mathrm{SL}}\|_{\mathcal{C}^{,\gamma}}^{2}+E_{2}\|\partial_{2}^{2}\tilde{\mathrm{SL}}\|_{\mathcal{C}^{,\gamma}}+E_{3}\|\partial_{1}\partial_{2}\tilde{\mathrm{SL}}\|_{\mathcal{C}^{,\gamma}})drds.\end{split}$

As
$\|\hat{\nabla}^{k}\Omega\|_{\mathcal{C}^{,\gamma}}\lesssim\tau^{k+\gamma}$,
we have

$\|\partial_{1}^{2}\tilde{\mathrm{SL}}\|_{\mathcal{C}^{,\gamma}}\lesssim\tau^{2+\gamma},\;\|\partial_{1}\partial_{2}\tilde{\mathrm{SL}}\|_{\mathcal{C}^{1,\gamma}}\lesssim\tau^{1+\gamma},\;\|\partial_{2}^{2}\tilde{\mathrm{SL}}\|_{\mathcal{C}^{,\gamma}}\lesssim\tau^{\gamma},$

which implies (75). ∎

####  7.5.3. Corresponding estimates for the family.

Let $[\tilde{L}_{t}]=(\tilde{L},\tilde{\iota}_{t})$ be the family of special
Lagrangian submanifolds we constructed in Theorem 6.1 with
$\tilde{g}_{t}:=\tilde{\iota}_{t}g$ be the induced $\mathcal{C}^{1,\gamma}$
metric and $\tau_{t}\geq 1$ be the maximal of the second fundamental form.

By Theorem 7.5, $\tilde{g}_{t}$ convergence to a cone metric with bounded
coefficients. If we take $g^{\prime}$ be a smooth background metric on
$\tilde{L}$, then there exists a $t$-independent constant $C$ such that
$\|\tilde{g}_{t}\|_{\mathcal{C}^{0}_{g^{\prime}}}\leq C$, where the
$\mathcal{C}^{0}$ norm is taken using $g^{\prime}$. For each $t$, we choose
$\hat{g}_{t}$ be a smooth Riemannian metric close to $\tilde{g}_{t}$ on
$\tilde{L}$ such that
$\|\tilde{g}_{t}-\hat{g}_{t}\|_{\mathcal{C}_{g^{\prime}}^{1,\gamma}}\leq\tau_{t}^{-1}$.
If we write $\hat{\tau}$ be the maximal of the second fundamental form of
$\hat{g}_{t}$, then we have

$|\hat{\tau}_{t}-\tau_{t}|\leq\|\tilde{g}_{t}\|_{\mathcal{C}^{0}_{g^{\prime}}}\|\hat{g}_{t}-\tilde{g}_{t}\|_{\mathcal{C}_{g^{\prime}}^{1}}+\|\hat{g}_{t}\|_{\mathcal{C}^{0}_{g^{\prime}}}\|\hat{g}_{t}-\tilde{g}_{t}\|_{\mathcal{C}_{g^{\prime}}^{1}}\leq
C\tau_{t}^{-1},$

where $C$ is a $t$-independent constant. Therefore, we obtain
$\hat{\tau}_{t}\leq\tau_{t}+C\tau_{t}^{-1}$. Our following estimates will be
independent of the choice $\hat{g}_{t}$ and $t$ as we will estimate using
$\hat{\tau}_{t}$, which is bounded by $C\tau_{t}$ for some constant $C$.

We write $\hat{\nabla}_{t}$, $\nabla_{t}$ be the Levi-Civita connections of
the Sasaki metric on $T^{\star}\tilde{L}$ defined by $\hat{g}_{t}$,
$\tilde{g}_{t}$. Let
$\mathcal{D}_{\theta_{t}}(f):=d^{\star}(\cos\theta_{t}df)$ and
$\mathrm{Vol}_{t}$ be the volume of $(\tilde{L},\tilde{g}_{t})$.

######  Proposition 7.19.

Let $f$ be a function on $\tilde{L}$ with
$\int_{\tilde{L}}fd\mathrm{Vol}_{t}=0$, we have

(79)
$t^{-2}\|df\|_{\mathcal{C}^{,\gamma}}+\|\nabla_{t}df\|_{\mathcal{C}^{,\gamma}}\lesssim(1+t^{-(4+2\gamma+n)})\|\mathcal{D}_{\theta_{t}}f\|_{\mathcal{C}^{,\gamma}}.$

For the holomorphic volume form, we have

$\|\hat{\nabla}_{t}^{k}\Omega\|_{\mathcal{C}^{,\gamma}}\lesssim
t^{-2(k+\gamma)}.$

###### Proof.

We write $\lambda_{t}$ be the 1-st eigenvalue on $\tilde{L}_{t}$, then by
Theorem 7.7, there exists $c_{0}>0$ such that $\lambda_{t}\geq c_{0}>0$. In
addition, by Theorem 7.5, $\mathrm{Vol}_{t}\leq 2\mathrm{Vol}(L)+1$ which
implies the claim. ∎

By Proposition 3.7, the special Lagrangian equation on $[\tilde{L}_{t}]$ could
be written as

$\mathrm{SL}_{t}(\alpha)=\sin\theta_{t}-d^{\star_{t}}(\cos\theta_{t}\alpha)+Q_{t}(\alpha).$

We have the following estimate for the quadratic terms $Q_{t}(\alpha).$

######  Corollary 7.20.

For 1-forms $\alpha_{1},\alpha_{2}$ on $[\tilde{L}_{t}]$ with
$\|\alpha_{1}\|_{\mathcal{C}^{0}}+\|\alpha_{1}\|_{\mathcal{C}^{0}}\leq Ct^{2}$
and
$\|\nabla_{t}\alpha_{1}\|_{\mathcal{C}^{0}}+\|\nabla_{t}\alpha_{2}\|_{\mathcal{C}^{0}}\leq
C$, we have

(80)
$\begin{split}\|Q_{t}(\alpha_{1})-Q_{t}(\alpha_{2})\|_{\mathcal{C}^{,\gamma}}\lesssim&t^{-2\gamma}(t^{-2}\|\alpha_{1}-\alpha_{2}\|_{\mathcal{C}^{,\gamma}}+\|\nabla_{t}\alpha_{1}-\nabla_{t}\alpha_{2}\|_{\mathcal{C}^{,\gamma}})\cdot\\\
&(t^{-2}\|\alpha_{1}\|_{\mathcal{C}^{,\gamma}}+t^{-2}\|\alpha_{2}\|_{\mathcal{C}^{,\gamma}}+\|\nabla_{t}\alpha_{1}\|_{\mathcal{C}^{,\gamma}}+\|\nabla_{t}\alpha_{2}\|_{\mathcal{C}^{,\gamma}}).\end{split}$

##  8\. Branched Deformation Family Construction and nearby Special Lagrangian
Theorem

In this section, we will prove our main theorem and finish our construction of
branched deformation family. We will first prove a nearby special Lagrangian
theorem in our setting, which is first obtained by Joyce [25], then we will
apply the nearby special Lagrangian theorem to the family we constructed in
Theorem 6.1 to obtain the branched deformations.

###  8.1. Nearby special Lagrangian theorem

Let $(X,J,\omega,\Omega)$ be a Calabi-Yau manifold, $[L]=(L,\iota:L\to X)$ be
an immersed Lagrangian submanfiold with $\mathcal{C}^{2,\gamma}$ structure and
$\theta$ be the Lagrangian angle. By Theorem 7.14, we could choose the
Weinstein neighborhood $U_{L}$ be

$U_{L}=\\{\alpha\in T^{\star}L|\|\alpha\|_{\mathcal{C}^{0}}\leq
C\tau^{-1}\\},$

where $C$ is a $\tau$ independent constant.

Let $f$ be a function on $L$, such that $df\in U_{L}$, then by Proposition
3.7, we could write the special Lagrangian equation for $df$ as

(81) $\sin\theta-\mathcal{D}_{\theta}f+Q(df)=0,$

where $\mathcal{D}_{\theta}f:=d^{\star}(\cos\theta df).$

We start with a lemma follows from Aubin [3, Thorem 4.7].

######  Lemma 8.1.

For each $h\in\mathcal{C}^{,\gamma}(L)$ with $\int_{L}hd\mathrm{Vol}=0$, there
exists $f\in\mathcal{C}^{2,\gamma}$ such that $\int_{L}fd\mathrm{Vol}=0$ and
$\mathcal{D}_{\theta}f=-h$.

We take $f:=\mathcal{D}_{\theta}^{-1}h$, then we are looking for suitable $h$
with $\int_{L}hd\mathrm{Vol}=0$ to solve

$h+Q(d\mathcal{D}_{\theta}^{-1}h)=-\sin\theta.$

We define the operator

$T:\mathcal{C}^{,\gamma}\to\mathcal{C}^{,\gamma},\;Th:=-Q(d\mathcal{D}_{\theta}^{-1}h)-\sin\theta,$

where $\mathcal{C}^{,\gamma}$ is the Hölder norm for functions on $L$, then we
have:

######  Proposition 8.2.

Let $\lambda$ be the first eigenvalue of $\mathcal{D}_{\theta}$, then there
exists a constant $C$ such that for
$N:=C\tau^{\gamma}(1+\lambda^{-1}\mathrm{Vol}(L)\tau^{2+\gamma+\frac{n}{2}})^{2}$
and
$B_{N^{-1}}:=\\{f\in\mathcal{C}^{,\gamma}|\|f\|_{\mathcal{C}^{,\gamma}}\leq
N^{-1}\\}$, suppose $\|\sin\theta\|_{\mathcal{C}^{,\gamma}}\leq\frac{1}{4N}$,
then $T:B_{N^{-1}}\to B_{N^{-1}}$ is a contraction mapping.

###### Proof.

For $h_{1},h_{2}\in B_{N}$, we compute

$\begin{split}\|Th_{1}-Th_{2}\|_{\mathcal{C}^{,\gamma}}=&\|Q_{g}(d\mathcal{D}_{\theta}^{-1}h_{1})-Q_{g}(d\mathcal{D}_{\theta}^{-1}h_{2})\|_{\mathcal{C}^{,\gamma}}\\\
\lesssim&{\tau}^{\gamma}(\tau\|d{\mathcal{D}_{\theta}}^{-1}(h_{1}-h_{2})\|_{\mathcal{C}^{,\gamma}}+\|\nabla
d{\mathcal{D}_{\theta}}^{-1}(h_{1}-h_{2})\|_{\mathcal{C}^{,\gamma}})\cdot\\\
&(\tau\|d\mathcal{D}^{-1}_{\theta}h_{1}\|_{\mathcal{C}^{,\gamma}}+\tau\|d\mathcal{D}^{-1}_{\theta}h_{2}\|_{\mathcal{C}^{,\gamma}}+\|\nabla
d{\mathcal{D}}^{-1}_{\theta}h_{1}\|_{\mathcal{C}^{,\gamma}}+\|\nabla
d{\mathcal{D}}^{-1}_{\theta}h_{2}\|_{\mathcal{C}^{,\gamma}}),\\\ \end{split}$

where for the second inequality we use Proposition 7.18. By Corollary 7.17, we
have

$\begin{split}&\tau\|d\mathcal{D}^{-1}_{\theta}h_{i}\|_{\mathcal{C}^{,\gamma}}+\|\nabla
d\mathcal{D}^{-1}_{\theta}h_{i}\|_{\mathcal{C}^{,\gamma}}\lesssim(1+\lambda^{-1}\mathrm{Vol}(L)\tau^{2+\gamma+\frac{n}{2}})\|h_{i}\|_{\mathcal{C}^{,\gamma}},\;\mathrm{and}\\\
&\tau\|d\mathcal{D}^{-1}_{\theta}(h_{1}-h_{2})\|_{\mathcal{C}^{,\gamma}}+\|\nabla
d\mathcal{D}^{-1}_{\theta}(h_{1}-h_{2})\|_{\mathcal{C}^{,\gamma}}\lesssim(1+\lambda^{-1}\mathrm{Vol}(L)\tau^{2+\gamma+\frac{n}{2}})\|h_{1}-h_{2}\|_{\mathcal{C}^{,\gamma}}.\end{split}$

Therefore, there exists a constant $C_{0}$ such that

$\begin{split}\|Th_{1}-Th_{2}\|_{\mathcal{C}^{,\gamma}}\leq&C_{0}\tau^{\gamma}(1+\lambda^{-1}\mathrm{Vol}(L)\tau^{2+\gamma+\frac{n}{2}})^{2}\|h_{1}-h_{2}\|_{\mathcal{C}^{,\gamma}}(\|h_{1}\|_{\mathcal{C}^{,\gamma}}+\|h_{2}\|_{\mathcal{C}^{,\gamma}})\\\
\leq&\frac{C_{0}}{C}\|h_{1}-h_{2}\|_{\mathcal{C}^{,\gamma}}.\end{split}$

Thus, for any $C\geq 4C_{0}$, we have
$\|Th_{1}-Th_{2}\|_{\mathcal{C}^{,\gamma}}\leq\frac{1}{4}\|h_{1}-h_{2}\|_{\mathcal{C}^{,\gamma}}$.
Moreover, we still need to check $T:B_{N}\to B_{N}$ is a well-defined map. By
a similar computation, we obtain

$\begin{split}\|Th\|_{\mathcal{C}^{,\gamma}}&\leq\|Q(d\mathcal{D}_{\theta}^{-1}h)\|_{\mathcal{C}^{,\gamma}}+\|\sin\theta\|_{\mathcal{C}^{,\gamma}}\\\
&\leq
C_{0}^{\prime}(\tau^{\gamma}(1+\lambda^{-1}\mathrm{Vol}(L)\tau^{2+\gamma+\frac{n}{2}})^{2}\|h\|^{2}_{\mathcal{C}^{,\gamma}}+\|\sin\theta\|_{\mathcal{C}^{,\gamma}})\\\
&\leq(C_{0}^{\prime}C^{-1}+\frac{1}{4})N^{-1}.\end{split}$

We take $C=\max\\{4C_{0}^{\prime},4C_{0}\\}$, then $T:B_{N^{-1}}\to
B_{N^{-1}}$ is a well-defined contraction map. ∎

######  Corollary 8.3.

There exists an unique solution to (81) which satisfies

$\tau\|df\|_{\mathcal{C}^{,\gamma}}+\|\nabla
df\|_{\mathcal{C}^{,\gamma}}\lesssim\tau^{-\gamma}(1+\lambda^{-1}\mathrm{Vol}(L)\tau^{2+\gamma+\frac{n}{2}})^{-1}.$

In particular, $df$ lies in the Weinstein neighborhood $U_{L}$.

###### Proof.

Let $h$ be the fixed point of the contraction map $T:B_{N^{-1}}\to
B_{N^{-1}}$, then $\|h\|_{\mathcal{C}^{,\gamma}}\leq N^{-1}$. By Corollary
7.17, we obtain

$\tau\|df\|_{\mathcal{C}^{,\gamma}}+\|\nabla
df\|_{\mathcal{C}^{,\gamma}}\lesssim(1+\lambda^{-1}\mathrm{Vol}(L)\tau^{2+\gamma+\frac{n}{2}})\|h\|_{\mathcal{C}^{,\gamma}}\lesssim
N^{-1}(1+\lambda^{-1}\mathrm{Vol}(L)\tau^{2+\gamma+\frac{n}{2}}).$

∎

Now, we will consider the regularity of the special Lagrangian submanifold we
constructed. Suppose that $\iota:L\to X$ is a smooth immersion, then we could
get a good regularity results of solution $f$ by standard elliptic estimate.
In particular, as $f$ satisfies $\tilde{\mathrm{SL}}(x,df,\nabla df)=0$, which
is a second order non-linear elliptic equation. As $\iota$ is a smooth
immersion, $\tilde{\mathrm{SL}}(x,y,z)$ is a smooth function w.r.t. the
variables. By [3, Theorem 3.56], any $\mathcal{C}^{2}$ solution $f$ to (81)
will be smooth.

Let $\iota_{f}:L\to U\subset T^{\star}L$ be the inclusion given by the graph
of $df$, then $\iota_{f}$ will be a $\mathcal{C}^{1,\gamma}$ immersion, then
$[L_{f}]=(L,\iota_{f})$ is a $\mathcal{C}^{1,\gamma}$ special Lagrangian
submanifold. Even $\iota$ is not a smooth immersion, we still could obtain
good regularity for $[L_{f}]$. We have the following regularity theorem for
minimal submanifold due to Morrey.

######  Theorem 8.4.

[38] Any $\mathcal{C}^{1}$ special Lagrangian submanifold is real analytic.

In particular, as $[L_{f}]$ we constructed above is a $\mathcal{C}^{1,\gamma}$
submanifold, $[L_{f}]$ is real analytic. In summary, we obtain the following
nearby special Lagrangian theorem, which will be a special case of Joyce [25].

######  Theorem 8.5.

Let $[L]=(L,\iota)$ be an immersed Lagrangian submanifold, let $\lambda$ be
the first eigenvalue of the linearization operator $\mathcal{D}$ and $\tau$ be
the maximal of the second fundamental form. There exists a constant $C$ such
that for
$\Upsilon:=C(1+\lambda^{-1}\mathrm{Vol}(L)\tau^{2+\gamma+\frac{n}{2}})^{2}\tau^{\gamma}$,
suppose
$\|\star\iota^{\star}\Im\Omega\|_{\mathcal{C}^{,\gamma}}\leq\Upsilon^{-1}$,
then there exists a function $f$ on $L$ with $\int_{L}fd\mathrm{Vol}_{L}=0$
and

$\|df\|_{\mathcal{C}^{1,\gamma}}\lesssim\tau^{-\gamma}(1+\lambda^{-1}\mathrm{Vol}(L)\tau^{2+\gamma+\frac{n}{2}})^{-1},$

such that the graph submanifold $[L_{f}]$ of $df$ in the Weinstein
neighborhood will be a real analytic special Lagrangian submanifold.

###  8.2. Existence of special Lagrangian submanifolds near the approximate
family

Let $(X,J,\omega,\Omega)$ be a Calabi-Yau manifold with a special Lagrangian
submanfiold $[L]=(L,\iota_{0})$. Suppose over $[L]$ there exists a
nondegenerate harmonic pair,
$[\tilde{L}_{t}^{\mathrm{app}}]=(\tilde{L},\tilde{\iota}_{t}^{\mathrm{app}})$
be the family of approximate special Lagrangian submanifolds constructed in
Theorem 6.1 and $\theta_{t}$ be the Lagrangian angle. In this subsection, we
will apply the nearby special Lagrangian theorem for the family of Lagrangian
submanifold $[\tilde{L}_{t}^{\mathrm{app}}]$ to construct special Lagrangian
submanifolds.

Let $\tau_{t}$ be the maximal of second fundamental form of
$[\tilde{L}_{t}^{\mathrm{app}}]$, then by Proposition 7.9, we have
$\tau_{t}\lesssim t^{-2}.$ We define
$\tilde{g}_{t}=(\tilde{\iota}_{t}^{\mathrm{app}})^{\star}g$ be the pull-back
Riemannian metric.

By Theorem 7.5, $\tilde{g}_{t}$ convergence to a cone metric which is quasi-
isomorphic to any smooth Riemannian metric on $\tilde{L}$. The
$\mathcal{C}^{,\gamma}$ norm defined using different $\tilde{g}_{t}$ or the
cone metric are equivalent. Therefore, for the rest of this subsection, we
don’t label $t$ for different $\mathcal{C}^{,\gamma}$ norms. Moreover, by
Theorem 7.14, for each $t$, we could choose the Weinstein neighborhood of
$[\tilde{L}_{t}^{\mathrm{app}}]$ to be

$U_{t}:=\\{\alpha\in T^{\star}\tilde{L}|\|\alpha\|_{\mathcal{C}^{,\gamma}}\leq
Ct^{2}\\}.$

Let $f_{t}$ be a function on $\tilde{L}$ such that $df_{t}\in U_{t}$, the
special Lagrangian equation for $df_{t}$ in this Weinstein neighborhood will
be

$\sin\theta_{t}-\mathcal{D}_{\theta_{t}}f_{t}+Q_{t}(df_{t})=0.$

We write $f_{t}=-\mathcal{D}_{\theta_{t}}^{-1}h_{t}$ with
$\int_{\tilde{L}}h_{t}d\mathrm{Vol}_{t}=0$, where $d\mathrm{Vol}_{t}$ is the
volume form w.r.t. the Riemannian metric $g_{t}$, then the special Lagrangian
equation for $h_{t}$ will be

$\sin\theta_{t}+h_{t}+Q_{t}(d\mathcal{D}_{\theta_{t}}^{-1}h_{t})=0.$

We write
$T_{t}h_{t}:=\sin\theta_{t}+Q_{t}(d\mathcal{D}_{\theta_{t}}^{-1}h_{t})$, by
Proposition 8.2 and Corollary 8.3, we obtain

######  Proposition 8.6.

For $N_{t}=t^{8+4\gamma+n}$, there exists a constant $C$ independent of $t$
such that suppose
$\|\sin\theta_{t}\|_{\mathcal{C}^{,\gamma}}\leq\frac{N_{t}}{4}$, then
$T_{t}:B_{t}\to B_{t}$ is a contracting map for
$B_{t}:=\\{\|\alpha\|_{\mathcal{C}^{,\gamma}}\leq C^{-1}N_{t}\\}.$ Moreover,
let $h_{t}$ be the fixed point of $T_{t}$ with
$f_{t}=\mathcal{D}_{\theta_{t}}h_{t}$, then $f_{t}$ satisfies

$t^{-2}\|df_{t}\|_{\mathcal{C}^{,\gamma}}+\|\nabla_{t}df_{t}\|_{\mathcal{C}^{,\gamma}}\leq
C^{-1}t^{4+n+4\gamma}.$

By Theorem 6.1, for any $k$, we can choose $[\tilde{L}_{t}^{\mathrm{app}}]$
such that $\|\sin\theta_{t}\|_{\mathcal{C}^{,\gamma}}\leq C_{0}t^{k}$. When
$t$ small enough and $t^{k}\geq t^{8+4\gamma+n+1}$, we could obtain the desire
approximate solutions.

From the above estimate, we know that $df_{t}\subset U_{t}$. We could write
the inclusion induces by $df_{t}$ as $\tilde{\iota}_{t}^{\prime}:\tilde{L}\to
U_{t}\subset T^{\star}\tilde{L}$, which is a special Lagrangian submanifold.
Using $\Phi_{t}:U_{t}\to U_{t}^{\prime}\subset X$, we define
$\tilde{\iota}_{t}:=\Phi_{t}\circ\tilde{\iota}_{t}^{\prime}$, then
$[\tilde{L}_{t}]=(\tilde{L},\iota_{t})$ will be a family of special Lagrangian
submanifolds on $X$.

Now, let’s discuss the rate of the convergence. By Proposition 8.6, we have

$\|\tilde{\iota}_{t}-\tilde{\iota}_{t}^{\mathrm{app}}\|_{\mathcal{C}^{,\gamma}}\lesssim\|df_{t}\|_{\mathcal{C}^{,\gamma}}\leq
Ct^{4+n+4\gamma}.$

In addition, by Theorem 6.1 (vi), we have
$\|\tilde{\iota}_{t}^{\mathrm{app}}-\iota_{0}\circ
p\|_{\mathcal{C}^{,\gamma}}\leq Ct.$ Therefore, we obtain

$\|\tilde{\iota}_{t}-\iota_{0}\circ p\|_{\mathcal{C}^{,\gamma}}\leq
C\|\tilde{\iota}_{t}-\tilde{\iota}_{t}^{\mathrm{app}}\|_{\mathcal{C}^{,\gamma}}+\|\tilde{\iota}_{t}^{\mathrm{app}}-\iota_{0}\circ
p\|_{\mathcal{C}^{,\gamma}}\leq Ct.$

On the other hand, let $\mathfrak{a}$ be the nondegenerate harmonic pair and
$\phi_{t}$ be the diffeomorphism constructed in Theorem 6.1. We write
$\tilde{\iota}_{\phi_{t}^{\star}(t\mathfrak{a})}:\tilde{L}\to U_{L}$ be the
inclusion map induces by the graph of $t\phi_{t}^{\star}(\mathfrak{a})$, then
by Theorem 6.1 (ii), we have
$\|\tilde{\iota}_{t}^{\mathrm{app}}-\tilde{\iota}_{t\phi_{t}^{\star}(\mathfrak{a})}\|_{\mathcal{C}^{,\gamma}}\leq
Ct^{2}$, which implies

$\|\tilde{\iota}_{t}-\tilde{\iota}_{t\phi_{t}^{\star}(\mathfrak{a})}\|_{\mathcal{C}^{,\gamma}}\leq
Ct^{2}.$

In summary, we could state our main theorem.

######  Theorem 8.7.

Let $(X,J,\omega,\Omega)$ be a Calabi-Yau manifold and $[L]=(L,\iota_{0}:L\to
X)$ be a special Lagrangian submanifold with induced metric $g_{L}$. Suppose
there exists a nondegenerate harmonic pair
$\mathfrak{a}=\alpha^{+}+\alpha^{-}$ over $[L]$, let $p:\tilde{L}\to L$ be the
branched covering of $L$ induces by $\alpha^{-}$, then there exists a positive
constant $T$ and a family of special Lagrangian submanifold
$\tilde{\iota}_{t}:\tilde{L}\to X$ for $|t|\leq T$ such that

  * (i)

$\lim_{t\to 0}\|\tilde{\iota}_{t}-\iota_{0}\circ
p\|_{\mathcal{C}^{,\gamma}}=0$.

  * (ii)

$\tilde{\iota}_{t}(\tilde{L})$ convergence to 2$\iota_{0}(L)$ as current.

  * (iii)

In a Weinstein neighborhood $U_{L}\subset T^{\star}L$ of $[L]$, there exists a
family of diffeomorphisms $\phi_{t}:L\to L$ such that suppose we write
$\iota_{t\phi_{t}^{\star}(\mathfrak{a})}:\tilde{L}\to U_{L}$ be the inclusion
map induces by the graph of $t\phi_{t}^{\star}(\mathfrak{a})$, then

$\|\tilde{\iota}_{t}-\tilde{\iota}_{t\phi_{t}^{\star}(\mathfrak{a})}\|_{\mathcal{C}^{,\gamma}}\leq
Ct^{2},$

where $C$ is a t-independent constant and $0<\gamma<\frac{1}{2}$,

where the $\mathcal{C}^{,\gamma}$ norms are defined using the metric
$p^{\star}g_{L}$.

Suppose $L$ is a Riemannian surface, by Proposition 5.2, the branched
deformations generates by quadratic differentials will be embedded. However,
in general, suppose $\iota_{0}$ is an embedding, $\tilde{\iota}_{t}$ we
constructed above will not be embedded. By Proposition 7.3,
$\tilde{\iota}_{t}^{\mathrm{app}}$ could have self intersections, as
$\tilde{\iota}_{t}$ could be regarded as a small deformation of
$\tilde{\iota}_{t}^{\mathrm{app}}$, thus we don’t expect $\tilde{\iota}_{t}$
is embedded.

######  Proposition 8.8.

Let $[L]=(L,\iota_{0}:L\to X)$ is an embedded special Lagrangian submanifold
and $\mathfrak{a}=\alpha^{+}+\alpha^{-}$ is a nondegenerate harmonic pair over
$[L]$, suppose $|\alpha^{-}|^{-1}(0)=\Sigma$, then there exists a positive
constant $T_{0}$ such that for $|t|<T_{0}$, $\tilde{\iota}_{t}$ constructed in
Theorem 8.7 is an embedding.

###### Proof.

We only need to prove this in a Weinstein neighborhood $U_{L}$. Let
$\tilde{\iota}_{t\mathfrak{a}}:\tilde{L}\to U_{L}$ be the inclusion induces by
the graph of $t\mathfrak{a}$, as $|\alpha^{-}|^{-1}(0)=\Sigma$,
$\tilde{\iota}_{t\mathfrak{a}}:\tilde{L}\to U_{L}$ is an embedding. Let
$\mathcal{A}_{t}=\mathcal{A}_{t}^{+}+\mathcal{A}_{t}^{-}$ be a nondegenerate
1-form we constructed in Theorem 6.1 with inclusion
$\tilde{\iota}_{t}^{\mathrm{app}}$, by Theorem 6.1, as
$\|t\mathfrak{a}-\mathcal{A}\|_{\mathcal{C}^{,\gamma}}\leq Ct^{2}$, then for
$t$ sufficiently small $\tilde{\iota}_{t}^{\mathrm{app}}$ is a embedding.
Moreover, as
$\|\tilde{\iota}_{t}-\tilde{\iota}_{t}^{\mathrm{app}}\|_{\mathcal{C}^{,\gamma}}\leq
Ct^{4+n+4\gamma}$, $\tilde{\iota}_{t}$ is also an embedding for sufficiently
small $t$. ∎

###  8.3. The branched deformation theorem for fixed real locus

Suppose the special Lagrangian is a fixed real locus, then the branched
deformations we constructed in Theorem 8.7 could also have extra symmetry.

######  Theorem 8.9.

Let $(X,J,\omega,\Omega)$ be a Calabi-Yau manifold and $[L]=(L,\iota_{0}:L\to
X)$ be a special Lagrangian submanifold which is locally a fixed real locus of
anti-holomorphic involution $R$. Suppose there exists a $\mathbb{Z}_{2}$
harmonic 1-form $\alpha^{-}$ over $[L]$, then the family of special Lagrangian
submanifolds $[\tilde{L}_{t}]=(\tilde{L},\tilde{\iota}_{t})$ constructed in
Theorem 8.7 will satisfy
$R\circ\tilde{\iota}_{t}=\tilde{\iota}_{t}\circ\sigma$, where $\sigma$ is the
involution on $\tilde{L}$.

###### Proof.

By Theorem 6.11 and Corollary 6.12, the special Lagrangian equation over
$[\tilde{L}_{t}^{\mathrm{app}}]$ for $df_{t}$ over $\tilde{L}$ will satisfy
$\sigma^{\star}\mathrm{SL}(df_{t})=-\mathrm{SL}(-\sigma^{\star}df_{t}).$
Therefore, suppose $f_{t}$ is a solution, then $-\sigma^{\star}df_{t}$ is also
a solution. By Corollary 8.3, the solution is unique for the contracting map,
therefore $df_{t}=-\sigma^{\star}df_{t}$. Recall that
$\tilde{\iota}_{t}:\tilde{L}\to X$ is given by the graph of $df_{t}$, which
defined as
$\tilde{\iota}_{t}(x)=\exp_{\tilde{\iota}_{t}^{\mathrm{app}}(x)}(J(\tilde{\iota}_{t}^{\mathrm{app}})_{\star}V_{df})$,
where $V_{df_{t}}$ is the dual vector of $df_{t}$ given by the Riemannian
metric. By the same computation as (61), we obtain
$R\circ\tilde{\iota}_{t}=\tilde{\iota}_{t}\circ\sigma.$ ∎

In general, the image of the branch locus $\tilde{\iota}_{t}(\Sigma)$ will not
lie in $[L]$. However, in this case, by
$R\circ\tilde{\iota}_{t}=\tilde{\iota}_{t}\circ\sigma,$ we obtain
$R\circ\tilde{\iota}_{t}(x)=\tilde{\iota}_{t}(x)$ for any $x\in\Sigma$.

######  Corollary 8.10.

The branch locus $\tilde{\iota}_{t}(\Sigma)$ is a subset of $\iota_{0}(L)$.

##  9\. Applications and Generalizations

In this section, we will discuss a few cases that our branched deformation
theorem could be applied and generalized.

###  9.1. Family of Calabi-Yau structures

Combining with the McLean’s deformation theorem, our main theorem could be
generalized to a smooth family of Calabi-Yau submanifolds. Let
$(X,J_{s},\omega_{s},\Omega_{s})$ be a family of Calabi-Yau manifolds for
$s\in(-\epsilon,\epsilon)$ and $[L_{0,0}]=(L,\iota_{0,0})$ be a special
Lagrangian submanifold on $(X,J_{0},\omega_{0},\Omega_{0})$. Suppose the
homology class of $\omega_{s}$ and $\Im\Omega_{s}$ restricted to $[L_{0}]$
vanishes, then by the deformation theorem for family of Calabi-Yau structures
[27, Theorem 5.3], $[L_{0}]$ could be extended smoothly to a family of special
Lagrangian submanifolds $[L_{s,0}]=(L,\iota_{s,0}).$ We write the induced
metric on $[L_{s,0}]$ as $g_{s,0}=\iota_{s,0}^{\star}g$.

Suppose over $[L_{0,0}]$, there exists a nondegenerate $\mathbb{Z}_{2}$
harmonic 1-form $\alpha_{0}$, then by Theorem 4.19, there exists
$\epsilon_{0}\leq\epsilon$ such that for each
$s\in(-\epsilon_{0},\epsilon_{0})$, there exists a family of nondegenerate
$\mathbb{Z}_{2}$ harmonic 1-forms $(g_{s,0},\Sigma_{s},\alpha_{s})$ over
$[L_{s,0}]$. Even $\alpha_{s}$ is branched along a different singular set
$\Sigma_{s}$, $\Sigma_{s}$ are close to $\Sigma$ and the branched covering map
$p_{s}:\tilde{L}\to L$ has the same diffeomorphism type. For each $v_{s}$, by
Theorem 8.7, we obtain a family of special Lagrangian submanifolds

$[\tilde{L}_{s,t}]=(\tilde{L},\tilde{\iota}_{s,t}:\tilde{L}\to X),$

such that $\lim_{t\to 0}\|\tilde{\iota}_{s,t}-\iota_{s,0}\circ
p_{s}\|_{\mathcal{C}^{,\gamma}}=0.$

######  Theorem 9.1.

Let $(X,J_{s},\omega_{s},\Omega_{s})$ be a family of Calabi-Yau manifold, let
$[L_{0,0}]=(L,\iota_{0,0})$ is a special Lagrangian submanifold for
$(X,J_{0},\omega_{0},\Omega_{0})$. Suppose

  * (i)

the homology class of $\omega_{s},\;\Im\Omega_{s}$ restricted to $[L_{0,0}]$
vanishes,

  * (ii)

over $[L_{0,0}]$, there exists a $\mathbb{Z}_{2}$ harmonic form $v_{0}$, with
branched covering $p_{0}:\tilde{L}\to L$.

then for small $\epsilon_{0}$ with $|s|\leq\epsilon_{0}$, there exists a
family of inclusions $\tilde{\iota}_{s,t}:\tilde{L}\to X$ such that

  * (i)

$[\tilde{L}_{s,t}]=(\tilde{L}_{s,t},\tilde{\iota}_{s,t})$ are special
Lagrangian submanifolds on $\mathbf{X}_{s}$,

  * (ii)

$\lim_{s\to 0}\lim_{t\to 0}\|\tilde{\iota}_{s,t}-\iota_{0,0}\circ
p\|_{\mathcal{C}^{,\gamma}}=0$.

###  9.2. The moduli space of the branched deformation family

The moduli space of special Lagrangian submanifold have been studied in [27,
36, 20]. In this subsection, we will briefly discuss the meaning of the
branched deformation theorem for the moduli space.

Let $[L]=(L,\iota_{0})$ be a special Lagrangian submanifold in a Calabi-Yau
manifold $(X,J,\omega,\Omega)$. Let $\mathcal{M}_{[L]}$ be the connected
component of the set of special Lagrangian submanifold containing $[L]$. Then
McLean’s deformation theorem 2.10 implies that $\mathcal{M}_{[L]}$ is a smooth
manifold with dimension $b^{1}(L)$, which is the first Betti number of $L$.

Suppose there exists a nondegenerate harmonic pair $(\alpha^{+},\alpha^{-})$
on $[L]$, and we write $[\tilde{L}_{s}]=(\tilde{L},\tilde{\iota}_{s})$ be the
family of branched deformation constructed in Theorem 8.7. We could consider
the moduli space $\mathcal{M}_{[\tilde{L}]}$, which is the connected
components containing all the $[\tilde{L}_{s}]$, then
$\mathcal{M}_{[\tilde{L}]}$ is also a smooth moduli space with dimension
$b^{1}(\tilde{L})$. By Proposition 4.5, $b^{1}(\tilde{L})=b^{1}_{-}+b^{1}(L)$.
Therefore, we might enlarge $\mathcal{M}_{[\tilde{L}]}$ to a compactifcation
$\overline{\mathcal{M}_{[\tilde{L}]}}$ by adding $\mathcal{M}_{[L]}$ as a
codimension $b^{1}_{-}$ boundary.

The inverse of the above discussions is incorrect. Let $[L]=(L,\iota_{0})$ be
an embedded special Lagrangian submanifold such that the induced metric on
$[L]$ is Ricci positive, then by [1, 48], $[L]$ doesn’t admit any deformation.
However, by Example 4.7, in the case when $L=S^{3}$, we could construct
branched covering $p:\tilde{L}\to S^{3}$ such that $b^{1}_{-}(\tilde{L})\geq
1$. We could also regard $(\tilde{L},\iota_{0}\circ p)$ be a special
Lagrangian submanfold, which is not immersed. The harmonic 1-forms on
$(\tilde{L},\iota_{0}\circ p)$ will not generate a deformation.

###  9.3. Branched deformation of the zero section

In this subsection, we will introduce examples that our deformation theorem
could be applied using the Calabi-Yau neighborhood theorem.

######  Definition 9.2.

Let $(L,g_{L})$ be a real analytic Riemannian manifold, let $U_{L}$ be a
neighborhood of the zero section in $T^{\star}L$. A Calabi-Yau neighborhood of
$L$ is a Calabi-Yau structure $(U_{L},J,\omega,\Omega)$ with Calabi-Yau metric
$g$ such that

  * (i)

$\omega$ is the canonical symplectic form,

  * (ii)

the restriction of $g$ to the zero section is $g_{L}$,

  * (iii)

the zero section is a special Lagrangian submanifold in this Calabi-Yau
structure.

By Stenzel [43], every compact rank 1 global symmetric space admits a Calabi-
Yau neighborhood. Moreover, there are more examples.

######  Theorem 9.3.

[4, 7] Let $(L,g_{L})$ be a real analytic Riemannian manifold, suppose
$\chi(L)=0$, then $L$ admits a Calabi-Yau neighborhood
$(U_{L},J,\omega,\Omega)$. In addition, let $R$ be the canonical involution on
$T^{\star}L$, then $R$ is an anti-holomorphic involution on $U_{L}$.

Using our existence theorem, we obtain:

######  Theorem 9.4.

Let $(L,g_{L})$ be a real analytic Riemannian manifold with $\chi(L)=0$, let
$(U_{L},J,\omega,\Omega)$ be a Calabi-Yau structure constructed in Theorem
9.3, suppose there exists a nondegenerate multivalued harmonic form on
$(L,g_{L})$ with associated branched covering $p:\tilde{L}\to L$, then there
exists a family of immersed special Lagrangian submanifolds
$\tilde{\iota}_{t}:\tilde{L}\to U_{L}$ such that $\lim_{t\to
0}\|\tilde{\iota}_{t}-\iota_{0}\circ p\|_{\mathcal{C}^{,\gamma}}=0$, where
$\iota_{0}$ is the inclusion of the zero section. In addition, $\iota_{t}$
will satisfy $R\circ\tilde{\iota}_{t}=\tilde{\iota}_{t}\circ\sigma.$

By [18], the author construct nondegenerate $\mathbb{Z}_{2}$ harmonic 1-forms
over some rational homology 3-spheres. As a rational homology 3-sphere has
vanishing first Betti number, it is rigid in McLean sense as in Theorem 2.10.
Therefore, we obtain the following.

######  Corollary 9.5.

There exist special Lagrangian submanifolds that are rigid in classical sense
but could have branched deformations.

###  9.4. Topological constraint of the existence of nondegenerate
$\mathbb{Z}_{2}$ harmonic 1-forms

Let $(L,g)$ be a Riemannian manifold and $(\Sigma,\mathcal{I},\alpha)$ be a
nondegenerate $\mathbb{Z}_{2}$ harmonic 1-form. By the work of Taubes [46,
45], nondegenerate $\mathbb{Z}_{2}$ harmonic 1-forms are the ideal boundary of
the gauge theory compactification of $\mathrm{PSL}(2,\mathbb{C})$ flat
connection and it is still very mystery. We would like to understand the
following question.

######  Question 9.6.

Is there any topological constraint to the existence of nondegenerate
$\mathbb{Z}_{2}$ harmonic 1-forms?

Using the branched deformation theorem we proved, we could actually give a
topological constraint for the existence of nondegenerate $\mathbb{Z}_{2}$
harmonic 1-forms. Our starting point come from the work of Abouzaid and Imagi
[1].

######  Theorem 9.7.

[1] Let $U_{L}$ be a neighborhood of the zero section of $T^{\star}L$, and
$(U_{L},J,\omega,\Omega)$ be a Calabi-Yau structure such that the zero section
$[L_{0}]$ a special Lagrangian submanifold, then for any unobstructed nearby
immersed special Lagrangian $[L^{\prime}]$ which is $\mathcal{C}^{0}$ close to
$[L_{0}]$,

  * (i)

suppose $\pi_{1}(L)$ is finite, then $[L^{\prime}]=[L_{0}]$,

  * (ii)

suppose $\pi_{1}(L)$ has no nonabelian free subgroup, then $[L^{\prime}]$ is
an unbranched deformation of $[L_{0}]$ that constructed in Corollary 2.11.

The definition of a Lagrangian submanifold is unobstructed is in [11], which
is not relevant to most of the parts in the paper, so we omit the definition.
We will list a few cases where the unobstructed condition is satisfied. Let
$(L,g)$ be a Riemannian manifold, suppose there exists a nondegenerate
$\mathbb{Z}_{2}$ harmonic 1-form on $L$ and suppose there exists a Calabi-Yau
neighborhood $(U_{L},J,\omega,\Omega)$. We write $[L_{0}]=(L,\iota_{0})$ be
the inclusion of the zero section. Then by Theorem 9.4, we obtain a family of
special Lagrangian submanifolds $[\tilde{L}_{t}]$ which will be
$\mathcal{C}^{0}$ close to $[L_{0}]$. If any $[\tilde{L}_{t}]$ is unobstructed
and suppose $\pi_{1}(L)$ is either finite or has no nonabelian free subgroup,
this will lead to a contradiction based on Theorem 9.7.

######  Proposition 9.8.

Let $(L,g_{L})$ be a real analytic Riemannian manifold and
$(\Sigma,\mathcal{I},\alpha)$ be a multivalued harmonic 1-form on $L$ with
associate branched covering $p:\tilde{L}\to L$. Suppose

  * (i)

$\pi_{1}(L)$ is finite or contains no nonablian free subgroup,

  * (ii)

$(L,g_{L})$ has a Calabi-Yau neighborhood $(U_{L},J,\omega,\Omega)$ in
$T^{\star}L$,

  * (iii)

every special Lagrangian submanifold over $U_{L}$ which is diffeomorphic to
$\tilde{L}$ is unobstructed,

then $(\Sigma,\mathcal{I},\alpha)$ is not nondegenerate. In addition, suppose
the canonical involution $R$ on $T^{\star}L$ induces an anti-holomorphic
involution, then (iii) could be changed to

  * (iii)′

every $R$-invariant special Lagrangian submanifold over $U_{L}$ which is
diffeomorphic to $\tilde{L}$ is unobstructed.

By [1, Proposition 2.8], the condition that $L$ has no nonabelian free
subgroup is a very strong assumption, $L$ only have the following four
possibility: (i) $\pi_{1}(L)$ is finite, (ii)$L\cong S^{1}\times S^{2}$,
(iii)$L\cong RP^{3}\sharp RP^{3}$, (iv) there exists a finite covering
$L^{\prime}\to L$ where $L^{\prime}$ is the mapping torus of an orientation
preserving diffeomorphism $T^{2}\to T^{2}$.

In general, it is very hard to check whether a Lagrangian submanifold is
unobstructed or not. By the work of FOOO [11], there is a topological
condition one could verify for the obstruction condition.

######  Proposition 9.9.

[11] Let $(\tilde{L},\tilde{\iota}:\tilde{L}\to U_{L})$ be a Lagrangian
submanifold, suppose $\tilde{\iota}^{\star}:H^{2}(U_{L};\mathbb{R})\to
H^{2}(\tilde{L};\mathbb{R})$ is surjective, then $(\tilde{L},\tilde{\iota})$
is unobstructed.

As a corollary, we obtain extra constraints of nondegenerate $\mathbb{Z}_{2}$
harmonic 1-form.

######  Theorem 9.10.

Let $(L,g_{L})$ be a real analytic Riemannian manifold and let
$(\Sigma,\mathcal{I},\alpha)$ be a nondegenerate $\mathbb{Z}_{2}$ harmonic
1-form on $L$ with branched covering $p:\tilde{L}\to L$. Suppose

  * (i)

$\pi_{1}(L)$ is either finite or contains no nonabelian free subgroup,

  * (ii)

$(L,g_{L})$ has a Calabi-Yau neighborhood on $T^{\star}L$,

then $b_{2}(\tilde{L})>b_{2}(L)$, where $b_{2}$ is the second Betti number.

Unfortunately, the above theorem is trivial when $\dim(L)=3$. By Corollary
4.6, the existence of $\mathbb{Z}_{2}$ harmonic 1-form will require that
$b_{1}(\tilde{L})>b_{1}(L)$ which implies $b_{2}(\tilde{L})>b_{2}(L)$ by
Poincare duality. However, this topological constraint is non-trivial in for
four dimensional or higher and we will introduce an example. Let
$\sigma_{n}:S^{n}\to S^{n}$ be the involution map fixing the equator and let
$\tilde{L}:=S^{1}\times S^{n-1}$. We take $\sigma:=(\sigma_{1},\sigma_{n-1})$,
then $\sigma$ is an involution action on $\tilde{L}$ with fixed point
$\mathrm{Fix}(\sigma)$ is diffeomorphic to $S^{0}\times S^{n-2}$ which is two
copies of $S^{n-2}$. The quotient $L:=\tilde{L}/\langle\sigma\rangle$ under
the involution, would be $L:=S^{n}$. Moreover, the quotient map
$p:\tilde{L}\to L$ will be a branched covering map. When $n\geq 4$ and $n$ is
odd, as $H^{1}(\tilde{L};\mathbb{R})^{-}\neq 0$, over $L$, there exists a
$\mathbb{Z}_{2}$ harmonic 1-form $\alpha$ over $\tilde{L}$. However, as
$H^{2}(\tilde{L};\mathbb{R})=0$ and $\chi(\tilde{L})=0$, by Theorem 9.10, this
$\mathbb{Z}_{2}$ harmonic 1-form will not be nondegenerate.

When $n$ is even, over $L=S^{n}$, as $\chi(L)\neq 0$, we don’t know the
existence of Calabi-Yau neighborhood for generic metric on $L$. However, over
$T^{\star}S^{n}$ we have the Stenzel metric [43], which extend the round
metric on $S^{n}$ to a complete Riemannian metric over the whole
$T^{\star}S^{n}$. Thus over $T^{\star}L$, we could associate a Calabi-Yau
structure. By Theorem 9.10, the $\mathbb{Z}_{2}$ harmonic 1-form will not be
nondegenerate. In summary, we obtain

######  Corollary 9.11.

Let $p:S^{1}\times S^{n-1}\to S^{n}$ be the branched covering map considered
above, $\Sigma$ be the branch locus, $\chi\in H^{1}(S^{1}\times
S^{n-1};\mathbb{R})$, then

  * (i)

If $n$ is odd, then for any Riemannian metric over $S^{n}$, the
$\mathbb{Z}_{2}$ harmonic 1-form corresponding to $\chi$ will not be
nondegenerate.

  * (ii)

If $n$ is even, then for the round Riemannian metric over $S^{n}$, the
$\mathbb{Z}_{2}$ harmonic 1-form corresponding to $\chi$ will not be
nondegenerate.

Here we provide another example. Let $M$ be a connected oriented closed
manifold, by Neofytidis [39], there exists a $\pi_{1}$ surjective branched
double covering

(82) $p:S^{1}\times\sharp_{k}(M\times S^{1})\to\sharp_{k}(M\times S^{2}).$

We write $\tilde{L}=S^{1}\times\sharp_{k}(M\times S^{1})$ and
$L=\sharp_{k}(M\times S^{2})$, suppose $M$ satisfies $b_{1}(M)=b_{2}(M)=0$,
then $b_{1}(L)=0,\;b_{2}(L)=k$ and
$b_{1}(\tilde{L})=1+k,\;b_{2}(\tilde{L})=k$. Then for any $\chi\in
H^{1}(\tilde{L};\mathbb{R})$, it will give a multivalued harmonic 1-form on
$L$. However, as $b_{2}(\tilde{L})=b_{2}(L)$, all these multivalued harmonic
1-forms could not be nondegenerate.

######  Corollary 9.12.

Let $M$ be a closed manifold which is odd dimensional and
$b_{1}(M)=b_{2}(M)=0$, then for the double branched covering in (82) and for
any Riemannian metric on $\sharp_{k}(M\times S^{2})$, the multivalued harmonic
1-form corresponding to $\chi\in H^{1}(S^{1}\times\sharp_{k}(M\times
S^{1});\mathbb{R})$ for (82) is not nondegenerate.

##  Appendix A Polyhomogeneous Expansions and Mellin Transformations

In this appendix, we will introduce the theory of polyhomogeneous expansions
and elliptic edge operators. All results surveyed here could be found in [34,
37].

###  A.1. Mellin transformation

Mellin transformation is a tool to study the expansion of functions along a
singular set, which gives a nice correspondence between meromorphic functions
and functions with expansions. We refer to [34, Section 2] and [37] for more
details.

Let $u(r)$ be a compact supported function defined on the positive real axis
$0<r<\infty$, the Mellin transformation $Mu:=u_{M}(\zeta)$ of $u$ is a
function defined on the complex plane defined as

$Mu(\zeta)=u_{M}(\zeta):=\int_{0}^{\infty}u(r)r^{i\zeta-1}dr,$

where $\zeta:=\xi+i\eta$ is a complex variable.

The Mellin transformation is related to the Fourier transformation. Let
$r=e^{s}$, then

$u_{M}(\zeta)=\int_{-\infty}^{+\infty}u(e^{s})e^{i\zeta s}ds,$

which is the Fourier transform for the function $u(e^{s})$. Therefore, there
is a Plancherel relationship extended the definition of Mellin transform to
$L^{2}$ function, or more generally, weighted $L^{2}$ space
$r^{\delta}L^{2}(\mathbb{R}^{+})$ such that the Mellin transform induces an
isomorphism

(83) $r^{\delta}L^{2}(\mathbb{R}^{+})\cong L^{2}(\eta=\delta-\frac{1}{2}),$

where $r^{\delta}L^{2}:=\\{r^{\delta}f|f\in L^{2}\\}$.

As for $\delta^{\prime}\leq\delta$, we have $r^{\delta}L^{2}\subset
r^{\delta^{\prime}}L^{2}$, the domain that Mellin transform exists will be
$\\{\Im\zeta<\beta\\}$. Analogy to the Fourier transform, the Mellin transform
also have an inverse formula.

$M^{-1}(u_{M}(\zeta))=\frac{1}{2\pi
i}\int_{\Im\zeta=C}u_{M}(\zeta)x^{-i\zeta}d\xi.$

In particular, if $u\in r^{\delta}L^{2}$ and $C=\delta-\frac{1}{2}$, we have
$M^{-1}(u_{M}(\zeta))=u(x)$.

Now, we will focus on the expansion at $r=0$. Let $\phi(r)$ be a smooth
function over $\mathbb{R}^{+}$ such that $\phi$ equals $1$ near $r=0$ and
$\phi$ vanishes for large $t$, we consider $u(r):=r^{s}(\log r)^{p}\phi(r)$.
In addition, as $r^{s}(\log r)^{p}\phi(r)\in x^{\delta}L^{2}$, for any
$\delta<\Re s+\frac{1}{2}$, $u_{M}(\zeta)$ is well defined on
$\\{\Im\zeta<\beta\\}$.

In addition, as

$\int_{0}^{1}r^{s}(\log
r)^{p}r^{i\zeta-1}dr=(-1)^{p+1}\frac{p!}{(i\zeta+s)^{p+1}},$

an integration by parts shows that $u_{M}$ can be extended to a meromorphic
function in the whole complex plane, with poles of order $p+1$ at $\zeta=is$.

In our situation, let $\Sigma$ be an embedded submanifold on a Riemannian
manifold, Let $U$ be a tubular neighborhood of $\Sigma$ and we choose local
coordinates $(z=re^{i\theta},t)$ along $U$ such that $\Sigma=\\{z=0\\}$. Let
$u(r,\theta,t)$ be a polyhomogeneous function near $\Sigma$ with index set
$E$, then the above arguments could be applied to $u(r,\theta,t)$. The Mellin
transform of $r$ coordinate and $(\phi(r)u)_{M}(\zeta,\theta,t)$ could be
understood as a function on $(\zeta,\theta,t)\in\mathbb{C}\times
S^{1}\times\Sigma$. In addition, for $s\in E$, $(\phi(r)u)_{M}$ has a
meromorphic extension at $\zeta=is$.

We have the following fundamental correspondence of Mellin transformation:

######  Theorem A.1.

[37, Proposition 5.27] The Mellin transformation gives an isomorphism between
the polyhomogeneous function with index set $E$ and meromorphic functions with
values in $\mathcal{C}^{\infty}(Y)$ having poles at $z=is$ for $s\in E$.

###  A.2. The expansion of $\mathbb{Z}_{2}$-harmonic function

Let $\Sigma$ be a codimension 2 submanifold of $L$ and
$\chi:\pi_{1}(L\setminus\Sigma)\to\Gamma$ be the representation satisfies (ii)
of Definition 4.1, with associative affine bundle $V^{-}$ and corresponding
vertical line bundle $\mathcal{I}$. Using the flat structure, we will study
the operator $\Delta:\Gamma(V^{-})\to\Gamma(\mathcal{I})$ near the branch
locus.

We could consider the following local model for a neighborhood near $\Sigma$.
Let $\mathbb{R}^{n}=\mathbb{C}\times\mathbb{R}^{n-2}$ and we will represents a
point in $\mathbb{R}^{n}$ by $(z,t)$ and write $z=re^{i\theta}$. Let $U$ be a
open set in a neighborhood of $\Sigma$ and we also regard $U$ as an open set
of $\mathbb{R}^{n}$ such that
$U\cap\Sigma\subset\\{0\\}\times\mathbb{R}^{n-2}$. For convenient, we might
think $V^{-}$ is defined over $\mathbb{C}^{\star}\times\mathbb{R}^{n-2}$ with
fiber $\mathbb{R}$ and holonomy $-1$, while the sections of $V^{-}$ could be
identified with 2-valued function $f$ such that $f(w^{2},t)$ is an odd
function on $w$.

We might write

$r^{2}\Delta=(r\partial_{r})^{2}+\partial_{\theta}^{2}+(r\partial_{t})^{2}+rE,$

where $E$ is smooth in $r\partial_{r},\partial_{\theta}$ or $r\partial_{t}$.
Here $r^{2}\Delta$ is called an elliptic edge operator, which considered in
[34]. The indicial family $I_{\zeta}:=\partial_{\theta}^{2}+\zeta^{2}$ of the
operator $r^{2}\Delta$ is given by

$r^{2}\Delta(r^{\zeta}(\log r)^{p}f(r,\theta,t))=r^{\zeta}(\log
r)^{p}I_{\zeta}(f(0,\theta,t))+\mathcal{O}(r^{\zeta}(\log r)^{p-1}),$

where the above formula works for $\forall f\in\mathcal{C}^{\infty}$,
$s\in\mathbb{C}$ and $p\in\mathbb{N}_{0}$. In local coordinates, we might
write $I:=(r\partial_{r})^{2}+\partial_{\theta}^{2}$, then under the Mellin
transformation, we have $(If)_{M}=I_{i\zeta}f_{M}$. We define

$L^{2}_{\mathrm{odd}}(S^{1}\times\Sigma)=\\{f\in\mathcal{I}|_{S^{1}\times\Sigma}|f(\theta+2\pi,t)=-f(\theta,t).\;f\in
L^{2}\\},$

which is the $V^{-}$ valued $L^{2}$ function, then we might regard

$I_{\zeta}:L^{2}_{\mathrm{odd}}(S^{1}\times\Sigma)\to
L^{2}_{\mathrm{odd}}(S^{1}\times\Sigma).$

######  Definition A.2.

The indicial roots of $r^{2}\Delta$ is the set of values $\zeta\in\mathbb{C}$
for which $I_{\zeta}$ fails to be invertible on
$L^{2}_{\mathrm{odd}}(S^{1}\times\Sigma)$ and we write $E_{\mathrm{ind}}$ be
the sets of all indicial roots.

######  Proposition A.3.

$E_{\mathrm{ind}}=\\{k+\frac{1}{2}|k\in\mathbb{Z}\\}$ is the collection of all
indicial roots for the operator $r^{2}\Delta$,

###### Proof.

Using the Fourier transform, we might write $f\in$
$L^{2}_{\mathrm{odd}}(S^{1}\times\Sigma)$ as

$f=\sum_{k}\Re(f_{k}(r,t)e^{i(k+\frac{1}{2})\theta}),$

where $f_{k}(r,t)$ is a complex function. Then we compute

$I_{\zeta}f=\sum_{k}(\zeta^{2}-(k+\frac{1}{2})^{2})\Re(f_{k}(r,t)e^{i(k+\frac{1}{2})\theta}).$

Therefore, the indicial roots will be $\zeta=k+\frac{1}{2}$ for
$k\in\mathbb{Z}$. ∎

Let $\rho\in L^{2}$ be a section of $\mathcal{I}$ such that $\rho$ is
polyhomogeneous with index set $\\{k+\frac{1}{2}|k\in\mathbb{N}\\}$. We now
consider the polyhomogeneous expansions of solutions, which will be a special
case of [34, Section 7].

######  Theorem A.4.

[34, Section 7] Let $f\in\Gamma(V^{-})$ be a solution to $\Delta f=\rho$, then
$f$ has a polyhomogeneous expansion with index set
$E^{\prime}=\\{k+\frac{1}{2}|k\in\mathbb{N}\\}$.

###### Proof.

We only need to show the Mellin transform $f_{M}(\zeta,\theta,t)$ has a
meromorphic extension to the whole $\zeta$ plane and the poles of $f_{M}$ lies
only on $i(k+\frac{1}{2})$. It is easier we write $r^{2}\Delta
f=\rho^{\prime}$ with $\rho^{\prime}:=r^{2}\rho$.

We might write $r^{2}\Delta f=I(f)+E^{\prime}(f)$, then for the equation
$I(f)+E^{\prime}(f)=\rho^{\prime}$, using the Mellin transform, we obtain

$I_{\zeta}f_{M}+(E^{\prime}f)_{M}=\rho_{M}.$

By Theorem A.1, $\rho^{\prime}_{M}(\zeta,\theta,t)$ has a meromorphic
extension to the whole $\zeta$ plane $\mathbb{C}$. By [8, Section 3], $f$ is
conormal, and we define $f_{\theta,t}(r):=f(r,\theta,t)$, then
$f_{\theta,t}(r)\in r^{\delta}L^{2}(\mathbb{R}^{+})$ for $\delta<\frac{3}{2}$
and $f_{\theta,t}$ smoothly depends on $\theta,t$. We also write
$f_{M,\theta,t}(\zeta)=f_{M}(\zeta,\theta,t)$, as $f_{\theta,t}(r)\in
r^{\delta}L^{2}(\mathbb{R}^{+})$, $f_{M,\theta,t}$ is holomorphic on
$\\{\Im\zeta<1\\}$. We will prove by induction that $f_{M,\theta,t}(\zeta)$
has the meromorphic extension. Suppose $f_{M,\theta,t}(\zeta)$ has a
meromorphic extension to $\\{\Im\zeta<N\\}$. Then we consider the term
$(E^{\prime}f)_{M}$. As $E^{\prime}$ consists of differentials with higher
order in $r$, such as $r\partial_{\theta}$, $r^{2}\partial_{r}$, etc. For
example, we might have
$(r\partial_{\theta}f)_{M,\theta,t}(\zeta)=(\partial_{\theta}f)_{M,\theta,t}(\zeta-i)$.
Therefore, $E^{\prime}f$ is holomorphic on $\\{\Im\zeta<N+1\\}$. In addition,
as $(\rho_{M,\theta,t})(\zeta)$ is meromorphic on $\\{\Im\zeta<N+1\\}$,
$(I_{\zeta}f_{M})(\cdot,\theta,t)$ is meromorphic on $\\{\Im\zeta<N+1\\}$. In
the region $\\{N\leq\Im\zeta<N+1\\}$, $I_{\zeta}$ is invertible except at
$\Im\zeta=\frac{1}{2}+N$, which is the indicial root set. Thus $f_{M}$ has a
meromorphic extension to $\\{\Im\zeta<N+1\\}$ will extra poles induces at
$\Im\zeta=\frac{1}{2}+N$. We keep on this induction, $f_{M}$ will has a
meromorphic extension to the whole $\zeta$ plane with extra poles at the
indicial roots set $\Im\zeta=\frac{1}{2}+k$. ∎

Under the previous assumption, for the solution $\Delta f=\rho$, we might
write

$f=\sum_{k\geq 0}\Re(f_{k}(r,t)e^{i(k+\frac{1}{2})\theta}),$

then from Theorem A.1 and Theorem A.4, in a suitable coordinate, $f$, thus
$f_{k}$, has a polyhomogeneous expansion

(84) $f_{k}(r,t)\sim\sum_{l\geq 0}\sum_{0\leq p\leq
p_{l}}r^{l+\frac{1}{2}}(\log r)^{p}f_{k,l,p}(t),$

where $l,\;p_{l}\in\mathbb{N}$.

The regularity of the derivatives of $f$ in $r$ direction depends on the
appearance of the $\log r$ terms in the expansion. However, from the general
theory, we could not avoid the appearance of $\log r$ terms in the expansions,
except special cases. One could step-by-step check the appearance of the $\log
r$ terms by comparing different terms in both side of the equation. If we have
extra regularity of $f$, for example, $f\in\mathcal{C}^{2,\alpha}$, then for
(84), we could conclude that $p_{0}=p_{1}=0$.

Using the language of polyhomogeneous, Theorem 4.14 could be rephrased to be
the following:

######  Proposition A.5.

Suppose $f$ is a $\mathbb{Z}_{2}$ valued harmonic function, $\Delta f=\rho$,
suppose $\rho\in L^{2}$ and $\rho$ has a polyhomogeneous expansion with index
set $E$, then $f$ has a polyhomogeneous expansion and the expansion of $f$
could be written as

$f=\Re(Az^{\frac{1}{2}}+Bz^{\frac{3}{2}})+E,$

where $E$ has expansions

$E\sim\sum_{k\geq 2}\sum_{0\leq p\leq
p_{k}}E_{k,p}(\theta,t)r^{k+\frac{1}{2}}(\log r)^{p}.$

######  Remark.

Suppose the coordinate $\theta$ is harmonic $\Delta\theta=0$, then for the
equation $\Delta f=0$, the solution $f$ will not have $\log r$ terms in its’
expansion. However, for the equation $\Delta f=\rho$, even suppose in the
expansion of $\rho$, it doesn’t contain $\log r$ terms, $f$ could still have
$\log r$ terms in its’ expansion.

## References

  * [1] Mohammed Abouzaid and Yohsuke Imagi.  Nearby special lagrangians.  arXiv preprint arXiv:2112.10385, 2021.
  * [2] Alfred Aeppli.  Mathematical Notes: Fenchel’s Theorem as a Consequence of Schur’s.  Amer. Math. Monthly, 72(3):283–285, 1965.
  * [3] Thierry Aubin.  Nonlinear analysis on manifolds. Monge-Ampère equations, volume 252 of Grundlehren der mathematischen Wissenschaften.  Springer-Verlag, New York, 1982.
  * [4] Robert L. Bryant.  Calibrated embeddings in the special Lagrangian and coassociative cases.  volume 18, pages 405–435. 2000.  Special issue in memory of Alfred Gray (1939–1998).
  * [5] Adrian Butscher.  Regularizing a singular special Lagrangian variety.  Comm. Anal. Geom., 12(4):733–791, 2004.
  * [6] Ana Cannas da Silva.  Lectures on symplectic geometry, volume 1764 of Lecture Notes in Mathematics.  Springer-Verlag, Berlin, 2001.
  * [7] Alexandru Doicu.  Calabi-Yau structures on cotangent bundles.  J. Differential Geom., 100(3):481–489, 2015.
  * [8] Simon Donaldson.  Deformations of multivalued harmonic functions.  arXiv preprint arXiv:1912.08274, 2019.
  * [9] Simon Donaldson.  Deformations of singular sets and nash-moser theory.  2021\.
  * [10] Birte Feix.  Hyperkähler metrics on cotangent bundles.  J. Reine Angew. Math., 532:33–46, 2001.
  * [11] Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, and Kaoru Ono.  Lagrangian intersection Floer theory: anomaly and obstruction. Part I, volume 46 of AMS/IP Studies in Advanced Mathematics.  American Mathematical Society, Providence, RI; International Press, Somerville, MA, 2009.
  * [12] Reese Harvey and H. Blaine Lawson, Jr.  Calibrated geometries.  Acta Math., 148:47–157, 1982.
  * [13] Mark Haskins.  The geometric complexity of special Lagrangian $T^{2}$-cones.  Invent. Math., 157(1):11–70, 2004.
  * [14] Mark Haskins and Nikolaos Kapouleas.  Special Lagrangian cones with higher genus links.  Invent. Math., 167(2):223–294, 2007.
  * [15] Andriy Haydys.  Seiberg-Witten monopoles and flat PSL(2;$\mathbb{R}$)-connections.  arXiv preprint arXiv:2001.07589, 2020.
  * [16] Andriy Haydys and Thomas Walpuski.  A compactness theorem for the seiberg–witten equation with multiple spinors in dimension three.  Geometric and Functional Analysis, 25(6):1799–1821, 2015.
  * [17] Siqi He.  The behavior of sequences of solutions to the Hitchin-Simpson equations.  arXiv preprint arXiv:2002.08109, 2020.
  * [18] Siqi He.  Existence of nondegenerate $\mathbb{Z}_{2}$ harmonic 1-forms via $\mathbb{Z}_{3}$ symmetry.  To appear.
  * [19] Nigel Hitchin.  The self-duality equations on a Riemann surface.  Proc. London Math. Soc. (3), 55(1):59–126, 1987.
  * [20] Nigel J. Hitchin.  The moduli space of special Lagrangian submanifolds.  volume 25, pages 503–515 (1998). 1997.  Dedicated to Ennio De Giorgi.
  * [21] Yohsuke Imagi, Dominic Joyce, and Joana Oliveira dos Santos.  Uniqueness results for special Lagrangians and Lagrangian mean curvature flow expanders in $\mathbb{C}^{m}$.  Duke Math. J., 165(5):847–933, 2016.
  * [22] Dominic Joyce.  Constructing special Lagrangian $m$-folds in $\mathbb{C}^{m}$ by evolving quadrics.  Math. Ann., 320(4):757–797, 2001.
  * [23] Dominic Joyce.  Special Lagrangian $m$-folds in $\mathbb{C}^{m}$ with symmetries.  Duke Math. J., 115(1):1–51, 2002.
  * [24] Dominic Joyce.  Special Lagrangian submanifolds with isolated conical singularities. I. Regularity.  Ann. Global Anal. Geom., 25(3):201–251, 2004.
  * [25] Dominic Joyce.  Special Lagrangian submanifolds with isolated conical singularities. II. Moduli spaces.  Ann. Global Anal. Geom., 25(4):301–352, 2004.
  * [26] Dominic Joyce.  Special Lagrangian submanifolds with isolated conical singularities. III. Desingularization, the unobstructed case.  Ann. Global Anal. Geom., 26(1):1–58, 2004.
  * [27] Dominic Joyce.  Lectures on special Lagrangian geometry.  In Global theory of minimal surfaces, volume 2 of Clay Math. Proc., pages 667–695. Amer. Math. Soc., Providence, RI, 2005.
  * [28] D. Kaledin.  A canonical hyperkähler metric on the total space of a cotangent bundle.  In Quaternionic structures in mathematics and physics (Rome, 1999), pages 195–230. Univ. Studi Roma “La Sapienza”, Rome, 1999.
  * [29] Dan A. Lee.  Connected sums of special Lagrangian submanifolds.  Comm. Anal. Geom., 12(3):553–579, 2004.
  * [30] Ronnie Lee and Steven H. Weintraub.  On the homology of double branched covers.  Proc. Amer. Math. Soc., 123(4):1263–1266, 1995.
  * [31] W. B. Raymond Lickorish.  An introduction to knot theory, volume 175 of Graduate Texts in Mathematics.  Springer-Verlag, New York, 1997.
  * [32] Chien-Hao Liu and Shing-Tung Yau.  Immersed lagrangian deformations of a branched covering of a special lagrangian 3-sphere in a calabi-yau 3-fold and its deviation from joyce’s criteria: Potential image-support rigidity of a-branes that wrap around a sl $S^{3}$.  arXiv preprint arXiv:1109.1878, 2011.
  * [33] Stephen P. Marshall.  Deformations of special lagrangian submanifolds.  Phd Thesis, 2002.
  * [34] Rafe Mazzeo.  Elliptic theory of differential edge operators. I.  Comm. Partial Differential Equations, 16(10):1615–1664, 1991.
  * [35] Rafe Mazzeo, Andriy Haydys, and Ryosuke Takahashi.  $\mathbb{Z}_{2}$ harmonic spinors on three-manifolds branching along graphs: the index problem.  to appear.
  * [36] Robert C. McLean.  Deformations of calibrated submanifolds.  Comm. Anal. Geom., 6(4):705–747, 1998.
  * [37] Richard B. Melrose.  The Atiyah-Patodi-Singer index theorem, volume 4 of Research Notes in Mathematics.  A K Peters, Ltd., Wellesley, MA, 1993.
  * [38] Charles B. Morrey, Jr.  Multiple integrals in the calculus of variations.  Die Grundlehren der mathematischen Wissenschaften, Band 130. Springer-Verlag New York, Inc., New York, 1966.
  * [39] Christoforos Neofytidis.  Branched coverings of simply connected manifolds.  Topology Appl., 178:360–371, 2014.
  * [40] Johannes Nordstrom.  Desingularising intersecting associatives.  in preparation.
  * [41] Sema Salur.  Deformations of special Lagrangian submanifolds; an approach via Fredholm alternative.  In Proceedings of Gökova Geometry-Topology Conference 2005, pages 157–164. Gökova Geometry/Topology Conference (GGT), Gökova, 2006.
  * [42] Jake P. Solomon.  Involutions, obstructions and mirror symmetry.  Adv. Math., 367:107107, 52, 2020.
  * [43] Matthew B. Stenzel.  Ricci-flat metrics on the complexification of a compact rank one symmetric space.  Manuscripta Math., 80(2):151–163, 1993.
  * [44] Ryosuke Takahashi.  The moduli space of S 1-type zero loci for Z/2-harmonic spinors in dimension 3.  Harvard University, 2015.
  * [45] Clifford Henry Taubes.  Compactness theorems for sl (2; c) generalizations of the 4-dimensional anti-self dual equations.  arXiv preprint arXiv:1307.6447, 2013.
  * [46] Clifford Henry Taubes.  ${\rm PSL}(2;\mathbb{C})$ connections on 3-manifolds with ${\rm L}^{2}$ bounds on curvature.  Camb. J. Math., 1(2):239–397, 2013.
  * [47] Clifford Henry Taubes.  The zero loci of z/2 harmonic spinors in dimension 2, 3 and 4.  arXiv preprint arXiv:1407.6206, 2014.
  * [48] Chung-Jun Tsai and Mu-Tao Wang.  Mean curvature flows in manifolds of special holonomy.  Journal of Differential Geometry, 108(3):531–569, 2018.
  * [49] Thomas Walpuski and Boyu Zhang.  On the compactness problem for a family of generalized seiberg-witten equations in dimension three.  arXiv preprint arXiv:1904.03749, 2019.
  * [50] Shuguang Wang.  Moduli spaces over manifolds with involutions.  Math. Ann., 296(1):119–138, 1993.
  * [51] Kentaro Yano and Shigeru Ishihara.  Tangent and cotangent bundles: differential geometry.  Pure and Applied Mathematics, No. 16. Marcel Dekker, Inc., New York, 1973\.