A new probabilistic analysis of the yard-sale model
Christoph Börgers1 and Claude Greengard2

1 Department of Mathematics, Tufts University, Medford, MA

2 Two Sigma Investments, LP, New York, NY, and

Courant Institute of Mathematical Sciences, New York University, New York, NY

> Abstract. In Chakraborti’s yard sale model of an economy [6], identical
> agents engage in trades that result in wealth exchanges, but conserve the
> combined wealth of all agents and each agent’s expected wealth. In this
> model, wealth condensation, that is, convergence to a state in which one
> agent owns everything and the others own nothing, occurs almost surely. We
> give a proof of this fact that is much shorter than existing ones and
> extends to a modified model in which there is a wealth-acquired advantage,
> i.e., the wealthier of two trading partners is more likely to benefit from
> the trade.

##  1 Background

The yard-sale model, first proposed in [6], is a caricature of a set of agents
trading with each other. The agents are identical, and there are $N$ of them.
Time proceeds in discrete steps labeled $0,1,2,\ldots$, and in the $n$-th step
the $i$-th agent has wealth $X_{n}^{i}\geq 0$. We will write

$X_{n}=\left[X_{n}^{i}\right]_{1\leq i\leq N}\in\mathbb{R}^{N}.$

Without loss of generality, we assume

$\sum_{i=1}^{N}X_{0}^{i}=1.$

We think of $X_{0}$ as deterministic. The $X_{n}$ with $n\geq 1$ are random,
defined inductively as follows. Given $X_{n-1}$, choose a random pair of
integers, $(\mu_{n},\nu_{n})$, uniformly distributed in the set of all pairs
$(i,j)$ with $1\leq i,j\leq N$, $i\neq j$. Without loss of generality, assume
$X_{n-1}^{\mu_{n}}\leq X_{n-1}^{\nu_{n}}$. So agent $\nu_{n}$ is, at time
$n-1$, at least as wealthy as agent $\mu_{n}$. Imagine that agents $\mu_{n}$
and $\nu_{n}$ now engage in an economic transaction. As a result of errors
(perhaps over- or underpayments occurring because of lack of complete
information for instance), a random fraction $B_{n}\in(0,1)$ of the wealth of
agent $\mu_{n}$ (the poorer of the two) is transferred either from agent
$\mu_{n}$ to agent $\nu_{n}$, or vice versa. The $B_{n}$ are assumed to be
identically distributed. The direction of the transfer is determined by a fair
coin flip. Formally, let $V_{n}=1$ with probability $1/2$, and $V_{n}=-1$
otherwise. Then

$X_{n}^{\mu_{n}}=X_{n-1}^{\mu_{n}}+V_{n}~{}\\!B_{n}~{}\\!X_{n-1}^{\mu_{n}}~{}~{}~{}~{}~{}~{}\mbox{and}~{}~{}~{}~{}~{}~{}X_{n}^{\nu_{n}}=X_{n-1}^{\nu_{n}}-V_{n}~{}\\!B_{n}~{}\\!X_{n-1}^{\mu_{n}}.$

The pairs $(\mu_{n},\nu_{n})$, the fractions $B_{n}$, and the signs $V_{n}$
are assumed to be independent of each other and of the $X_{k}$, $B_{k}$, and
$V_{k}$ with $k\leq n-1$. In reference [6], and in much of the literature on
the yard-sale model, $B_{n}$ is assumed to be a deterministic number
$\beta\in(0,1)$. We use a random $B_{n}\in(0,1)$ more generally. Notice that
wealth is conserved:

$\sum_{i=1}^{N}X_{n}^{i}=1$

for all $n$. The model is known to have the following surprising property,
which we will call the yard-sale theorem.

Yard-Sale Theorem. (a) There almost surely exists an $i\in\\{1,\ldots,N\\}$
with $\displaystyle{\lim_{n\rightarrow\infty}X_{n}^{i}=1.}$

(b) For all $i\in\\{1,\ldots,N\\}$,

$P\left(\lim_{n\rightarrow\infty}X_{n}^{i}=1\right)=X_{0}^{i}.$ (1)

In the limit, one agent owns everything. This sort of maximal inequality is
called wealth condensation in the literature. In the yard-sale model, wealth
condensation is the inescapable result of random, statistically unbiased
interactions.

For a version of the model in which there is a continuum of agents, rather
than a finite number, a result of this kind was proved by Boghosian et al [3].
Chorro [7] pointed out that the theorem as stated above is an immediate
consequence of Doob’s martingale convergence theorem: For a fixed $i$, the
sequence $\\{X_{n}^{i}\\}_{n=0,1,2,\ldots}$ is a bounded martingale, and
therefore must converge. It is clear from the definition of the model that the
$X_{n}^{i}$ cannot all converge unless one converges to 1 and the others
converge to 0. Equation (1) follows from the fact that
$E(X_{n}^{i})=E(X_{0}^{i})$ for all $n$ and $i$.

Some interesting variations can be handled immediately using the same
reasoning. For instance, different agents can be assumed to have different
degrees of risk tolerance [5]. The amount of wealth transferred during the
trade between agents $\mu_{n}$ and $\nu_{n}$ might be taken to be
$V_{n}~{}\\!B_{n}~{}\\!\lambda_{\mu_{n}}~{}\\!X_{n-1}^{\mu_{n}}$, where the
$\lambda_{i}$, $1\leq i\leq N$, are fixed numbers in $(0,1)$; if $\lambda_{i}$
is smaller, agent $i$ is more risk averse. Obviously but remarkably, eq. (1)
still holds for the modified model; risk aversion does not make an agent less
or more likely to end up owning everything.

Cardoso et al. [5] have recently proposed a different argument, based on the
Gini index, to derive related results for a broader class of models. Their
analysis relies on what they call the fair rule hypothesis [5, Equation (8)].
For wealth-conserving models, it is the martingale property. Most, but not
all, examples in [5] are wealth-conserving. The result in [5] is that the Gini
index is monotonically increasing, and stationary if and only if it is 1.

An interesting extension is obtained by adding a wealth-acquired advantage
[4]. The coin flips that determine in which direction the wealth will flow in
each interaction is biased in favor of the wealthier agent: $V_{n}=1$ with
probability $p$, and $V_{n}=-1$ otherwise, where $p$ is no longer required to
be $\frac{1}{2}$, but is allowed to be anywhere in
$\left[\frac{1}{2},1\right)$. One should certainly expect wealth condensation
for $p>\frac{1}{2}$ if there is wealth condensation for $p=\frac{1}{2}$.
However, proofs that rely on the martingale property no longer work; the model
is still wealth-conserving, but $\\{X_{n}^{i}\\}_{n\geq 0}$ is no longer a
martingale, nor a sub- or super-martingale.

To the model with a (possible) wealth-acquired advantage, Boghosian et al. [2,
1] added wealth taxation. Each agent is taxed a fraction $\chi\in(0,1)$ of
their fortune in each time step, uniformly re-distributing the total amount
taken in:

$\tilde{X}_{n}^{\mu_{n}}=X_{n-1}^{\mu_{n}}+V_{n}~{}\\!B_{n}~{}\\!X_{n-1}^{\mu_{n}}~{}~{}~{}~{}~{}~{}\mbox{and}~{}~{}~{}~{}~{}~{}\tilde{X}_{n}^{\nu_{n}}=X_{n-1}^{\nu_{n}}-V_{n}~{}\\!B_{n}~{}\\!X_{n-1}^{\mu_{n}},$
$X_{n}^{i}=(1-\chi)\tilde{X}_{n}^{i}+\frac{\chi}{N}~{}~{}~{}\mbox{for all
$i\in\\{1,\ldots,N\\}$}.$

It is clear that wealth taxation, no matter how small $\chi\in(0,1)$ may be,
prevents wealth condensation: We now have

$X_{n}^{i}\geq\frac{\chi}{N}$

for all $n\geq 1$ and $i\in\\{1,\ldots,N\\}$, which precludes the existence of
the limits $\lim_{n\rightarrow\infty}X_{n}^{i}$.

For a version of the model with wealth-acquired advantage and taxation in
which there is a continuum of agents, Boghosian et al [4] have shown that for
each $p\in\left(\frac{1}{2},1\right)$, there is a threshold value $\chi_{c}$
depending on $p$, such that there will be oligarchy for $\chi<\chi_{c}$, but
not for $\chi>\chi_{c}$. Here oligarchy means that a vanishingly small
fraction of the population will own, in the long run, a non-vanishing fraction
of total wealth.

In this brief note, we propose a new probabilistic proof of part (a) of the
Yard-Sale Theorem, which applies also if $p\in\left(\frac{1}{2},1\right)$. Our
proof does not use the martingale property, and therefore applies when
$p\in\left(\frac{1}{2},1\right)$. The tools used in our analysis are much
lighter than those used in previously published proofs of the Yard-Sale
Theorem or similar results. Instead of the Gini index, we use $\|X_{n}\|^{2}$
as the measure of concentration, where $\|\cdot\|$ denotes the Euclidean norm.
We note that the idea of using the Euclidean norm of a probability vector as a
measure of concentration is not new; it appears in quantum physics [8],
political science [9], ecology [10], and antitrust regulation [11].

##  2 Proof of almost sure wealth condensation

We consider, from here on, the yard-sale model, possibly with a wealth-
acquired advantage: $\frac{1}{2}\leq p<1$, but without taxation. Write
$M_{n}=\max_{i}X_{n}^{i}$. We will prove $M_{n}\rightarrow 1$ almost surely,
which is part (a) of the Yard-Sale Theorem. We have no analogue of part (b)
for $p>\frac{1}{2}$.

We begin with a preliminary calculation in which we determine the expected
change in the concentration measure $\|X_{n}\|^{2}$ in a single step. (As
before, $\|\cdot\|$ is the Euclidean norm.) Suppose that $X_{n-1}^{i}$, $1\leq
i\leq N$, are given, $\mu_{n}$ and $\nu_{n}$ have been chosen, with
$X_{n-1}^{\mu_{n}}\leq X_{n-1}^{\nu_{n}}$, and $B_{n}$ has been chosen as
well. We write $\omega_{n}=B_{n}~{}\\!X_{n-1}^{\mu_{n}}$; this is the amount
of wealth at stake in the trade between agents $\mu_{n}$ and $\nu_{n}$. We
also write $p=\frac{1}{2}+\delta$, with $0\leq\delta\leq\frac{1}{2}$. Then the
(conditional) expectation of $\|X_{n}\|^{2}-\|X_{n-1}\|^{2}$ equals

$\left(\frac{1}{2}+\delta\right)\left(\left(X_{n-1}^{\mu_{n}}-\omega_{n}\right)^{2}-\left(X_{n-1}^{\mu_{n}}\right)^{2}+\left(X_{n-1}^{\nu_{n}}+\omega_{n}\right)^{2}-\left(X_{n-1}^{\nu_{n}}\right)^{2}\right)+$
$\left(\frac{1}{2}-\delta\right)\left(\left(X_{n-1}^{\mu_{n}}+\omega_{n}\right)^{2}-\left(X_{n-1}^{\mu_{n}}\right)^{2}+\left(X_{n-1}^{\nu_{n}}-\omega_{n}\right)^{2}-\left(X_{n-1}^{\nu_{n}}\right)^{2}\right)=$
$2\omega_{n}^{2}+4\delta\omega_{n}\left(X_{n-1}^{\nu_{n}}-X_{n-1}^{\mu_{n}}\right)\geq
2\omega_{n}^{2}.$

This implies

$E\left(\|X_{n}\|^{2}\right)-E\left(\|X_{n-1}\|^{2}\right)\geq
2E(\omega_{n}^{2})$ (2)

where $E$ denotes unconditional expectations.

We now proceed to the main argument, which makes use of inequality (2) at the
end. To show $M_{n}\rightarrow 1$ almost surely, it is enough to show that
$\omega_{n}\rightarrow 0$ almost surely. Therefore we have to show that for
any $\epsilon>0$, it almost surely happens only finitely many times that
$\omega_{n}\geq\epsilon$. By the Borel-Cantelli lemma, it is enough to show

$\sum_{n=1}^{\infty}P(\omega_{n}\geq\epsilon)<\infty$

for any $\epsilon>0$. Since $E(\omega_{n}^{2})\geq
P(\omega_{n}\geq\epsilon)\epsilon^{2},$ we have

$\sum_{n=1}^{\infty}P(\omega_{n}\geq\epsilon)\leq\frac{1}{\epsilon^{2}}\sum_{n=1}^{\infty}E(\omega_{n}^{2}).$

Using (2),

$\sum_{n=1}^{\infty}E(\omega_{n}^{2})\leq\frac{1}{2}\sum_{n=1}^{\infty}\left(E(\|X_{n}\|^{2}-E(\|X_{n-1}\|^{2})\right)\leq\frac{1}{2}\left(\limsup_{n\rightarrow\infty}E(\|X_{n}\|^{2})-E(\|X_{0}\|^{2})\right)\leq\frac{1}{2}.$

This completes the proof.

## References

  * [1] B. Boghosian.  Fokker-Planck description of wealth dynamics and the origin of Pareto’s law.  Int. J. Mod. Phys. C, 25(12):1441008, 2014.
  * [2] B. Boghosian.  Kinetics of wealth and the Pareto law.  Phs. Rev. E, 89, 042804, 2014.
  * [3] B. Boghosian, M. Johnson, and J. Marcq.  An H theorem for Boltzmann’s equation for the yard-sale model of asset exchange.  J. Stat. Phys., 161:1339–1350, 2015.
  * [4] B. M. Boghosian, A. Devitt-Lee, M. Johnson, J. A. Marcq, and H. Wang.  Oligarchy as a phase transition: The effect of wealth-attained advantage in a Fokker-Planck description of asset exchange.  Physica A, 476:15–37, 2017.
  * [5] B.-H. F. Cardoso, S. Gonçalves, and J. R. Iglesias.  Why equal opportunities lead to maximum inequality? The wealth condensation paradox generally solved.  Chaos Solit. Fractals, 168, 113181, 2023.
  * [6] A. Chakraborti.  Distributions of money in model markets of economy.  Int. J. Mod. Phs. C, 13(10):1315–1321, 2002.
  * [7] C. Chorro.  A simple probabilistic approach of the yard-sale model.  Stat. Probab. Lett., 112:35–40, 2016.
  * [8] B. Kramer and A. MacKinnon.  Localization: theory and experiment.  Rep. Prog. Phys., 56:1469–1564, 1993.
  * [9] M. Laakso and R. Taagepera.  “Effective” number of parties: a measure with applications to West Europe.  Comp. Polit. Stud., 12(1):3–27, 1979.
  * [10] E. H. Simpson.  Measurement of diversity.  Nature, 163, 1949.
  * [11] United States Department of Justice.  Herfindahl-Hirschmann Index.  https://www.justice.gov/atr/herfindahl-hirschman-index.


# Compactness of Hamiltonian Stationary Lagrangian submanifolds in symplectic
manifold

Jingyi Chen, John Man Shun Ma  Jingyi Chen
Department of Mathematics
The University of British Columbia
1984 Mathematics Rd, Vancouver, BC V6T 1Z2, Canada<EMAIL_ADDRESS>John Man Shun Ma
Department of Mathematical Sciences
University of Copenhagen
Universitetsparken 5
DK-2100 Copenhagen Ø, Denmark<EMAIL_ADDRESS>
###### Abstract.

In this work, we prove a compactness theorem on the space of all Hamiltonian
stationay Lagrangian submanifolds in a compact symplectic manifold with
uniform bounds on area and total extrinsic curvature. This generalizes the
compactness theorems in [8] and [6].

Chen is partially supported by an NSERC Discovery Grant (22R80062). Ma is
partially supported by DFF Sapere Aude 7027-00110B, by CPH-GEOTOP-DNRF151 and
by CF21-0680 from respectively the Independent Research Fund Denmark, the
Danish National Research Foundation and the Carlsberg Foundation.

##  1\. Introduction

Let $(M,\omega,h,J)$ be a $2n$-dimensional symplectic manifold with a
symplectic 2-form $\omega$, an almost complex structure $J$ and a compatible
metric $h$. An immersed submanifold $L$ in $M$ is Lagrangian if $\dim L=n$ and
$\omega|_{L}=0$, and a Lagrangian immersion is Hamiltonian stationary (HSL) if
it is a critical point of the volume functional among all Hamiltonian
variations. The notion was first introduced in [15], [16]. A compact and
graded Lagrangian in a Calabi-Yau manifold is Hamiltonian stationary if and
only if it is special [19]. Thus HSL submanifolds are natural generalization
of special Lagrangians submanifolds for general symplectic manifolds.

Examples of HSL submanifolds include the totally geodesic $\mathbb{RP}^{n}$ in
$\mathbb{CP}^{n}$ and the flat tori
$\mathbb{S}^{1}(a_{1})\times\cdots\times\mathbb{S}^{1}(a_{n})$ in
$\mathbb{C}^{n}$ [15], [16]. In two dimension, Schoen and Wolfson constructed
in [17] HSL surfaces with conical singularity in any Kähler surfaces, which
are also area minimizer in its Lagrangian homology class. On the other hand,
using perturbation methods, Joyce, Lee and Schoen proved in [11] the existence
of closed HSL submanifolds in every compact symplectic manifolds (see also the
previous works [4, 13] and a family version in [14]). A large classes of
examples of HSLs are constructed using techniques in integrable system. We
refer the readers to the bibliography of [6] for a more comprehensive list of
references.

Using the regularity theory developed in [7], Chen and Warren obtain the
smoothness estimates and small Willmore energy regularity in [8] and prove a
compactness theorem for HSL submanifolds in $\mathbb{C}^{n}$ with uniformly
bounded areas and total extrinsic curvatures in $\mathbb{C}^{n}$. The
regularity and compactness results in [7, 8] rely on the assumption that the
ambient space is $\mathbb{C}^{n}$ since it is used, in an essential way, that
the Lagrangian phase angle $\Theta$ can be written as
$\arctan\lambda_{1}+\dots+\arctan\lambda_{n}$ for the graphic representation
$(x,Du)$, where $\lambda_{i}$’s are the eigenvalues of $D^{2}u$.

In another direction, using the bubble tree convergence for conformal mappings
with uniformly bounded area and Willmore energy, we proved in [6] a
compactness theorem on the space of compact HSL surfaces with bounded area,
genus and Willmore energy in a Kähler surfaces. A sequence of such HSL
surfaces would converges smoothly away from finitely many points to a branched
HSL immersion. Unlike the case for $\mathbb{C}^{n}$ where one make use of the
Lagrangian angle, in [6] we exploit the fact that the mean curvature one form
of a HSL immersion in a Kähler manifold satisfies a first order elliptic
system of ”Hodge type”; this is also essential in the regularity results in
[17] for a Lagrangian area minimizers in two dimensions.

Our main result is

######  Theorem 1.1.

Let $(M,\omega,h)$ be a compact $2n$-dimensional symplectic manifold, where
$h$ is a Hermitian metric compatible with $\omega$. Let $C_{V}$, $C_{A}$ be
positive constants and let $(L_{k})_{k=1}^{\infty}$ be a sequence of connected
compact Hamiltonian stationary Lagrangian immersion in $M$ so that

(1.1) $\displaystyle\operatorname{Vol}(L_{k})\leq C_{V},\ \ \
\|A_{k}\|_{L^{n}}\leq C_{A},\ \ \forall k\in\mathbb{N}.$

Then either $(L_{k})$ converges to a point, or there is a finite set $S$ so
that a subsequence $(L_{k_{i}})$ of $(L_{k})$ converges smoothly locally
graphically to a Hamiltonian stationary Lagrangian immersion $L$ on
$M\setminus S$ and

(1.2) $\operatorname{Vol}(L)=\lim_{i\to\infty}\operatorname{Vol}(L_{k_{i}}).$

Also, the closure $\overline{L}$ in $M$ is connected and admits a structure of
a Lagrangian varifold, and is Hamiltonian stationary in the sense that the
generalized mean curvature vector $\vec{H}$ of the varifold
$(\overline{L},\mu_{L})$ satisfies

(1.3) $\displaystyle\int_{M}h(J\nabla^{M}f,\vec{H})d\mu_{L}=0,\ \ \forall f\in
C^{\infty}_{c}(M).$

This generalizes the compactness theorems in [8], [6] to HSLs in any compact
symplectic manifolds with metrics compacitble with the symplectic structures
and the almost complex structures. The main step in proving compactness is to
derive a local a-priori estimates for HSLs with small total curvature. In [8],
[6], this is done by representating the Hamiltonian stationary conditions as a
coupled elliptic systems of lower order. This simplification is not available
for HSLs in a general symplectic manifold. Therefore, the methods in [8], [6]
do not directly yield a proof of Theorem 1.1. In a Darboux coordinates, if one
represents the HSL as a gradient graph $(x,Du(x))$ of a function $u$, then the
Hamiltonian stationary condition is captured by a fourth order PDE on $u$. In
a recent work [2], Bhattachaya, Chen and Warren have studied a fourth order
elliptic PDE of double divergence form and derive a regularity results.
Together with the constructions of Darboux coordinates with estimates in [11]
(See also subsecion 2.2), it is proved that a $C^{1}$ HSL in a general
symplectic manifolds is smooth. In this work, we apply the results in [2] to
derive a-priori estimates (Theorem 4.2) and a $\epsilon$-regularity result
(Theorem 4.7) for HSL submanifolds, which is essential to the proof of Theorem
1.1.

The paper is organized as follows. In section 2, we discuss some background in
Lagrangian submanifolds, which includes the definition of HSL and the Darboux
coordinates with estimates constructed in [11]. In section 3, we use the
regularity results in [2] to derive local $C^{k}$ estimates for functions
which satisfy a fourth order elliptic equation in double divergence form. In
section 4, we prove a $\epsilon$-regularity theorem. Theorem 1.1 is proved in
sections 5 and 6.

In the following, given any vector space $W$ with a metric $h$, we denote
$|w|_{h}=\sqrt{h(w,w)}$ for $w\in W$. When $W$ is the Euclidean space, we
write $|w|=|w|_{h_{0}}$, where $h_{0}$ is the Euclidean metric on $W$. Given
any open subset $\Omega$ of $\mathbb{R}^{n}$ and any function
$u:\Omega\to\mathbb{R}$ on $\Omega$, we use $\|u\|_{C^{k,\alpha}(\Omega)}$,
$\|u\|_{L^{p}(\Omega)}$ and $\|u\|_{W^{k,p}(\Omega)}$ to denote respectively
the $C^{k,\alpha}$, $L^{p}$ and $W^{k,p}$ norms of $u$. For any
$k\in\mathbb{N}$, $x\in\mathbb{R}^{k}$ and $r>0$, the open ball with radius
$r$ and center $x$ in $\mathbb{R}^{k}$ is denoted $B^{k}_{r}(x)$. We also
write $B^{k}_{r}$ for $B^{k}_{r}(0)$.

##  2\. Background

###  2.1. Hamiltonian stationary Lagrangian submanifolds

Let $(M,h)$ be a Riemannian manifold. Unless otherwise specified, throughout
this paper a submanifold of $M$ is an immersed submanifold, that is, a
submanifold of $M$ is a pair $(L,\iota)$, where $L$ is a subset of $M$ and
$\iota:N\to M$ is an immersion with $\iota(N)=L$. An immersed submanifold is
compact (resp. connected), if $N$ is compact (resp. connected), and is proper
if $\iota$ is a proper map.

Assume that $M$ is compact. By the Nash embedding theorem, there is an
isometric embedding of $(M,h)$ into $\mathbb{R}^{K}$ for some positive integer
$K$. Fixing such an embedding, the definition of varifolds on $M$ is given in
[1]. Let $U$ be an open subset of $M$, and let $L$ be a properly immersed
$k$-dimensional submanifold of $U$ given by the immersion $\iota:N\to U$. Let
$g=\iota^{*}h$ be the induced metric on $N$ and let $dV_{g}$ be the volume
form. The volume of a submanifold $L$ is

(2.1) $\operatorname{Vol}(L)=\int_{N}dV_{g}.$

Let $\pi_{k}:G_{k}M\to M$ be the Grassmann bundle on $M$, where each fiber at
$x\in M$ is the Grassmann manifold of $k$-dimensional subspaces of $T_{x}M$.
Given any $k$-dimensional immersed submanifold $L$ of $M$, let $G_{\iota}:N\to
G_{k}M$ be the Gauss map given by $x\mapsto(\iota_{*})_{x}T_{x}N$. Then
$(L,\iota)$ is given a structure of a varifold $\mu_{L}$ by pushing forward:
$|L|:=(G_{\iota})_{\sharp}dV_{g}$, or

(2.2) $|L|(f):=\int_{N}f\circ G_{\iota}\ dV_{g},\ \ \forall f\in C(G_{k}M).$

For any integral varifold $V$, the weight measure $\mu_{V}$ on $M$ is given by

(2.3) $\int_{M}\phi d\mu_{V}:=\int_{G_{k}M}\phi(x)dV,\ \ \forall\phi\in C(M).$

Let $\mu_{L}:=mu_{|L|}$. Clearly $\mu_{L}$ and $|L|$ depend not only on $L$
but also on $\iota$.

Let $(M,\omega,h,J)$ be a $2n$-dimensional symplectic manifold, where $\omega$
is the symplectic form, $J$ is an almost complex structure and $h$ is a
Riemannian metric. We assume that $\omega,J,h$ satisfy the compatibility
conditions

(2.4) $h(X,Y)=\omega(X,JY),\ \ \ h(JX,JY)=h(X,Y)$

for all tangent vectors $X,Y$ at the same point.

An $n$-dimensional immersed submanifold $L$ in $M$ given by an immersion
$\iota$ is called Lagrangian if $\iota^{*}\omega=0$. Given an immersed
Lagrangian submanifold $L$, (2.4) implies that $J$ maps tangent vectors of $L$
to normal vectors. Thus the second fundamental form of $L$ can be written as

(2.5) $\displaystyle A:T_{p}L\times T_{p}L\times T_{p}L\to\mathbb{R},\ \ \
A(X,Y,Z)=h(\overline{\nabla}_{X}\widetilde{Y},JZ)=\omega(Z,\overline{\nabla}_{X}\widetilde{Y}),$

where $\widetilde{Y}$ is any local extension of $Y$. The mean curvature one
form of $L$ is a one form on $N$ defined by

(2.6) $\alpha_{x}(X)=\omega_{\iota(x)}(\iota_{*}X,\vec{H}(x))$

for any $x\in N$ and $X\in T_{x}N$. Here $\vec{H}$ is the mean curvature
vector of the immersion $\iota:N\to M$.

Next we give the definition of a Hamiltonian stationary Lagrangian immersion.
As in [8], we first give the definition for general Lagrangian integral
$n$-varifolds (An integral $n$-varifold $V$ on $(M,\omega,h)$ is called
Lagrangian if $\omega|_{T_{x}V}=0$ for $V$-almost every $x$).

######  Definition 2.1.

A Lagrangian integral $n$-varifold $V$ in $U$ is called Hamiltonian stationary
Lagrangian (HSL) in $U$ if

(2.7) $\displaystyle\int_{U}h(J\nabla^{M}f,\vec{H})d\mu_{V}=0,\ \ \forall f\in
C^{\infty}_{c}(U),$

where $\vec{H}$ is the generalized mean curvature vector of $V$.

######  Definition 2.2.

Let $L$ be a Lagrangian immersion into an open subset $U$ of $M$ given by a
proper immersion $\iota$, and let $|L|$ be the integral $n$-varifold
associated with $L$ as in (2.2). $L$ is called a Hamiltonian stationary
Lagrangian (HSL) immersion in $U$ if $|L|$ is HSL as in Definition 2.1.

Let $\iota:N\to U$ be a proper Lagrangian immersion. For any $f\in
C^{\infty}_{c}(U)$, let $\bar{f}=f\circ\iota$. By (2.3), (2.4) and (2.6), if
the immersed submanifold $(L,\iota)$ is HSL in $U$, then

$\int_{N}(d\bar{f},\alpha)_{g}dV_{g}=0,$

where $\alpha$ is the mean curvature one form of $L$ defined in (2.6). Let $L$
be an embedded Lagrangian submanifold on $M$. Since every smooth function with
compact support $\psi$ on $L$ can be extended to $f\in C^{\infty}_{c}(M)$, $L$
is HSL if and only if

$\int_{N}(d\psi,\alpha)_{g}dV_{g}=0,\ \ \ \forall\psi\in C^{\infty}_{c}(N).$

This implies that $\alpha$ is co-closed. That is,

(2.8) $d^{*}\alpha=0,$

where $d^{*}=\pm*d*$ is the formal adjoint of the exterior derivative $d$ with
respect to the metric $g$.

Let $L$ be an immersed submanifold in $M$ given by an immersion $\iota:N\to M$
and let $U\subset M$ be an open set in $M$. Decompose $\iota^{-1}(U)$ as
connected components

(2.9) $\displaystyle\iota^{-1}(U)=\bigsqcup_{i}V_{i}.$

If $\iota|_{V_{i}}$ is an embedding for each $i$, we say that each
$L_{i}=\iota(V_{i})$ is an embedded connected component of $U\cap L$ and that
$\iota$ splits into embedded connected components on $U$.

As in [8, Proposition 2.2], if $L$ is a proper immersion, for each $p\in L$
there is an open neighborhood $U$ of $p$ so that $L\cap U$ splits into
finitely many embedded connected components. We will have a more precise
statement in Proposition 4.6.

######  Proposition 2.3.

Let $L$ be a Lagrangian submanifold in an open subset $U$ of $M$ given by a
proper immersion $\iota:N\to U$. Then $L$ is HSL as in Definition 2.2 if and
only if its mean curvature one form $\alpha$ is co-closed.

###### Proof.

The proof is similar to that of [8, Proposition 2.5 (1)] and we only sketch
the proof for the direction $(\Rightarrow)$. Let $p\in L$. By [8, Proposition
2.2], let $W$ be an open neighborhood of $p$ in $U$ so that $\iota^{-1}(W)$
splits into finitely many connected components
$\iota|_{E_{i}}:E_{i}\to\Sigma_{i}$, where $\Sigma_{i}=\iota(E_{i})$. We
recall that $q\in W\cap L$ is called an embedded point of $L$ if there is an
open neighborhood $V_{q}$ of $q$ in $W$ so that $L\cap V_{q}$ is an embedded
submanifold. Let $\mathscr{E}\subset L\cap W$ be the set of all embedded
points of $L$. Using embeddedness, one can argue as in above that
$d^{*}\alpha=0$ on $\iota^{-1}(\mathscr{E})$ using Definition 2.2. For each
$i$, one can prove that $\mathscr{E}\cap\Sigma_{i}$ is dense in $\Sigma_{i}$.
Hence $d^{*}\alpha=0$ on $\iota^{-1}(W)$. Since $p\in L$ is arbitrary,
$\alpha$ is co-closed. ∎

In particular, the Hamiltonian stationary condition holds on each embedded
connected components.

######  Corollary 2.4.

Let $L$ be a properly immersed HSL in an open subset $U$ of $M$. Let $W$ be an
open subset of $U$ such that $W\cap L$ splits into embedded connected
components $\iota|_{E_{i}}:E_{i}\to\Sigma_{i}$. Then for each $i$,

(2.10) $\int_{W}h(J\nabla^{M}f,\vec{H}_{i})d\mu_{\Sigma_{i}}=0,\ \ \ \forall
f\in C^{\infty}_{c}(W).$

Here $\vec{H}_{i}$ is the mean curvature vector of $\Sigma_{i}$.

###  2.2. Darboux coordinates with estimates

In this subsection we recall the Darboux coordinates with estimates
constructed in [11]. Let $(M,\omega)$ be a compact symplectic $2n$-manifold,
and let $h$ be a Riemannian metric on $M$ compatible with $\omega$. Let
$\mathbb{U}$ be the $U(n)$ frame bundle of $M$: that is, for each $p\in M$,
the fiber $\mathbb{U}_{p}$ consists of unitary linear mappings
$v:\mathbb{R}^{2n}\to T_{p}M$.

We recall Proposition 3.2 in [11]:

######  Proposition 2.5.

For small $\epsilon>0$, we can choose a family of embeddings
$\Upsilon_{p,v}:B^{2n}_{\epsilon}\to M$ depending smoothly on
$(p,v)\in\mathbb{U}$ such that for all $(p,v)\in\mathbb{U}$, we have

  * (i)

$\Upsilon_{p,v}(0)=p$ and $d\Upsilon_{p,v}|_{0}=v:\mathbb{R}^{2n}\to T_{p}M$;

  * (ii)

$\Upsilon_{p,v\circ\gamma}\cong\Upsilon_{p,v}\circ\gamma$ for all $\gamma\in
U(n)$;

  * (iii)

$\Upsilon^{*}_{p,v}(\omega)=\omega_{0}=\frac{i}{2}\sum_{j=1}^{n}dz_{j}\wedge
d\bar{z}_{j}$; and

  * (iv)

$\Upsilon^{*}_{p,v}(h)=h_{0}+O(|z|)=\sum_{j=1}^{n}|dz_{j}|^{2}+O(|z|)$.

From now on, by scaling $\omega,h$ if necessary, we assume $\epsilon=2$ in
Proposition 2.5.

######  Definition 2.6.

Let $p\in M$ and $0<r\leq 1$. The Darboux ball at $p$ with radius $r$ is

(2.11) $\displaystyle\mathscr{B}_{r}(p):=\Upsilon_{p,v}(B^{2n}_{r}).$

By (ii) in Proposition 2.5, $\mathscr{B}_{r}(p)$ is well-defined, independent
of $v\in\mathbb{U}_{p}$. For each $R\geq 1$ and $0<t\leq R^{-1}$, we define
$t:B^{2n}_{R}\to B^{2n}_{1}$ as $z\mapsto tz$. The following is proved in [11,
Proposition 3.4].

######  Proposition 2.7.

There exists $K_{0},K_{1},K_{2},\cdots,$ depending only on $(M,\omega,h)$ such
that for all $(p,v)\in U$ and $t\in(0,R^{-1}]$, the metric
$h^{t}_{p,v}=t^{-2}(\Upsilon_{p,v}\circ t)^{*}h$ satisfies on $B^{2n}_{R}$

(2.12) $\displaystyle|h^{t}_{p,v}-h_{0}|\leq K_{0}t\ \text{ and }\
|\partial_{j}h^{t}_{p,v}|\leq K_{j}t^{j}\ \text{ for }j=1,2,\cdots.$

Again, by scaling $\omega,h$ if necessary, we assume that

(2.13)
$\displaystyle\max\\{K_{0},K_{1}\\}\leq\epsilon_{1}<\max\\{1/2,\epsilon_{1}(1/2,n)\\},$

where $\epsilon_{1}(1/2,n)$ is given in [2, Lemma 3.2]. By the first
inequality in (2.12) and $K_{0}<1/2$,

(2.14) $\displaystyle B^{M}_{r/2}(p)\subset\mathscr{B}_{r}(p)\subset
B^{M}_{2r}(p)$

for all $p\in M$ and $r\in(0,1)$, where $B^{M}_{s}(p)$ is the metric ball with
radius $s$ in $(M,h)$.

###  2.3. Convergence of immersed submanifolds

We recall the definition of smooth convergence as local graphs of a sequence
of immersed submanifolds introduced in [8] (see also [9], [18] for similar
definitions for sequences of embedded hypersurfaces).

######  Definition 2.8.

Let $(\Sigma_{k})_{k=1}^{\infty}$ be a sequence of properly immersed
submanifolds in an open subset $U$ of a smooth manifold $M$ given by a proper
immersion $\iota_{k}:N_{k}\to U$ for each $k$. Let $S$ be an integral varifold
in $(M,h)$ and $m\in\mathbb{N}\cup\\{\infty\\}$. We say that
$(\Sigma_{k})_{k=1}^{\infty}$ converges graphically in $U$ to $S$ in
$C^{m}$-topology, if there is $I\in\mathbb{N}\cup\\{0\\}$ such that

  * •

$\Sigma_{k}$ splits into $I$ embedded connected components
$\iota_{k}|_{E^{i}_{k}}:E^{i}_{k}\to\Sigma^{i}_{k}$, $i=1,\cdots I$. Here
$E_{k}^{i}$ is open in $N_{k}$ and $\iota_{k}|_{E^{i}_{k}}$ is an embedding
onto $\Sigma^{i}_{k}$, and

  * •

$S\cap U=\Sigma^{1}\cup\cdots\cup\Sigma^{I}$, where each $\Sigma^{i}$,
$i=1,\cdots,I$, is an embedded submanifold in $U$.

Moreover, for each $i=1,\cdots,I$, there is $p_{i}\in\Sigma^{i}$ such that
$\Sigma^{i}_{k}$, $k\in\mathbb{N}$, and $\Sigma^{i}$ are graphs of a smooth
mappings $X^{i}_{k}$ and $X^{i}$ respectively, where $X^{i}_{k}$ and $X^{i}$
are defined on some open subsets of $T_{p_{i}}\Sigma^{i}$, such that
$X^{i}_{k}$ converges in $C^{m}$ norm to $X^{i}$ as $k\to\infty$.

We remark when $I=0$, it is understood that $\Sigma_{k}=S\cap U=\emptyset$.

######  Definition 2.9.

Let $(\Sigma_{k})_{k=1}^{\infty}$ be a sequence of properly immersed
submanifolds in $M$, and let $V$ be an open subset of $M$. We say that
$(\Sigma_{k})_{k=1}^{\infty}$ converges in $V$ smoothly as local graphs to an
integral varifold $S$, if for all $q\in V$, there is an open neighborhood $U$
of $q$ contained in $V$ such that $(\Sigma_{k}\cap U)_{k=1}^{\infty}$
converges graphically to $S\cap U$ in $C^{\infty}$-topology.

##  3\. Estimates for double divergence equation

In [2], the authors study the following fourth order equation in double
divergence form

(3.1)
$\partial_{x^{i}}\partial_{x^{j}}F^{jl}(x,Du,D^{2}u)=\partial_{x^{k}}a^{k}(x,Du,D^{2}u)-b(x,Du,D^{2}u)$

where $F^{ij}$, $a^{k}$ and $b$ are smooth functions in $x,Du,D^{2}u$, defined
in a convex neighborhood $U\subset B^{n}_{r}\times\mathbb{R}^{n}\times
S^{n\times n}$, where $S^{n\times n}$ is the space of $n\times n$ symmetric
matrices. A $W^{2,\infty}$ function $u$ on $B^{n}_{r}$ is called a weak
solution to (3.1) if

(3.2)
$\displaystyle\int_{B_{r}}\big{[}F^{ij}\eta_{ij}+a^{k}\eta_{k}+b\eta\big{]}dx=0,\
\ \text{ for all }\eta\in C^{\infty}_{c}(B^{n}_{r}).$

######  Definition 3.1.

Let $\Lambda>0$. We say that (3.1) is $\Lambda$-uniform on $U$ if the standard
Legendre ellipticity condition

(3.3) $\displaystyle\frac{\partial F^{jl}}{\partial
u_{ik}}(\xi)\sigma_{ij}\sigma_{kl}\geq\Lambda|\sigma|^{2},\ \forall\sigma\in
S^{n\times n}$

is satisfied for any $\xi\in U$.

For each $p=1,\cdots,n$, let $h_{p}=he_{p}$,

(3.4) $\displaystyle\eta^{-h_{p}}(x):=\frac{\eta(x-h_{p})-\eta(x)}{h},$

and $f:=u^{h_{p}}$. Define

$\displaystyle\xi_{0}$ $\displaystyle=(x,Du(x),D^{2}u(x)),$
$\displaystyle\xi_{h}$ $\displaystyle=(x+h_{p},Du(x+h_{p}),D^{2}u(x+h_{p})),$
$\displaystyle\vec{V}$ $\displaystyle=\xi_{h}-\xi_{0}.$

One can check that (3.2) implies

(3.5)
$\displaystyle\int_{B_{r}}\left(\beta^{ij,kl}f_{ik}\eta_{jl}+\gamma^{il}\eta_{jl}+\psi^{k}\eta_{k}^{-h_{p}}+\zeta\eta^{-h_{p}}\right)dx=0\
\ \ \ \forall\eta\in C^{\infty}_{c}(B^{n}_{r-h}),$

where

$\displaystyle\beta^{ij,kl}(x)$ $\displaystyle=\int_{0}^{1}\frac{\partial
F^{jl}}{\partial u_{ik}}(\xi_{0}+tV)dt,$ $\displaystyle\gamma^{jl}(x)$
$\displaystyle=\int_{0}^{1}\left(\frac{\partial F^{jl}}{\partial
u_{k}}(\xi_{0}+tV)f_{k}+\frac{\partial F^{jl}}{\partial
x_{p}}(\xi_{0}+tV)\right)dt,$ $\displaystyle\psi^{k}(x)$
$\displaystyle=a^{k}(x,Du,D^{2}u),$ $\displaystyle\zeta(x)$
$\displaystyle=b(x,Du,D^{2}u).$

In the following, we assume that $u$ is at least $C^{2}$ and satisfies (3.2).
The following propositions are the key steps in proving the regularity results
in [2].

######  Proposition 3.2.

Assume that $u\in C^{2,\alpha}(B^{n}_{r})$ satisfies (3.5), then for all
$0<r^{\prime}<r$ and $0<\alpha^{\prime}<\alpha$ we have $u\in
C^{3,\alpha^{\prime}}(B^{n}_{r^{\prime}})$ and

(3.6) $\displaystyle\|D^{3}u\|_{C^{\alpha^{\prime}}(B^{n}_{r^{\prime}})}\leq
C(\Lambda,\alpha,\alpha^{\prime},r,r^{\prime},\|u\|_{W^{2,\infty}(B^{n}_{r})},\|\Psi\|_{C^{\alpha}(B^{n}_{r})})$

where for simplicity we write

$\displaystyle\Psi=\big{(}(\beta_{0}^{ij,kl})_{i,j,k,l},(\gamma_{p}^{jl})_{j,l,p},(\psi^{k})_{k},\zeta\big{)}.$

and

(3.7) $\displaystyle\beta^{ij,kl}_{0}(x)$ $\displaystyle=\frac{\partial
F^{jl}}{\partial u_{ik}}(x,Du,D^{2}u),$ (3.8)
$\displaystyle\gamma^{jl}_{p}(x)$ $\displaystyle=\frac{\partial
F^{jl}}{\partial u_{k}}(x,Du,D^{2}u)u_{pk}+\frac{\partial F^{jl}}{\partial
x^{p}}(x,Du,D^{2}u).$

###### Proof.

We first assume $r^{\prime}=1/5$ and $r=1$. Without rewriting the whole proof
again, we merely point out that in the proof of [2, Proposition 2.3], the
constants used have the following dependence:

$\displaystyle C_{3}$
$\displaystyle=\|\beta^{ij,kl}\|_{C^{\alpha}(B^{n}_{r})},$ $\displaystyle
C_{4}$ $\displaystyle=\|\beta^{ij,kl}\|_{C^{\alpha}(B^{n}_{r})},$
$\displaystyle C_{5}$
$\displaystyle=C(n)\|\gamma^{jl}\|_{C^{\alpha}(B^{n}_{r})},$ $\displaystyle
C_{6}$ $\displaystyle=C(n)\|\psi^{k}\|_{C^{\alpha}(B^{n}_{r})},$
$\displaystyle C_{7}$ $\displaystyle=C(n)\|\zeta\|_{C^{\alpha}(B^{n}_{r})},$
$\displaystyle C_{2}$ $\displaystyle=C(\Lambda)$ $\displaystyle C_{1}$
$\displaystyle=C_{1}(n)$ $\displaystyle\tilde{\alpha}$
$\displaystyle=1-\delta,$ $\displaystyle\tilde{q}$
$\displaystyle=\frac{n}{2-2\tilde{\alpha}},$ $\displaystyle\tilde{K}$
$\displaystyle=C(n)(\|\gamma^{jl}\|^{2}_{L^{2\tilde{q}}(B^{n}_{1})}+\|\psi^{k}\|^{2}_{L^{2\tilde{q}}(B^{n}_{1})}+\|\zeta\|_{L^{2\tilde{q}}(B^{n}_{1})}),$
$\displaystyle r_{0}$ $\displaystyle=r_{0}(C_{1},\Lambda,n,\delta),$
$\displaystyle C_{8}$ $\displaystyle=C_{8}(C_{1},n,\delta),$ $\displaystyle
C_{9}$
$\displaystyle=C_{9}(C_{1},C_{8},r_{0},\Lambda,\|D^{2}f\|_{L^{2}(B^{n}_{1/2})}),$
$\displaystyle C_{10}$
$\displaystyle=C_{10}(C_{4},C_{5},C_{7},\Lambda,\alpha,\delta,r_{0},\|D^{2}f\|^{2}_{L^{2}(B^{n}_{1/2})},C_{2})$

Thus by Proposition 2.1 and Proposition 2.3 in [2],

(3.9) $\displaystyle\|D^{3}u\|_{C^{\alpha-\delta/2}(B_{1/5})}\leq
C(\Lambda,\alpha,\delta,\|u\|_{W^{2,\infty}(B^{n}_{1})},\|\Psi\|_{C^{\alpha}(B^{n}_{1})}),$

In general, given $0<r^{\prime}<r\leq 1$ and $x_{0}\in B^{n}_{r^{\prime}}$, we
scale the ball $B^{n}_{r-r^{\prime}}(x_{0})$ to radius $1$ and obtain, via
(3.9),

$\displaystyle\|D^{3}u\|_{C^{\alpha-\delta/2}(B^{n}_{(r-r^{\prime})/5}(x_{0}))}\leq
C(\Lambda,\alpha,\delta,r,r^{\prime},\|u\|_{W^{2,\infty}(B^{n}_{r})},\|\Psi\|_{C^{\alpha}(B^{n}_{r})}),$

Since $x_{0}\in B^{n}_{r^{\prime}}$ is arbitrary, (3.6) is proved. ∎

From the above proposition, we obtain

######  Proposition 3.3.

For each $0<r^{\prime}<r\leq 1$, $0<\alpha^{\prime}<\alpha<1$ and $k\geq 3$,
assume that $u\in C^{2,\alpha}(B^{n}_{r})$ satisfies (3.2) and $(F^{jl})$
satisfies the Legendre ellipticity condition (3.3), then $u$ is smooth in
$B_{r}$ and

(3.10) $\displaystyle\|D^{k}u\|_{C^{\alpha^{\prime}}(B^{n}_{r^{\prime}})}\leq
C(k,\Lambda,\alpha,\alpha^{\prime},r,r^{\prime},\|u\|_{W^{2,\infty}(B^{n}_{r})},\|(\vec{F},\vec{a},b)\|_{C^{k-3}(B^{n}_{r})}).$

###### Proof.

That $u$ is smooth is proved in [2]. Taking the limit $h\to 0$ in (3.5), we
obtain (writing $f=u_{p}$)

(3.11)
$\displaystyle\int_{B^{n}_{r}}\big{[}(\beta^{ij,kl}_{0}f_{ij}+\gamma^{jl}_{p})\eta_{jl}+\partial_{p}\psi^{k}\eta_{k}+\partial_{p}\zeta\eta\big{]}dx=0,\
\ \forall\eta\in C^{\infty}_{c}(B^{n}_{r}).$

This is an equation of the form (3.2) with the leading term

(3.12) $\displaystyle
F^{ij}_{(p)}(x,Df,D^{2}f)=\beta^{ij,kl}_{0}f_{ij}+\gamma^{jl}_{p}$

and in particular

(3.13) $\displaystyle\frac{\partial F^{jl}_{(p)}}{\partial
f_{ik}}=\beta^{ij,kl}_{0}.$

Hence (3.11) also satisfies the $\Lambda$-uniform condition (3.3).

In general, one can inductively show that for any $p_{1},\cdots,p_{m}$, the
function $f=u_{p_{1}\cdots p_{m}}$ satisfies

(3.14)
$\displaystyle\int_{B^{n}_{r}}\big{[}(\beta^{ij,kl}_{0}f_{ik}+\gamma^{jl}_{(p_{1}\cdots
p_{m})})\eta_{jl}+\psi_{p_{1}\cdots p_{m}}^{k}\eta_{k}+\zeta_{p_{1}\cdots
p_{m}}\eta\big{]}dx=0,\ \ \forall\eta\in C^{\infty}_{c}(B^{n}_{r}),$

here $\gamma^{jl}_{(p_{1}\cdots p_{m})}$ is defined inductively by

(3.15) $\displaystyle\gamma^{jl}_{(p_{1}\cdots
p_{m})}=\big{(}\gamma^{jl}_{(p_{1}\cdots
p_{m-1})}\big{)}_{x_{p_{m}}}+\big{(}\beta^{ij,kl}_{0}\big{)}_{x_{p_{m}}}u_{ikp_{1}\cdots
p_{m}}.$

By Proposition 3.2, we obtain

(3.16) $\displaystyle\|D^{3}f\|_{C^{\alpha^{\prime}}(B^{n}_{r^{\prime}})}\leq
C(\Lambda,\alpha,\alpha^{\prime},r,r^{\prime},\|f\|_{W^{2,\infty}(B^{n}_{r})},\|\Psi_{(p_{1}\cdots
p_{m})}\|_{C^{\alpha}(B^{n}_{r})}),$

where

(3.17) $\displaystyle\Psi_{(p_{1}\cdots
p_{m})}=\big{(}(\beta_{0}^{ij,kl})_{i,j,k,l},(\gamma_{(p_{1}\cdots
p_{m})}^{jl})_{j,l,p_{1},\cdots p_{m}},(\partial_{p_{1}\cdots
p_{m}}\psi^{k})_{k},\partial_{p_{1}\cdots p_{m}}\zeta\big{)}.$

Arguing inductively from (3.8), (3.15) that

$\displaystyle\|\Psi_{p_{1}\cdots p_{m}}\|_{C^{\alpha}(B^{n}_{r})}\leq
C(m,\|\vec{a}\|_{C^{m}(B^{n}_{r})},\|F\|_{C^{m}(B^{n}_{r})},\|b\|_{C^{m}(B^{n}_{r})},\|u\|_{C^{m+2,\alpha}(B^{n}_{r})})$

Hence we have for any $0<r^{\prime}<r<1$ and $0<\alpha^{\prime}<\alpha<1$,

(3.18) $\displaystyle\|u\|_{C^{m+3,\alpha^{\prime}}(B^{n}_{r^{\prime}})}\leq
C(m,\Lambda,r,r^{\prime},\alpha,\alpha^{\prime},\|(\vec{F},\vec{a},b)\|_{C^{m}(B^{n}_{r})},\|u\|_{C^{m+2,\alpha}(B^{n}_{r})})$

The proposition is then proved by choosing

$\displaystyle r^{\prime}$ $\displaystyle=r_{m}<r_{m-1}<\cdots<r_{1}<r_{0}=r,$
$\displaystyle\alpha^{\prime}$
$\displaystyle=\alpha_{m}<\alpha_{m-1}<\cdots<\alpha_{1}<\alpha_{0}=\alpha$

and applying (3.18) $m-1$ times (with $r,r^{\prime},\alpha,\alpha^{\prime}$
replaced by $r_{j-1},r_{j},\alpha_{j-1},\alpha_{j}$ respectively) and lastly
Proposition 3.2. ∎

##  4\. Local calculations (Symplectic)

In this section, we represent locally a Lagrangian immersion as a gradient
graph in a Darboux coordinates chart given in Proposition 2.5. We apply the
estimates in the previous section. Under a smallness assumption on the
$L^{n}$-norm of the second fundamental form, we derive a $\epsilon$-regularity
result (Theorem 4.7).

Let $h=h^{t}_{p,v}$ be a Riemannian metric on $B^{2n}_{R}$ which satisfies
(2.12) for some $t\in(0,R^{-1}]$ and $K_{0},K_{1}$ satisfy (2.13).

Let $(x^{1},\cdots x^{2n})$ be the standard coordinates on $B^{2n}_{R}$ and
$e_{1},\cdots,e_{2n}$ be the standard basis of $\mathbb{R}^{2n}$. The
Christoffel symbols for the Levi-Civita connection of $h$ is given by

$\overline{\nabla}_{e_{i}}e_{j}=\Gamma_{ij}^{k}e_{k}+\Gamma_{ij}^{\bar{k}}e_{\bar{k}},$

where $\bar{k}:=n+k$ and the repeated indices are summed from $1$ to $n$.

From (2.12) we have

(4.1) $\displaystyle|\Gamma_{ij}^{k}|+|\Gamma_{ij}^{\bar{k}}|\leq
K_{1}R^{-1},\ \ \ \forall\ i,j,k.$

Let $L$ be a Lagrangian submanifold in $(B^{2n}_{R},\omega_{0})$ given by a
gradient graph

(4.2) $\displaystyle\Gamma(u):=\\{\Gamma_{u}(x):x\in U_{R}\subset
B^{n}_{R}\subset\mathbb{R}^{n}\\},$

where

(4.3) $\Gamma_{u}(x)=(x,Du(x)).$

for some smooth function $u:U_{R}\to\mathbb{R}$, where $U_{R}\subset
B^{n}_{R}$. In this case, $L=\Gamma(u)$ satisfies (2.2) if it is a critical
point of the area functional

(4.4) $\displaystyle\operatorname{Vol}(u)=\int_{U_{R}}\sqrt{\det g_{ij}}dx,$

where $g=\Gamma_{u}^{*}h$, among all Hamiltonian variations $t\mapsto
u+t\eta$, where $\eta\in C^{\infty}_{c}(U_{R})$. The Euler-Lagrange equation
has been computed in [2, Lemma 3.1], which is

(4.5)
$\displaystyle\int_{U_{R}}\big{[}F^{jl}\eta_{jl}+c^{k}\eta_{k}\big{]}dx=0,\ \
\forall\eta\in C^{\infty}_{c}(U_{R})$

where

(4.6) $\displaystyle F^{jl}(x,Du,D^{2}u)$
$\displaystyle=\sqrt{g}g^{ij}\big{(}(\delta^{kl}+\mathcal{B}_{lk})u_{ik}+\mathcal{C}_{li}\big{)},$
(4.7) $\displaystyle c^{k}(x,Du,D^{2}u)$
$\displaystyle=\frac{1}{2}\sqrt{g}g^{ij}\big{(}D_{y_{k}}\mathcal{A}_{ij}+2u_{im}D_{y_{k}}\mathcal{C}_{mj}+u_{im}u_{jl}D_{y_{k}}\mathcal{B}_{kl}\big{)},$

here $\mathcal{A}$, $\mathcal{B}$ and $\mathcal{C}$ are the components of
$(h-I_{2n})\circ\Gamma_{u}$ (see (3.2) in [2]). Note that $\sqrt{g}$, $g^{ij}$
can also be represented by $D^{2}u$, $\mathcal{A}$, $\mathcal{B}$ and
$\mathcal{C}$. By (2.12) we obtain the following lemma.

######  Lemma 4.1.

For each $m\in\mathbb{N}$, there is a constant
$C=C(r,m,K_{0},\cdots,K_{m},\|D^{2}u\|_{L^{\infty}(B^{n}_{r})})$ such that

(4.8) $\displaystyle\|F^{jl}\|_{C^{m}(B^{n}_{r})}$ $\displaystyle\leq C,$
(4.9) $\displaystyle\|c^{k}\|_{C^{m-1}(B^{n}_{r})}$ $\displaystyle\leq C,$

where $F^{jl}$ and $c^{k}$ are defined in (4.6), (4.7).

By (2.13) we have $|Dh|<\epsilon_{1}$. By [2, Lemma 3.2], if

(4.10) $\|D^{2}u\|_{C^{0}(B^{n}_{r})}\leq\epsilon_{1},$

then (4.5) is $\Lambda$-uniform with $\Lambda=1/2$. Hence we can apply
Proposition 3.3 and Lemma 4.1 to obtain

######  Theorem 4.2.

Let $u:B^{n}_{r}\to\mathbb{R}$ be a smooth function which satisfies
$u(0)=Du(0)=D^{2}u(0)=0$ and (4.10). If the graph $\Gamma(u)$ is Hamiltonian
stationary Lagrangian in $(B_{r}^{2n},\omega,h)$, where $h$ satisfies (2.12)
and $K_{0},K_{1}$ satisfy (2.13). Then for each $k\geq 3$ and $x\in
B^{n}_{r/2}$,

(4.11) $\displaystyle|D^{k}u(x)|\leq C(k,r,K_{0},\cdots,K_{k-3}).$

###  4.1. Curvature estimates, $\epsilon$-regularity

Next we use the result in the previous subsection to derive local curvature
estimates for HSL immersions in $(M,\omega,J,h)$ with small $L^{n}$-norm of
the second fundamental form.

Using the Darboux coordinates in subsection 2.2, one can think of
$L\cap\mathscr{B}_{r}(p)$ as a Lagrangian immersion in
$(B^{2n}_{r},\omega_{0})$. We first derive an inequality comparing the second
fundamental form of $L$ computed using two different Riemannian metrics.

######  Proposition 4.3.

Let $k,K\in\mathbb{N}$ and $k<K$. Let $L$ be a $k$-dimensional immersed
submanifold in an open subset $U$ of $\mathbb{R}^{K}$ and let $h_{1}$ be a
Riemannian metrics on $U$ so that
$\|h_{1}-I\|_{C^{0}(U)}+\|Dh_{1}\|_{C^{0}(U)}<\epsilon$ for some positive
number $\epsilon<1/2$. Let $A_{1},A_{0}$ be the second fundamental form of $L$
calculated with respect to $h_{1}$ and the Euclidean metric $h_{0}=I$
respectively. Then there is $C=C(K)$ so that

$|A_{0}(X,X)|\leq(1+C\epsilon)|A_{1}(X,X)|_{h_{1}}+C\epsilon|X|_{h_{1}}^{2}$

for all $x\in L$ and $X\in T_{x}L$.

###### Proof.

Let $\nabla^{1},\nabla^{0}$ be the Levi-Civita connections of $h_{1},h_{0}$
respectively. Let $\mathscr{C}$ be the tensor
$\mathscr{C}(X,Y)=\nabla^{1}_{X}Y-\nabla^{0}_{X}Y$. Let $x\in L$ and let
$\pi^{\perp}_{\alpha}$ be the orthogonal projection onto the orthogonal
complement of $T_{x}L$ with respect to $h_{\alpha}$ for $\alpha=1,0$. Let
$\\{e_{1},\cdots,e_{K}\\}$ be a basis of $\mathbb{R}^{K}$ so that
$\\{e_{1},\cdots,e_{k}\\}$ forms a basis for $T_{x}L$. Then

$\pi^{\perp}_{\alpha}(Z)=Z-\sum_{a,b=1}^{k}H^{ab}_{\alpha}h_{\alpha}(Z,e_{a})e_{b},\
\ \ \alpha=1,0.$

Here $(H_{\alpha})_{ab}:=h_{\alpha}(e_{a},e_{b})$ and $(H^{ab}_{\alpha})$ is
the inverse matrix of $(H_{\alpha})_{ab}$. Hence

$\displaystyle(\pi^{\perp}_{1}-\pi^{\perp}_{0})Z$
$\displaystyle=\sum_{a,b=1}^{k}\big{(}H^{ab}_{1}h_{1}(Z,e_{a})-H^{ab}_{0}h_{0}(Z,e_{a})\big{)}e_{b}$
$\displaystyle=\sum_{a,b=1}^{k}\big{(}(H^{ab}_{1}-H_{0}^{ab})h_{1}(Z,e_{a})+H^{ab}_{0}(h_{1}(Z,e_{a})-h_{0}(Z,e_{a}))\big{)}e_{b}.$

Write $e_{a}=(e_{a}^{1},\cdots,e_{a}^{K})$ and $Z=(Z^{1},\cdots,Z^{K})$, then

$\displaystyle(H_{\alpha})_{ab}=\sum_{i,j=1}^{K}e_{a}^{i}e_{b}^{j}(h_{\alpha})_{ij},\
\ h_{\alpha}(Z,e_{a})$
$\displaystyle=\sum_{i,j=1}^{K}Z^{i}e_{a}^{j}(h_{\alpha})_{ij}.$

This implies

$\displaystyle|H_{1}^{ab}-H_{0}^{ab}|$ $\displaystyle\leq
C\|h_{1}-h_{0}\|_{C^{0}(U)},$ $\displaystyle|h_{1}(Z,e_{a})-h_{0}(Z,e_{a})|$
$\displaystyle\leq C\|h_{1}-h_{0}\|_{C^{0}(U)}|Z|$

and

(4.12) $|(\pi^{\perp}_{1}-\pi^{\perp}_{2})Z|\leq
C\|h_{1}-h_{0}\|_{C^{0}(U)}|Z|.$

for some $C=C(K)$. Let $X,Y$ be any tangent vector fields of $L$ defined on
some open neighborhood of $x$. Then

$\displaystyle A_{0}(X,Y)$ $\displaystyle=\pi^{\perp}_{0}(\nabla^{0}_{X}Y)$
$\displaystyle=\pi^{\perp}_{0}(-\mathscr{C}(X,Y)+\nabla^{1}_{X}Y)$
$\displaystyle=-\pi^{\perp}_{0}\mathscr{C}(X,Y)+\pi^{\perp}_{1}\nabla^{1}_{X}Y+(\pi^{\perp}_{0}-\pi^{\perp}_{1})(\nabla^{1}_{X}Y)$
$\displaystyle=A_{1}(X,Y)-\pi^{\perp}_{0}\mathscr{C}(X,Y)+(\pi^{\perp}_{0}-\pi^{\perp}_{1})(\nabla^{1}_{X}Y).$

Now let $X$ be chosen such that $\nabla^{1}_{X}X=A_{1}(X,X)$ at $x$. Using
(4.12), at $x$ we have

$\displaystyle|A_{0}(X,X)|$
$\displaystyle\leq|A_{1}(X,X)|+|\mathscr{C}(X,X)|+C\|h_{1}-h_{2}\|_{C^{0}(U)}|A_{1}(X,X)|$
$\displaystyle\leq(1+C\epsilon)|A_{1}(X,X)|_{h_{1}}+|\mathscr{C}(X,X)|.$

For some $C=C(K)$. To estimates $\mathscr{C}(X,X)$, note that locally

$\mathscr{C}_{ij}^{k}=(\Gamma_{1})_{ij}^{k}-(\Gamma_{0})_{ij}^{k}=(\Gamma_{1})_{ij}^{k},$

where $(\Gamma_{1})_{ij}^{k}$ are the Christoffel symbols of $h_{1}$. Since
$\|Dh_{1}\|_{C^{0}(U)}<\epsilon$ by assumption and
$(\Gamma_{1})_{ij}^{k}=h^{-1}_{1}*Dh_{1}$,

$|\mathscr{C}(X,X)|\leq C\epsilon|X|_{h_{1}}^{2}$

for some $C=C(K)$. This finishes the proof of Proposition 4.3. ∎

######  Lemma 4.4.

Let $U$ be an open subset in $\mathbb{R}^{2n}$ and let $L$ be an immersed
Lagrangian submanifold in $(U,\omega_{0},h,J)$, so that $h$ satisfies (2.12)
and the $C^{0}$-norm of the second fundamental form of $L$ is bounded by
$C_{A}$. Then there is $C_{1}$ depending on $n,K_{0},K_{1}$ so that the
following holds: for any $\epsilon\in(0,1]$, write
$r=\epsilon/(C_{1}(C_{A}+1))$. For any $p\in L\cap U$ so that
$B^{2n}_{r}(p)\subset U$, any embedded connected component $L^{i}$ of $L\cap
B^{2n}_{r}(p)$ containing $p$ can be written as a gradient graph $\Gamma_{u}$,
where $u$ is a function defined on $U_{i}\subset T_{p}L^{i}$,
$B^{n}_{r/2}\subset U_{i}$ and $u$ satisfies

(4.13) $u(0)=Du(0)=D^{2}u(0)=0$

and

(4.14) $\|D^{2}u\|_{C^{0}(U_{i})}\leq\epsilon.$

###### Proof.

Apply Proposition 4.3 with $h_{1}=h$, we conclude that $|A_{0}|\leq
c_{2}(C_{A}+1)$, where $A_{0}$ is the second fundamental form of $L$ in
$(U,h_{0})$ and $c_{2}$ depends only on $n,K_{0},K_{1}$. By [3, Theorem 2.6]
(see also [12, Theorem 2.4]), there is a dimensional constant $c$ so that if
$L^{i}$ is an embedded connected component of $L\cap
B^{2n}_{c\epsilon(C(C_{A}+1))^{-1}}(p)$ containing $p$, then $L^{i}$ is
graphical: there is $X_{i}:U_{i}\to\mathbb{R}^{n}$, where
$B^{n}_{{c\epsilon(2c_{2}(C_{A}+1))^{-1}}}(p)\subset U_{i}$, so that

$L^{i}=\\{(x,X_{i}(x)):x\in U_{i}\\}$

with $X_{i}(0)=DX_{i}(0)=0$ and $|X_{i}(x)|\leq\epsilon$ for all $x\in U_{i}$.
Since $L^{i}$ is Lagrangian, $X_{i}=Du_{i}$ for some smooth function
$u_{i}:U_{i}\to\mathbb{R}$ with $u_{i}(0)=0$. This finishes the proof of the
lemma, by choosing $C_{1}=2c_{2}/c$. ∎

The following lemma can be proved by a direct computation, using the bounds on
$D^{2}u$, $h-I$ and $Dh$.

######  Lemma 4.5.

With the same assumption as in Lemma 4.4, there is $C_{2}>0$ depending only on
$n$, $K_{0},K_{1}$ such that

(4.15) $\displaystyle|D^{3}u(x)|\leq C_{2}(|A(x)|+1),\ \ \ |A(x)|\leq
C_{2}(|D^{3}u(x)|+1)$

whenever $|x|\leq\epsilon_{1}(C_{1}(C_{A}+1))^{-1}$.

From the above lemmas we obtain the following proposition.

######  Proposition 4.6.

Let $L$ be a properly immersed Lagrangian submanifold in $M$. Let $p\in M$ and
let $\delta,C_{A}$ be positive numbers. Assume that the $C^{0}$-norm of the
second fundamental form $A$ of $L$ is bounded above by $C_{A}$ in the Darboux
ball $\mathscr{B}_{\delta}(p)$. Then there is $C_{2}>0$ depending only on
$n,K_{0},K_{1}$ such that the following holds: for any
$\epsilon\in(0,\epsilon_{1}]$, $L\cap\mathscr{B}_{r}(p)$ splits into embedded
connected components, where
$r=\min\\{\delta/2,\epsilon(C_{2}(C_{A}+1))^{-1}\\}$. Moreover, for each
$v\in\mathbb{U}_{p}$, each embedded connected component $L_{i}$ of
$L\cap\mathscr{B}_{r}(p)$,
$z\in\widetilde{L}^{i}:=\Upsilon_{p,v}^{-1}(L_{i})$, up to a unitary action,
$\widetilde{L}^{i}$ is the gradient graph of a function $u_{i}$ with
$z=(0,0)$,

  * (i)

$u_{i}(0)=Du_{i}(0)=D^{2}u_{i}(0)$,

  * (ii)

$|D^{2}u_{i}(x)|\leq\epsilon$, and

  * (iii)

$|D^{3}u_{i}(x)|\leq C_{1}(C_{A}+1)$.

###### Proof.

For any $r<\delta$ and $v\in\mathbb{U}_{p}$, let
$\tilde{L}:=\Upsilon_{p,v}^{-1}(L\cap\mathscr{B}_{p}(r))$. Then $\tilde{L}$ is
a immersed Lagrangian submanifold in $B^{2n}_{r}$. For any
$\epsilon\in(0,\epsilon_{1}]$, let
$r_{1}(\epsilon)=\epsilon/(C_{1}(C_{A}+1))$, where $C_{1}$ is defined in Lemma
4.4 and let $r_{2}=r_{1}(\epsilon)/4$. Then for all $q\in B^{2n}_{r_{2}}$ we
have $B^{2n}_{r_{2}}\subset B^{2n}_{2r_{1}}(q)\subset B^{2n}_{r_{1}}$. By
Lemma 4.4, when $r=r_{1}(\epsilon)$, $\tilde{L}$ splits into finitely many
embedded connected components. Let $\bar{L}^{1},\cdots,\bar{L}^{I}$ be those
components that intersects $B^{2n}_{r_{2}}$. For any $i=1,\cdots,I$ and
$z\in\bar{L}^{i}\cap B^{2n}_{r_{2}}$, apply Lemma 4.4 to the ball
$B^{2n}_{2r_{2}}(z)$. Thus up to a unitary action, $\bar{L}^{i}\cap
B^{2n}_{2r_{2}}$ is the gradient graph of a function $u_{i}$. (i), (ii), (iii)
follows from Lemma 4.4 and Lemma 4.5. Since $B^{2n}_{r_{2}}\subset
B^{2n}_{2r_{2}}$, the Proposition is proved by choosing $C_{2}=4C_{1}$. ∎

The following $\epsilon$-regularity theorem is essential to the proof of
Theorem 1.1.

######  Theorem 4.7.

There are positive numbers $\epsilon_{0},C_{\epsilon_{0}}$ depending only on
$K_{0},K_{1},K_{2}$ such that the following holds: if $L$ is an
$n$-dimensional properly immersed HSL submanifold in a symplectic manifold
$M$, $p\in L$ and

(4.16)
$\displaystyle\int_{L\cap\mathscr{B}_{r_{0}}(p)}|A|^{n}d\mu_{L}\leq\epsilon_{0},$

where $r_{0}\leq 1$. Then for all $0<\sigma\leq r_{0}$ and
$y\in\mathscr{B}_{r_{0}-\sigma}(p)\cap L$,

(4.17) $\displaystyle\sigma|A(y)|\leq C_{\epsilon_{0}}.$

###### Proof.

Let $L$ be given by an immersion $\iota:N\to M$. Let
$\Upsilon_{p,v}:B^{2n}_{r_{0}}\to M$ be given by Proposition 2.5, where
$v\in\mathbb{U}_{p}$. Let $z_{0}$ be the maximum of the function

$z\mapsto(r_{0}-|z|)^{2}\max_{s\in\iota^{-1}(\Upsilon_{p,v}(z))}|A(s)|^{2}$

defined on $\Upsilon_{p,v}^{-1}(L\cap\mathscr{B}_{r_{0}}(p))$. Note that
maximum exists since $\iota^{-1}(\Upsilon_{p,v}(z))$ is finite for each $z$.
We assume this maximum is positive, or otherwise the result is trivial. So
$|z_{0}|<r_{0}$. Let $s_{0}\in\iota^{-1}(\Upsilon_{p,v}(z_{0}))$ such that

$|A(s_{0})|^{2}=\max_{s\in\iota^{-1}(\Upsilon_{p,v}(z_{0}))}|A(s)|^{2}.$

For all $z\in B^{2n}_{\frac{r_{0}-|z_{0}|}{2}}(z_{0})$ and
$s\in\iota^{-1}(\Upsilon_{p,v}(z))$,

$\displaystyle|A(s)|^{2}$
$\displaystyle\leq\frac{(r_{0}-|z_{0}|)^{2}}{(r_{0}-|z|)^{2}}|A(s_{0})|^{2}$
$\displaystyle\leq\frac{(r_{0}-|z_{0}|)^{2}}{\left(\frac{r_{0}-|z_{0}|}{2}\right)^{2}}|A(s_{0})|^{2}$
$\displaystyle\leq 4|A(s_{0})|^{2}.$

Let $p_{0}=\Upsilon_{p,v}(z_{0})$. By (2.14),
$U:=\mathscr{B}_{\frac{r_{0}-|z_{0}|}{8}}(p_{0})\subset\Upsilon_{p,v}(B^{2n}_{\frac{r_{0}-|z_{0}|}{2}}(z_{0}))$.
By Proposition 4.6, there is $C_{1}>0$ such that $L\cap
U\cap\mathscr{B}_{r}(p_{0})$ splits into embedded connected components, where
$r=(C_{1}(2|A(s_{0})|+1))^{-1}\epsilon_{1}$ and each of the components is HSL
by Corollary 2.4. Moreover, let $L_{i}$ be a connected component so that
$\iota^{-1}(L_{i})$ contains $s_{0}$. Choose $v_{0}\in\mathbb{U}_{p_{0}}$ so
that $v_{0}(\mathbb{R}^{n}\times\\{0\\})=T_{p_{0}}L_{i}$. Then
$\Upsilon_{p_{0},v_{0}}^{-1}L_{i}$ is the gradient graph of a smooth function
$u$ with $u(0)=Du(0)=D^{2}(0)=0$ and

$|D^{2}u(x)|\leq\epsilon_{1},\ \ \ |D^{3}u(x)|\leq C_{1}(2|A(s_{0})|+1)$

for all $x$ in the domain of $u$.

Let

$R=\frac{r_{0}-|z_{0}|}{8}|A(s_{0})|,\ \ \ t=\frac{1}{|A(s_{0})|}$

and let

$t:B^{2n}_{R}\to B^{2n}_{\frac{r_{0}-|z_{0}|}{8}}$

be the scaling $z\mapsto tz$. Then $\tilde{L}=(\Upsilon_{p_{0},v_{0}}\circ
t)^{-1}(L_{i})$ is an immersed HSL in $(B^{2n}_{R},\omega_{0},\tilde{h})$
given by

$\tilde{\iota}=t^{-1}\circ\iota:V\to B^{2n}_{R}$

with metric $\tilde{h}=t^{-2}(\Upsilon_{p_{0},v_{0}}\circ t)^{*}h$. By the
choice of $R$, the second fundamental form $\widetilde{A}$ of $\widetilde{L}$
satisfies $\|\widetilde{A}\|_{0}\leq 2$ and $|\widetilde{A}(0)|=1$. Also,
$\widetilde{L}$ is a gradient graph of a smooth function $\tilde{u}$ with

$|D^{2}\tilde{u}(x)|\leq\epsilon_{1},\ \ \
|D^{3}\tilde{u}(x)|\leq\frac{C_{1}(2|A(s_{0})|+1)}{|A(s_{0})|}\leq 3C_{1}$

for all $x\in B^{2n}_{(3C_{1})^{-1}}\cap B^{2n}_{R}$. Moreover,

(4.18)
$\displaystyle\int_{\widetilde{L}}|\widetilde{A}|^{n}d\mu_{\tilde{L}}\leq\int_{B_{\frac{r_{0}-|z_{0}|}{8}}}|A|^{n}d\mu<\epsilon_{0}$

since the quantity is scale-invariant. Using Proposition 3.3 with $m=4$ and
(4.1)

$|D^{4}\tilde{u}(x)|\leq C(K_{0},K_{1},K_{2})$

for all $x\in B^{2n}_{R}$ with $|x|\leq(4C_{1})^{-1}$. This implies

$|\widetilde{\nabla}\widetilde{A}|\leq C(K_{0},K_{1},K_{2})$

for all $x\in B^{2n}_{R}$ with $|x|\leq(4C_{1})^{-1}$. Since
$|\widetilde{A}(0)|=1$, one concludes

$|\widetilde{A}(x)|\geq\frac{1}{2}$

for all $x\in\widetilde{L}^{1}$ with $|x|\leq\tilde{r}$, where
$\tilde{r}=\tilde{r}(K_{0},K_{1},K_{2})$. Now choose
$\epsilon_{0}=\omega^{-1}_{n}(4\tilde{r})^{n}$. If $R>\tilde{r}$, then

$\int_{B^{2n}_{R}}|\tilde{A}|^{n}d\mu_{\tilde{L}}\geq\int_{B^{2n}_{\tilde{r}}}|\tilde{A}|^{n}d\mu_{\tilde{L}}\geq\frac{1}{2^{n}}\mu_{\tilde{L}}(B^{2n}_{\tilde{r}})\geq\frac{1}{\omega_{n}(4\tilde{r})^{n}}=\epsilon_{0},$

which is a contradiction. Thus

$R=\frac{r_{0}-|z_{0}|}{8}|A(s_{0})|\leq\tilde{r},$

which implies for all $z\in\mathscr{B}_{r_{0}}(p)$ and
$s\in\iota^{-1}(\Upsilon_{p,v}(z))$,

$|A(s)|\leq\frac{8\tilde{r}}{r_{0}-|z|}.$

∎

##  5\. Proof of Theorem 1.1 for $n=1$

In this section we prove Theorem 1.1 for $n=1$. The general case is proved in
the next section. We remark that it is necessary to split the proof into two
cases: when $n\geq 2$ we use [8, Corollary 4.5], which says that if $L$ is a
Hamiltonian stationary Lagrangian in $M\setminus S$ and $S$ is finite, then
$\overline{L}$ is also Hamiltonian stationary Lagrangian varifold in $M$. As
pointed out in the introduction in [8], this does not hold when $n=1$.

###### Proof of Theorem 1.1 for $n=1$.

In this case, $(L_{k})$ is a sequence of compact connected immersed curves in
an oriented Riemannian surface $(M,h,J)$ and the Lagrangian condition is
automatically satisfied by any smooth curve. For each $k$, $L_{k}$ is given by
an immersion $\gamma_{k}:\mathbb{S}^{1}\to M$. We assume each $\gamma_{k}$ to
be parameterized proportional to arc length.

Since $J$ is compatible to $h$, one checks that an immersed curve
$\gamma:\mathbb{S}^{1}\to M$ in $(M,h)$ is Hamiltonian stationary if and only
if its curvature is constant. We remark that if $h$ is generic then $(M,h)$
admits infinitely many embedded circles with constant curvatures [20].

For each $k\in\mathbb{N}$, let $\kappa_{k}$ be the (constant) curvature of
$\gamma_{k}$. Let $\operatorname{Length}(L)$ denotes the length of the
immersed curves $L$ defined in (2.1). The condition (1.1) implies

(5.1) $\operatorname{Length}(\gamma_{k})=\operatorname{Length}(L_{k})\leq
C_{V}$

and

(5.2)
$|\kappa_{k}|\operatorname{Length}(\gamma_{k})=\int_{L_{k}}|\kappa_{k}|\leq
C_{A}.$

Taking a subsequence of $(L_{k})$ if necessary, we may assume that

(5.3)
$\lim_{k\to\infty}\operatorname{Length}(\gamma_{k})=\ell_{\infty}\in[0,C_{V}]$

and

(5.4) $\lim_{k\to\infty}\kappa_{k}=\kappa_{\infty}\in[-\infty,\infty].$

By compactness of $M$, we may also assume that

(5.5) $\lim_{k\to\infty}\gamma_{k}(0)=p,\ \ \
\lim_{k\to\infty}\gamma_{k}^{\prime}(0)=v$

for some $p\in M$ and $v\in T_{p}M$.

If $\ell_{\infty}=0$, then $(\gamma_{k})$ converges to the point $p$. If
$\ell_{\infty}>0$, then $\kappa_{\infty}\neq\pm\infty$ by (5.2). Using (5.5)
and the smooth dependence of ODE on initial data and parameters [10, Theorem
4.1], $(\gamma_{k})$ converges smoothly to an immersed curve $\gamma_{\infty}$
with $\gamma_{\infty}(0)=p$, $\gamma_{\infty}^{\prime}(0)=v$ and constant
curvature $\kappa_{\infty}$. This finishes the proof of Theorem 1.1 when
$n=1$.

∎

##  6\. Proof of Theorem 1.1 for $n\geq 2$

First we prove a simple covering lemma by Darboux balls.

######  Proposition 6.1.

Let $\lambda\in(0,1/4)$. There is $b$ depending only on $n$ and $\lambda$ such
that the following holds. For any $r<1$, there is a finite open covering
$\mathscr{U}$ of $M$ by Darboux balls $\mathscr{B}_{r}(a)$ so that (i) each
member of $\mathscr{U}$ intersect with at most $b-1$ other members of
$\mathscr{U}$, and (ii) the open cover $\\{\mathscr{B}_{\lambda
r}(p):\mathscr{B}_{r}(p)\in\mathscr{U}\\}$ still covers $M$.

###### Proof.

Let $\mathscr{V}=\\{\mathscr{B}_{\lambda
r/8}(p_{1}),\cdots,\mathscr{B}_{\lambda r/8}(p_{N})\\}$ be a maximal
collection of disjoint Darboux balls of radius $\lambda r/8$ in $M$. Then
$\\{\mathscr{B}_{\lambda r}(p_{1}),\cdots,\mathscr{B}_{\lambda r}(p_{N})\\}$
covers $M$: to see this, assume $p\in M$ and is not in one of
$\mathscr{B}_{\lambda r/8}(p_{i})$. Since $\mathscr{V}$ is maximal, there is
$i$ so that $\mathscr{B}_{\lambda r/8}(p)\cap\mathscr{B}_{\lambda r/8}(p_{i})$
contains an element $y$. By (2.14) and triangle inequality,

$d(p,p_{i})\leq d(p,y)+d(y,p_{i})<\frac{\lambda r}{4}+\frac{\lambda
r}{4}=\frac{\lambda r}{2}.$

Hence $p\in B^{M}_{\lambda r/2}(p_{i})\subset\mathscr{B}_{\lambda r}(p_{i})$
by (2.14) again. Let $\mathscr{U}$ be the open cover

$\mathscr{U}=\\{\mathscr{B}_{r}(p_{1}),\cdots,\mathscr{B}_{r}(p_{N})\\}.$

Note that $\mathscr{U}$ satisfies (ii) by construction. To show (i), let
$p_{i}$ be fixed and assume that $y_{i_{j}}$ lies in the intersection of
$\mathscr{B}_{r}(p_{i})$ and $\mathscr{B}_{r}(p_{i_{j}})$, for some
$j=1,\cdots,k$. Then $d(p_{i},p_{i_{j}})\leq 4r$, or
$p_{i_{j}}\in\mathscr{B}_{8r}(p_{i})$ for all $i_{j}$. However, since
$\mathscr{V}$ is a collection of disjoint Darboux balls with radius $\lambda
r/8$, one has $d(p_{i_{j}},p_{i_{l}})\geq\lambda r/8$. Let
$v_{i}\in\mathbb{U}_{p_{i}}$ be fixed and let
$z_{j}=\Upsilon_{p_{i},v_{i}}^{-1}(p_{i_{j}})$. Then
$\\{z_{1},\cdots,z_{k}\\}$ be a collection of points in $B^{2n}_{8r}$ so that
$|z_{i}-z_{j}|\geq\lambda r/16$ whenever $i\neq j$. This implies $k\leq b$,
where $b$ depends only on $n$, $\lambda$ but not on $r$. This finishes the
proof of (i). ∎

###### Proof of Theorem 1.1 for $n\geq 2$.

Choose $\lambda=\frac{\epsilon_{1}}{27C_{2}C_{\epsilon_{0}}}$, where
$\epsilon_{1}$, $C_{2}$ and $C_{\epsilon_{0}}$ are defined in (4.10),
Proposition 4.6 and Theorem 4.7 respectively. By Proposition 6.1, there is a
constant $b=b(n,\lambda)$ so that the following holds: for any
$m\in\mathbb{N}$, there is a finite open cover

$\mathscr{U}_{m}=\\{\mathscr{B}_{2^{-m}}(\tilde{p}^{m}_{1}),\cdots,\mathscr{B}_{2^{-m}}(\tilde{p}^{m}_{j(m)})\\}$

of $M$ by Darboux balls of radius $2^{-m}$, so that

  1. (i)

each elements in $\mathscr{U}$ intersects at most $b-1$ other members in
$\mathscr{U}$, and

  2. (ii)

the collection

$\mathscr{U}_{m,\lambda}=\\{\mathscr{B}_{\lambda
2^{-m}}(\tilde{p}^{m}_{1}),\cdots,\mathscr{B}_{\lambda
2^{-m}}(\tilde{p}^{m}_{j(m)})\\}$

still covers $M$.

The first condition implies that for each $k\in\mathbb{N}$,

(6.1)
$\displaystyle\sum_{j=1}^{j(m)}\int_{\mathscr{B}_{2^{-m}}(\tilde{p}^{m}_{j})\cap
L_{k}}|A_{k}|^{n}d\mu_{k}<b\int_{L_{k}}|A_{k}|^{n}d\mu_{k}<bC_{A},$

where $C_{A}$ is give in Theorem 1.1 and $\mu_{k}:=\mu_{L_{k}}$. For any fixed
$k$ and $m$, let $J_{k,m}=\\{p^{m}_{1,k},\cdots,p^{m}_{\ell,k}\\}$ be the
subset of $\\{\tilde{p}^{m}_{1},\cdots,\tilde{p}^{m}_{j(m)}\\}$ so that

$\int_{\mathscr{B}_{2^{-m}}(p^{m}_{j,k})\cap
L_{k}}|A_{k}|^{n}d\mu_{k}\geq\epsilon_{0}.$

Here $\epsilon_{0}$ is given in Theorem 4.7 and $\ell=\ell(m,k)$ is less than
$bC_{A}/\epsilon_{0}$ by (6.1). Taking a diagonal subsequence of $(L_{k})$ if
necessary, we may assume that $\ell(k,m)=\ell(m)$ and $J_{k,m}=J_{m}$ are both
independent of $k$ for all $m\in\mathbb{N}$. Write

$J_{m}=\\{p^{m}_{1},\cdots,p^{m}_{\ell(m)}\\}.$

Using $\ell(m)<bC_{A}/\epsilon_{0}$, there is a subsequence $(m_{i})$ such
that $\ell(m_{i})=\ell$ for all $i\in\mathbb{N}$. Using the compactness of $M$
and taking a further subsequence of $(m_{i})$ if necessary, we may assume that

$\lim_{i\to\infty}p^{m_{i}}_{j}=p_{j}$

for all $j=1,\cdots,\ell$.

Let $S=\\{p_{1},\cdots,p_{\ell}\\}$ and $r_{i}=2^{-m_{i}}$. Fix any
$p\in\\{\tilde{p}^{m_{i}}_{1},\cdots,\tilde{p}^{m_{i}}_{j(m)}\\}\setminus
J_{m_{i}}$. By definition of $J_{m_{i}}$ this implies

$\int_{\mathscr{B}_{r_{i}}(p)\cap L_{k}}|A_{k}|^{n}d\mu_{k}<\epsilon_{0}$

for all $k\in\mathbb{N}$. By Theorem 4.7,

(6.2) $|A_{k}(q)|\leq 2C_{\epsilon_{0}}/r_{i},\ \ \ \forall
k\in\mathbb{N}\text{ and }q\in\mathscr{B}_{r_{i}/2}(p).$

Setting $C_{A}=2C_{\epsilon_{0}}/r_{i}$ and note that

$\frac{\epsilon_{1}}{C_{2}(C_{A}+1)}=\frac{r_{i}\epsilon_{1}}{C_{2}(2C_{\epsilon_{0}}+r_{i})}\geq\frac{\epsilon_{1}}{3C_{2}C_{\epsilon_{0}}}r_{i}=9\lambda
r_{i}.$

By Proposition 4.6, for each $k\in\mathbb{N}$, $\mathscr{B}_{8\lambda
r_{i}}(p)\cap L_{k}$ splits into finitely many embedded connected components

$\mathscr{B}_{8\lambda r_{i}}(p)\cap L_{k}=L_{k}^{1}\cup\cdots\cup
L_{k}^{I_{k}}.$

Let $v\in\mathbb{U}_{p}$ be fixed and for each $a=1,\cdots,I_{k}$, denote
$\tilde{L}_{k}^{a}:=\Upsilon_{p,v}^{-1}(L^{a}_{k})$.

######  Lemma 6.2.

For each $i\in\mathbb{N}$, there is $M_{i}\in\mathbb{N}$ depending on $i$,
$\lambda$, $n$ and $C_{V}$ in (1.1) only so that $\mathscr{B}_{8\lambda
r_{i}}(p)\cap L_{k}$ has at most $M_{i}$ embedded connected components which
intersect $\mathscr{B}_{\lambda r_{i}}(p)$.

###### Proof of Lemma 6.2.

Let $L^{a}_{k}$ be one of the components $\mathscr{B}_{8\lambda r_{i}}(p)\cap
L_{k}$ which intersects $\mathscr{B}_{\lambda r_{i}}(p)$. Let $z^{a}_{k}\in
B^{2n}_{\lambda r_{i}}\cap\tilde{L}_{k}^{a}$. Then there is a unitary matrix
$A^{a}_{k}$ and a smooth function $u^{a}_{k}$ defined on $U^{a}_{k}$ so that

$(\Upsilon^{a}_{k})^{-1}(\tilde{L}_{k}^{a}\cap B^{2n}_{8\lambda
r_{i}}(z^{a}_{k}))=\\{(x,Du^{a}_{k}(x)):x\in U^{a}_{k}\\},$

where $\Upsilon^{a}_{k}(z)=A^{a}_{k}z+z^{a}_{k}$. Using (2.12), we obtain
$\mu(L^{a}_{k})\geq C(n)(\lambda r_{i})^{n}$ for some dimensional constant
$C(n)$. Together with the assumption on the area bound in (1.1), the lemma is
proved. ∎

By Lemma 6.2 and taking a diagonal subseqeunce of $(L_{k})$ if necessary, we
may assume that for any $i\in\mathbb{N}$ and
$p\in\\{\tilde{p}^{m_{i}}_{1},\cdots,\tilde{p}^{m_{i}}_{j(m)}\\}\setminus
J_{m_{i}}$, there is an integer $n_{i}(p)$ such that $\mathscr{B}_{8\lambda
r_{i}}(p)\cap L_{k}$ has exactly $n_{i}(p)$ connected components which
intersect $\mathscr{B}_{\lambda r_{i}}(p)$, where $n_{i}(p)\leq M_{i}$. Let
$L^{1}_{k},\cdots,L^{n_{i}(p)}_{k}$ be such connected components.

Let $z^{a}_{k}$, $A^{a}_{k}$, $\Upsilon^{a}_{k}$ and
$u^{a}_{k}:U^{a}_{k}\to\mathbb{R}$ be defined as in the proof of Lemma 6.2,
where $u=u^{a}_{k}$ satisfies (4.13) and (4.14) with $\epsilon=\epsilon_{1}$.
By Theorem 4.2, for any $m\geq 3$, there are $\overline{C}_{m}$ depending on
$r_{i},\lambda,m,K_{0},\cdots,K_{m-3}$ so that

(6.3) $\displaystyle|D^{m}u^{a}_{k}(x)|\leq\overline{C}_{m},\ \ \text{ for all
}x\in B^{n}_{4\lambda r_{i}}.$

Taking a diagonal subsequence of $(L^{k})_{k=1}^{\infty}$ if necessary, we may
assume that for each $i\in\mathbb{N}$ and $p$,

  1. (1)

the sequence of points $(z^{a}_{k})_{k=1}^{\infty}$ converges to
$z^{a}\in\overline{B^{2n}_{\lambda r_{i}}}$ as $k\to\infty$,

  2. (2)

the sequence of unitary matrices $(A^{a}_{k})_{k=1}^{\infty}$ converges to
some $A^{a}\in U(n)$ as $k\to\infty$, and

  3. (3)

the sequence of functions $(u^{a}_{k})_{k=1}^{\infty}$ converges in
$C^{m}(B^{n}_{3\lambda r_{i}})$ for all $m\in\mathbb{N}$ to a smooth function
$u^{a}:B^{n}_{r}\to\mathbb{R}$. This is possible by the higher order estimates
(6.3).

Then the sequence of immersions
$(\Upsilon^{a}_{k}\circ\Gamma_{u^{a}_{k}})_{k}$ converges to the immersion
$\Upsilon^{a}\circ\Gamma_{u^{a}}$, where $\Upsilon^{a}(z)=A^{a}z+z^{a}$. In
particular, there is $\tilde{u}^{a}_{k}$ defined on $B^{n}_{2\lambda r_{i}}$
so that $\tilde{L}^{a}_{k}$ is locally given by the immersion
$\Upsilon^{a}\circ\Gamma_{\tilde{u}^{a}_{k}}$.

Next we define the Lagrangian integral $n$-varifold $L$ on $M\setminus S$. Let

$\mathscr{B}=\\{\mathscr{B}_{\lambda r_{i}}(p):i\in\mathbb{N},\
p\in\\{\tilde{p}^{m_{i}}_{1},\cdots,\tilde{p}^{m_{i}}_{j(m_{i})}\\}\setminus
J_{m_{i}}\\}.$

Clearly $\mathscr{B}$ is an open cover of $M\setminus S$. For each
$K\in\mathbb{N}$, let

$\mathscr{B}_{K}=\\{\mathscr{B}_{\lambda
r_{i}}(p)\in\mathscr{B}:\mathscr{B}_{\lambda r_{i}}(p)\cap
S_{1/K}=\emptyset\\},$

where for any $\rho>0$ we set $S_{\rho}=\\{q\in M:d^{M}(q,s)<\rho\text{ for
some }s\in S\\}$. Then $\tilde{\mathscr{B}}=\cup_{K}\mathscr{B}_{K}$ is an
open cover of $M\setminus S$. For each $q\in M\setminus S$, then
$q\in\mathscr{B}_{\lambda r_{i}}(p)$ for some $U=\mathscr{B}_{\lambda
r_{i}}(p)\in\tilde{\mathscr{B}}$. Define $|L|$ on $\mathscr{B}_{\lambda
r_{i}}(p)$ by

$|L|=|L^{1}|+\cdots+|L^{n_{i}(p)}|,$

where for each $a=1,\cdots,n_{i}(p)$, $L^{a}$ is the limits of
$(L^{a}_{k}\cap\mathscr{B}_{\lambda r_{i}}(p))_{k=1}^{\infty}$ constructed in
the previous paragraph. Since $z^{a}\in B^{2n}_{\lambda r_{i}}$ for each
$a=1,\cdots,n_{i}(p)$, $B^{2n}_{\lambda r_{i}}\subset B^{2n}_{2\lambda
r_{i}}(z^{a})$. Hence $(L^{a}_{k}\cap\mathscr{B}_{\lambda
r_{i}}(p))_{k=1}^{\infty}$ converges graphically in $\mathscr{B}_{\lambda
r_{i}}(p)$ to $L\cap U$ in $C^{\infty}$-topology. Since $q\in M\setminus S$ is
arbitrary, $(L_{k})_{k=1}^{\infty}$ converges in $M\setminus S$ smoothly
locally graphically to $L$. Since the convergence is locally smooth and each
$L_{k}$ is HSL, $L$ is also HSL on $M\setminus S$. By [8, Corollary 4.4],
$\overline{L}$ admits a structure of an $n$-integral Lagrangian varifold so
that (4.5) holds (although Theorem 4.3 and Corollary 4.4 in [8] are stated
only for Lagrangian immersions in $\mathbb{R}^{2n}$, by isometrically embed
$(M,h)$ into some Euclidean space, the proofs work for immersions in
$(M,\omega,J,h)$). Connectedness of $\overline{L}$ can be shown as in the
proof of [8, Theorem 1.1].

Lastly, it remains to show (1.2). By the Nash embedding theorem, one can
assume that $(M,h)$ is an embedded submanifold of $\mathbb{R}^{n+K}$ for some
natural number $K$. Let $L$ be an immmersed submanifold $M$ and let
$\tilde{A},\tilde{H}$ be the second fundamental form and mean curvature vector
of $L$ respectively as a submanifold in $\mathbb{R}^{n+K}$. Then
$\tilde{A}=A+A^{M}$, where $A^{M}$ is the second fundamental form of $M$ in
$\mathbb{R}^{n+K}$. Together with the fact that $|A^{M}|\leq C(M)$ since $M$
is compact, we obtain

$\left(\int_{L}|\tilde{H}|^{n}d\mu_{L}\right)^{1/n}\leq\left(\int_{L}|H|^{n}d\mu_{L}\right)^{1/n}+C(M)\mu(L).$

Thus we can apply [8, Proposition 4.1] to $L=L_{k}$ and conclude that

(6.4) $\operatorname{Vol}(B_{\rho}(p)\cap L_{k})\leq
C(|\ln\rho|+1)^{n}\rho^{n}$

for all $p\in M$ and $\rho>0$ small enough, here $C$ depends only on $M$ and
$C_{V}$, $C_{A}$ in Theorem 1.1. For any $\rho>0$, by (6.4) we have

(6.5)
$\operatorname{Vol}(L_{k})-C(|\ln\rho|+1)^{n}\rho^{n}\leq\operatorname{Vol}(L_{k}\setminus
S_{\rho})\leq\operatorname{Vol}(L_{k}).$

Since $(L_{k})$ converges smoothly graphically to $L$ away from $S_{\rho}$,
taking $k\to\infty$ in (6.5) gives

$\lim_{k\to\infty}\operatorname{Vol}(L_{k})-C(|\ln\rho|+1)^{n}\rho^{n}\leq\operatorname{Vol}(L\setminus
S_{\rho})\leq\lim_{k\to\infty}\operatorname{Vol}(L_{k}).$

Thus (1.2) is shown by taking $\rho\to 0$ and this finishes the proof of
Theorem 1.1.

∎

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