# A Geometric flow towards hamiltonian stationary submanifolds

Jingyi Chen and Micah Warren  Department of Mathematics
The University of British Columbia, Vancouver, BC V6T 1Z2<EMAIL_ADDRESS>Department of Mathematics
University of Oregon, Eugene, OR 97403<EMAIL_ADDRESS>
###### Abstract.

In this paper, we introduce a geometric flow for Lagrangian submanifolds in a
Kähler manifold that stays in its initial Hamiltonian isotopy class and is a
gradient flow for volume. The stationary solutions are the Hamiltonian
stationary Lagrangian submanifolds. The flow is not strictly parabolic but it
corresponds to a fourth order strictly parabolic scalar equation in the
cotangent bundle of the submanifold via Weinstein’s Lagrangian neighborhood
theorem. For any compact initial Lagrangian immersion, we establish short-time
existence, uniqueness, and higher order estimates when the second fundamental
forms are uniformly bounded up to time $T$.

Chen is partially supported by NSERC Discovery Grant GR010074

##  1\. Introduction

The objective of this paper is to introduce a fourth order flow of Lagrangian
submanifolds in a Kähler manifold as a gradient flow of volume within a
Hamiltonian isotopy class and establish basic properties such as short-time
existence, uniqueness, and extendibility with bounded second fundamental form.

Our setting includes a Kähler manifold $(M^{2n},h,\omega,J)$ with symplectic
form $\omega$ and compatible Kähler metric $h$ satisfying
$h(JV,W)=\omega(V,W)$ where $J$ is a complex structure, and a given compact
Lagrangian immersion $\iota:L^{n}\rightarrow M^{2n}.$ We propose to find
$F:L\times[0,T)\rightarrow M^{2n}$ satisfying

(1.1) $\displaystyle\frac{dF}{dt}=J\nabla\operatorname{div}\left(JH\right)$
(1.2) $\displaystyle F\left(\cdot,0\right)=\iota\left(\cdot\right)$

where $H$ is the mean curvature vector of $L_{t}=F(\cdot,t)$ in $M$ and
$\nabla,\operatorname{div}$ are along $L_{t}$ in the induced metric from $h$.

The stationary solutions of the above evolution equation are the so-called
Hamiltonian stationary submanifolds, which are fourth order generalizations of
special Lagrangians, and exist in more abundance. Within a Hamiltonian isotopy
class it is possible for a compact Lagrangian submanifold of $\mathbb{C}^{n}$
to minimize volume (for example, a Clifford torus). Meanwhile, compact special
Lagrangian submanifolds of $\mathbb{C}^{n}$ do not exist. Recall that two
manifolds are Hamiltonian isotopic if they can be joined by the flow generated
by a vector field of the form $J\nabla f$ for a function $f$.

We will show that the initial value problem (1.1) - (1.2):

  1. (1)

stays within the Hamiltonian isotopy class - that is, the flow is generated by
a vector field of the form $J\nabla f,$

  2. (2)

is the gradient flow of volume with respect to an appropriate metric, so
decreases volume along the flow,

  3. (3)

enjoys short time existence given smooth initial conditions, and

  4. (4)

continues to exist as long as a second fundamental form bound is satisfied.

The flow (1.1) is degenerate parabolic but not strictly parabolic. Our proof
of existence and uniqueness involves constructing global solutions (locally in
time) using Weinstein’s Lagrangian neighbourhood theorem, which results in a
nice fourth order parabolic scalar equation. This equation has a good
structure, satisfying conditions required in [MM12] so we can conclude
uniqueness and existence within a given Lagrangian neighbourhood. We then
argue that the flows described by solutions to the scalar fourth order
equations are in correspondence with the normal flows of the form (1.1)
leading to uniqueness and extendability provided the flow remains smooth.

After proving existence and uniqueness using Weinstein neighborhoods, we turn
to local Darboux charts to prove higher regularity from second fundamental
form bounds. It’s not immediately clear how to extract regularity from
arbitrary Weinstein neighbourhoods as the submanifolds move, but using a
description of Darboux charts in [JLS11] we can fix a finite set of charts and
perform the regularity theory in these charts. Unlike mean curvature flow,
there is no maximum principle for fourth order equations. We deliver a
regularity theory using Sobolev spaces.

The newly introduced flow already exhibits nice properties in special cases:

(1) For Calabi-Yau manifolds, Harvey and Lawson [HL82] showed that, for a
Lagrangian submanifold, the Lagrangian angle $\theta$ generates the mean
curvature via

$H=J\nabla\theta.$

In this case (1.1) becomes

$\frac{dF}{dt}=-J\nabla\Delta\theta,$

while $\theta$ satisfies the pleasant fourth order parabolic equation

$\frac{d\theta}{dt}=-\Delta_{g}^{2}\theta$

where $g$ is the induced metric on $L_{t}$.

(2) The case $n=1$ involves curves in $\mathbb{C}$. Hamiltonian isotopy
classes bound a common signed area. Higher order curvature flows have been
studied and are called polyharmonic heat flow or curve diffusion flow (cf.
[PW16], [EI05], [May01]), dealing with evolution in the form
$\gamma^{\prime}=(-1)^{p-1}\kappa_{s^{2p}}$ where $\kappa_{s^{2p}}$ is the
$2p$-th order derivative of the curvature $\kappa$ of a plane curve $\gamma$
with respect to the arclength $s$. For $p=0$ it is the standard curve
shortening flow. The flow discussed here corresponds to $p=1$.

The case $n=1$ arises in material science [Mul99]. For more work on the pure
side, see [Whe13] and the many references therein. In [Whe13] it is shown that
curves near a circle have flows which exist forever and converge to a round
circle, as well as a short-time existence theorem that requires only $L^{2}$
initial curvature. It is not difficult to argue that for immersed figure 8
type curves singularities must develop in finite time. It is curious to know
whether there exist embedded curves which develop singularities in finite time
(cf. [EI05], [PW16], [May01]).

##  2\. Gradient Flow

We begin by setting up the equation as a formal gradient flow over an $L^{2}$
metric space. Given an embedded Lagrangian submanifold $L^{n}\subset M^{2n}$,
we can consider Hamiltonian deformations of $L$, these will be flows of vector
fields $J\nabla f$ for scalar functions $f$ on $M$. At $x\in L$, the normal
component of $J\nabla f$ is given by $J\nabla_{L}f$ where $\nabla_{L}f$ is the
gradient of $f$ as a function restricted to $L$; conversely, given any smooth
function $f$ on an embedded $L,$ we can extend $f$ to a function on $M$ so
that $\nabla_{L}f$ is not changed and is independent of the extension of $f$
(see section 2.1 below).

In other words, given a family of $C^{1}$ functions $f\left(\cdot,t\right)$
along $L,$ one can construct a family of embeddings

(2.1) $F:L\times\left(-\varepsilon,\varepsilon\right)\rightarrow M$

satisfying

(2.2) $\frac{d}{dt}F(x,t)=J\nabla f\left(x,t\right)$

and conversely, given any path (2.1) within a Hamiltonian isotopy class, there
will be a function $f$ so that (possibly after a diffeomorphism to ensure the
deformation vector is normal) the condition (2.2) is satisfied.

Let $\mathcal{I}_{L_{0}}$ be the set of smooth manifolds that are Hamiltonian
isotopic to $L_{0}$. The (smooth) tangent space at any
$L\in\mathcal{I}_{L_{0}}$ is parameterized via

(2.3) $T_{L}\mathcal{I}_{L_{0}}\mathcal{=}\left\\{f\in
C^{\infty}(L):\int_{L}fdV_{g}=0\right\\}.$

We use the $L^{2}$ metric on $T_{L}\mathcal{I}_{L_{0}}$: For $JX_{i}=\nabla
f_{i}$ $\in T_{L}\mathcal{I}_{L_{0}},i=1,2$

(2.4) $\langle X_{1},X_{2}\rangle=\int_{L}f_{1}f_{2}\,dV_{g}.$

With volume function given by

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

the classical first variation formula gives

$d\operatorname{Vol}(L)(W)=-\int_{L}\langle W,H\rangle dV_{g}$

where $W$ is the deformation vector field and $H$ is the mean curvature
vector. In the situation where allowable deformation vectors are of the form
$W=J\nabla f,$ we get

$\displaystyle d\operatorname{Vol}(L)(f)$ $\displaystyle=-\int_{L}\langle
J\nabla f,H\rangle dV_{g}$ $\displaystyle=\int_{L}\langle\nabla f,JH\rangle
dV_{g}$ $\displaystyle=-\int_{L}f\operatorname{div}\left(JH\right)dV_{g}$

using the fact that $J$ is orthogonal, then integrating by parts. Note that
$-\operatorname{div}\left(JH\right)$ belongs to $T_{L}\mathcal{I}_{L_{0}}$
since it integrates to 0 on $L$. Therefore, it is the gradient of the volume
function with respect to the metric. Thus a (volume decreasing) gradient flow
for volume would be a path satisfying

(2.5) $\frac{dF}{dt}=J\nabla\operatorname{div}\left(JH\right).$

######  Remark 2.1.

The metric (2.3) is not the usual $L^{2}$ metric for deformations of a
submanifold, which would measure the length of the tangent vector by
$\int\left|J\nabla f\right|^{2}dV_{g}.$ It is better suited than the standard
metric on vector fields. Suppose instead we take the “standard” $L^{2}$ metric
on deformation fields:

$d\operatorname{Vol}(L)(J\nabla f)=-\int_{L}\langle J\nabla f,H\rangle dV_{g}$

The gradient with respect to this metric would be $J\nabla\eta$ for some
$\eta\in C^{\infty}(L)$ such that

$-\int_{L}\langle J\nabla f,H\rangle=\int_{L}\langle J\nabla
f,J\nabla\eta\rangle.$

While the Lagrangian angle $\theta$ (in the Calabi-Yau case) does produce this
gradient locally, typically $\theta$ is not globally defined on $L$. So
instead, we must find a function $\eta$ solving the equation

$\Delta\eta=-\operatorname{div}(JH)$

which one can solve uniquely up to additive constants since $L$ is compact and
$\nabla\eta=-JH$ \+ $X$ (divergence free vector field on $L$). A gradient flow
would be $dF/dt=H-JX$ but there is no canonical way to determine $X$.

It is also worth noting that gradient flow with respect to the $L^{2}$ metric
(sometimes called $\mathcal{H}^{-1}$) is not new: It has been used for example
in mechanics to describe the flow of curves [Fif00].

###  2.1. Related definitions of Hamiltonian deformations

Traditionally, Hamiltonian isotopies are defined as flows of the entire
manifold along the direction of a time-dependent vector field $J\nabla f$ for
some $f$ a smooth function on $M$. Two submanifolds are Hamiltonian isotopic
if the one submanifold is transported to the other via the isotopy. In order
to use the description (2.3), we note the following standard result:

######  Lemma 2.2.

For a smooth flow of embedded Lagrangian submanifolds satisfying

(2.6) $\frac{dF}{dt}=J\nabla f(\cdot,t)\text{ }$

for some function $f$ defined on $L$ for each $t$, there exists a function
$\tilde{f}$ on $M\times[0,1]$ that defines a Hamiltonian isotopy on $M$ and
determines the same Hamiltonian isotopy of the submanifolds.

Conversely, given a global Hamiltonian isotopy determined by $\tilde{f}$, the
function $\tilde{f}$ restricted to $L_{t}$ determines a flow of the form
(2.6), possibly up to reparameterization by diffeomorphisms of $L.$

###### Proof.

The function $f$ is defined on a smooth compact submanifold of $M\times[0,1]$.
We can use the Whitney extension theorem to extend a smooth function off this
set in which the normal derivatives vanish. Thus along the Lagrangian
submanifolds, $J\nabla f=J\nabla\tilde{f}$.

Conversely, given any function $\tilde{f}$ its gradient decomposes into the
normal and tangential parts on the Lagrangian submanifold. By the Lagrangian
condition, $J\nabla^{T}\tilde{f}$ is normal and $J\nabla^{\perp}\tilde{f}$ is
tangential with the latter component describing merely a reparameterization of
$L$. So the flow is completely determined by the component
$J\nabla^{T}\tilde{f}$ which is determined by the restriction to $L$. ∎

###  2.2. Immersed Lagrangian submanifolds and their Hamiltonian deformations

Along the evolution equation (1.1), it is feasible that a submanifold which is
initially embedded will become merely immersed. Thus we would like the
equation to behave well even when the submanifold is immersed.

Weinstein’s Lagrangian neighborhood Theorem for immersed Lagrangian
submanifolds [EM02, Theorem 9.3.2] states that any Lagrangian immersion
$F_{0}:L\to M$ extends to an immersion $\Psi$ from a neighborhood of the
0-section in $T^{*}L$ to $M$ with $\Psi^{*}\omega_{M}=\omega_{\text{can}}$.

Sections of the cotangent bundle $T^{*}L$ are clearly embedded as graphs over
the 0-section of $T^{*}L$ which is identified with $L$, so by factoring the
immersion through $T^{*}L$, we get immersed submanifolds in $M$, in
particular, immersed Lagrangian submanifolds in $M$ for sections defined by
closed 1-forms on $L$.

Even though the deformation of an immersed manifold is not properly
Hamiltonian (that is, velocity vector $J\nabla f$ determined by a global
function $f$ on $M$) one can define deformations by using a function $f$
defined on the submanifold, and $J\nabla f$ makes sense within $T^{*}L$ as
pullback by the immersion. For example, the figure 8 is not problematic
because the two components of a neighborhood of the crossing point can have
different velocity vectors; these are separated within the cotangent bundle.

###  2.3. The evolution equation in terms of $\theta$

By [HL82], for a Lagrangian submanifold $L$ in a Calabi-Yau manifold
$(M^{n},\omega,J,\Omega)$ with a covariant constant holomorphic $n$-form
$\Omega$, the mean curvature of $L$ satisfies $H=J\nabla\theta$ where

$\Omega|_{L}=e^{i\theta}d\operatorname{Vol}_{L}.$

Now (2.5) leads to

$\frac{dF}{dt}=J\nabla\operatorname{div}\left(JH\right)=J\nabla\operatorname{div}\left(JJ\nabla\theta\right)=-J\nabla\Delta\theta.$

Differentiating the left-hand side (cf. [Woo20, Prop 3.2.1]):

$\displaystyle\frac{d}{dt}\Omega|_{L}$
$\displaystyle=\frac{d}{dt}\left(F_{t}^{\ast}\Omega\right)=F_{0}^{\ast}\mathcal{L}_{-J\nabla\Delta\theta}\Omega$
$\displaystyle=F_{0}^{\ast}d\left(\iota_{-J\nabla\Delta\theta}\Omega\right)=d\left(F_{0}^{\ast}\left(\iota_{-J\nabla\Delta\theta}\Omega\right)\right)$
$\displaystyle=d\left(F_{0}^{\ast}i\left(\iota_{-\nabla\Delta\theta}\Omega\right)\right)$
$\displaystyle=d\left(ie^{i\theta}dVol_{L}(-\nabla\Delta\theta,\cdot,...,\cdot)\right)$
$\displaystyle=-d\left(ie^{i\theta}\ast d\Delta\theta\right)$
$\displaystyle=\left(e^{i\theta}\ast d\Delta\theta-ie^{i\theta}d\left(\ast
d\Delta\theta\right)\right).$

Then differentiating the right hand side:

$\frac{d}{dt}e^{i\theta}d\operatorname{Vol}_{L}=e^{i\theta}\frac{d}{dt}d\operatorname{Vol}_{L}+ie^{i\theta}\frac{d\theta}{dt}d\operatorname{Vol}_{L}.$

Comparing the imaginary parts (after multiplying by $e^{-i\theta}$) of the
above two gives

(2.7) $\frac{d\theta}{dt}=-\Delta_{g(t)}^{2}\theta.$

##  3\. Existence and Uniqueness via a scalar equation on a Lagrangian
Neighborhood

The system of equations (1.1) is not strictly parabolic as given. Our approach
is to make good use of the Lagrangian property, in particular, by setting up
the equation as a scalar, uniformly parabolic equation via Weinstein’s
Lagrangian neighborhood theorem. For the convenience of using common
terminologies, we make our discussion for embeddings but the conclusions hold
for immersions in view of subsection 2.2.

###  3.1. Accompanying flow of scalar functions

Let $L$ be an embedded compact Lagrangian submanifold in a symplectic manifold
$(M,\omega)$. By Weinstein’s Lagrangian neighborhood [Wei71, Corollary 6.2]
theorem, there is a diffeomorphism $\Psi$ from a neighborhood $U\subset
T^{*}L$ of the 0-section (identified with $L$) to a neighborhood $V\subset M$
of $L$ such that $\Psi^{\ast}\omega=d\lambda_{can}$ and $\Psi$ restricts to
the identity map on $L$.

Let $\varphi(x,t)$ be a smooth function on $L\times[0,\delta)$ with
$\varphi(\cdot,0)=0$. Then $d\varphi$ is a $t$-family of exact (hence closed)
1-forms on $L$ hence a family of sections of $T^{\ast}L$ and each is a graph
over the 0-section. The symplectomorphism $\Psi$ yields a $t$-family of
Lagrangian submanifolds $L_{t}$ in $M$ near $L$:

(3.1) $F=\Psi(x,d\varphi(x,t))=\Psi\left(x,\frac{\partial\varphi}{\partial
x^{k}}dx^{k}\right).$

######  Proposition 3.1.

Suppose that $d\varphi(x,t)$ is an exact section describing an evolution of
Lagrangian submanifolds which satisfy the equation

(3.2)
$\left(\frac{dF}{dt}\right)^{\perp}=J\nabla\operatorname{div}\left(JH\right).$

Then there is a function $G$ (depending on $\Psi$) such that $\varphi$
satisfies

$\frac{\partial\varphi}{\partial
t}=-g^{ap}g^{ij}\frac{\partial^{4}\varphi}{\partial x^{a}\partial
x^{j}\partial x^{i}\partial x^{p}}+G(x,D\varphi,D^{2}\varphi,D^{3}\varphi).$

The coordinate free expression is

$\frac{\partial\varphi}{\partial
t}=\operatorname{div}J\Delta_{g}\Psi(x,d\varphi).$

###### Proof.

Taking $(x,v)$ for coordinates of $T^{\ast}L,$ let $y^{\alpha}$ be coordinates
in $M$, $\alpha=1,...,2n$. This gives a frame

(3.3) $e_{i}:=\frac{\partial F}{\partial
x^{i}}=\frac{\partial\Psi^{\alpha}}{\partial x^{i}}\frac{\partial}{\partial
y^{\alpha}}+\frac{\partial\Psi^{\alpha}}{\partial
v^{k}}\frac{\partial^{2}\varphi}{\partial x^{i}\partial
x^{j}}\delta^{jk}\frac{\partial}{\partial
y^{\alpha}}=\Psi_{i}+\varphi_{ij}\delta^{jk}\Psi_{k+n}$

where

$\displaystyle\Psi_{i}$
$\displaystyle:=D_{x^{i}}\Psi=\frac{\partial\Psi^{\alpha}}{\partial
x^{i}}\frac{\partial}{\partial y^{\alpha}}$ $\displaystyle\Psi_{j+n}$
$\displaystyle:=D_{v^{j}}\Psi=\frac{\partial\Psi^{\alpha}}{\partial
v^{j}}\frac{\partial}{\partial y^{\alpha}}.$

Letting also

$F_{i}^{\alpha}:=\frac{\partial\Psi^{\alpha}}{\partial
x^{i}}+\frac{\partial\Psi^{\alpha}}{\partial
v^{k}}\frac{\partial^{2}\varphi}{\partial x^{i}\partial x^{j}}\delta^{jk}$

we have

$e_{i}=F_{i}^{\alpha}\frac{\partial}{\partial y^{\alpha}}.$

Now suppose

$h=h_{\alpha\beta}dy^{a}dy^{\beta}$

is the Riemannian metric on $M$. We are also assuming $\omega(V,W)=h(JV,W)$.
We compute the induced metric $g$ from the immersion:

(3.4) $\displaystyle g_{ij}$ $\displaystyle=h(\partial_{i}F,\partial_{j}F)$
$\displaystyle=\left(\frac{\partial\Psi^{\alpha}}{\partial
x^{i}}\frac{\partial\Psi^{\beta}}{\partial
x^{j}}+\sum_{k}\left(\frac{\partial\Psi^{\alpha}}{\partial
x^{i}}\frac{\partial\Psi^{\beta}}{\partial
v^{k}}\frac{\partial^{2}\varphi}{\partial x^{j}\partial
x^{k}}+\frac{\partial\Psi^{\alpha}}{\partial
x^{j}}\frac{\partial\Psi^{\beta}}{\partial
v^{k}}\frac{\partial^{2}\varphi}{\partial x^{i}\partial
x^{k}}\right)+\sum_{k.l}\frac{\partial\Psi^{\alpha}}{\partial
v^{k}}\partial_{ik}^{2}\varphi\frac{\partial\Psi^{\beta}}{\partial
v^{l}}\partial_{jl}^{2}\varphi\right)h_{\alpha\beta}.$
$\displaystyle=h(\Psi_{i}+\varphi_{ik}\Psi_{k+n},\Psi_{j}+\varphi_{jl}\Psi_{l+n})$
$\displaystyle=h(\Psi_{i},\Psi_{j})+\sum_{k}\left(\varphi_{ik}h(\Psi_{k+n},\Psi_{j})+\varphi_{jk}h(\Psi_{i},\Psi_{k+n})\right)+\sum_{k,l}\varphi_{ik}\varphi_{jl}h(\Psi_{k+n},\Psi_{l+n}).$

Since $\Psi:T^{\ast}L\rightarrow M$ is a symplectomorphism with
$\Psi^{\ast}\omega=dx\wedge dv$, we have

(3.5) $\displaystyle\delta_{ij}$ $\displaystyle=dx\wedge dv(\partial/\partial
x^{i},\partial/\partial v^{j})=\Psi^{\ast}\omega\,(\partial/\partial
x^{i},\partial/\partial v^{j})$
$\displaystyle=h(J\Psi_{\ast}(\partial/\partial
x^{i}),\Psi_{\ast}(\partial/\partial v^{j}))=h(J\Psi_{i},\Psi_{j+n}).$

Similarly

(3.6) $h(J\Psi_{i+n},\Psi_{j+n})=\omega(\Psi_{\ast}(\partial/\partial
v^{i}),\Psi_{\ast}(\partial/\partial v^{j}))=0.$

Now as $F$ describes a Lagrangian manifold, (summing repeated indices below)

(3.7) $\displaystyle 0$
$\displaystyle=\omega(e_{i},e_{j})=h(J\Psi_{i}+\partial_{k}\varphi_{i}J\Psi_{k+n},\Psi_{j}+\partial_{l}\varphi_{j}\Psi_{l+n})$
$\displaystyle=h(J\Psi_{i},\Psi_{j})+\partial_{l}\varphi_{j}h(J\Psi_{i},\Psi_{l+n})+\partial_{k}\varphi_{i}h(J\Psi_{k+n},\Psi_{j})+\partial_{k}\varphi_{i}\partial_{l}\varphi_{j}h(J\Psi_{k+n},\Psi_{l+n})$
$\displaystyle=h(J\Psi_{i},\Psi_{j})-\varphi_{jk}h(\Psi_{i},J\Psi_{k+n})+\varphi_{ik}h(J\Psi_{k+n},\Psi_{j})+\varphi_{ik}\varphi_{jl}h(J\Psi_{k+n},\Psi_{l+n})$
$\displaystyle=h(J\Psi_{i},\Psi_{j})+\varphi_{ik}\varphi_{jl}h(J\Psi_{k+n},\Psi_{l+n})\hskip
142.26378pt\mbox{by \eqref{psi}}$ $\displaystyle=h(J\Psi_{i},\Psi_{j}).$

Now, $\\{\Psi_{i},J\Psi_{j}:1\leq i,j\leq n\\}$ is a basis for the ambient
tangent space at a point in the image of $F$. So is
$\\{\Psi_{i},\Psi_{j+n}:1\leq i,j\leq n\\}$ (as $\Psi$ is a local
diffeomorphism). We represent the latter vectors by

(3.8) $\Psi_{i+n}=a^{ij}\Psi_{j}+b^{ij}J\Psi_{j}.$

Computing the pairing $h\left(J\Psi_{j},\Psi_{i+n}\right)$ using (3.5) on the
left and (3.8) on the right yields $b^{ij}=h^{ij}$ as the inverse of the
positive definite matrix $h_{ij}:=h(\Psi_{i},\Psi_{j})$. Now recalling (3.1)

$\frac{\partial F}{\partial t}=\left(\frac{\partial}{\partial
t}\frac{\partial\varphi}{\partial x^{k}}\right)\frac{\partial\Psi}{\partial
v^{k}},$

project onto the normal space:

$\displaystyle\left(\frac{\partial F}{\partial t}\right)^{\perp}$
$\displaystyle=\frac{\partial\varphi_{t}}{\partial
x^{k}}h(\Psi_{k+n},Je_{p})Je_{q}g^{pq}$
$\displaystyle=\frac{\partial\varphi_{t}}{\partial
x^{k}}h(\Psi_{k+n},J\Psi_{p}+\varphi_{pj}J\Psi_{j+n})Je_{q}g^{pq}$
$\displaystyle=\frac{\partial\varphi_{t}}{\partial
x^{k}}h(\Psi_{k+n},J\Psi_{p})Je_{q}g^{pq}$
$\displaystyle=\frac{\partial\varphi_{t}}{\partial
x^{k}}\delta_{kp}Je_{q}g^{pq}\,\,\,\,\,\,{\mbox{ by \eqref{psi}}}$
$\displaystyle=\frac{\partial\varphi_{t}}{\partial
x^{k}}\left(Je_{q}g^{kq}\right)$ $\displaystyle=J\nabla\varphi_{t}.$

Let $H=H^{m}Je_{m}$ for the Lagrangian $L$. As $JH$ is tangential its
divergence on $L$ is

(3.9) $\displaystyle\operatorname{div}(JH)$
$\displaystyle=-\operatorname{div}\left(H^{m}e_{m}\right)$
$\displaystyle=-g^{ab}h(\nabla_{e_{a}}\left(H^{m}e_{m}\right),e_{b})$
$\displaystyle=-g^{ab}h\left(\frac{\partial}{\partial
x^{a}}H^{m}e_{m}+H^{m}\Gamma_{am}^{p}e_{p},e_{b}\right)$
$\displaystyle=-g^{ab}\frac{\partial}{\partial
x^{a}}H^{m}g_{mb}-g^{ab}H^{m}\Gamma_{am}^{p}g_{pb}$
$\displaystyle=-\frac{\partial}{\partial x^{a}}H^{a}-H^{m}\Gamma_{am}^{a}$

where the Christoffel symbols are for the induced metric $g$. The components
of $H$ are given by

(3.10)
$H^{a}=h\left(H,Je_{p}\right)g^{ap}=h\left(g^{ij}\left(\frac{\partial^{2}F^{\beta}}{\partial
x^{i}\partial
x^{j}}+F_{i}^{\alpha}F_{j}^{\gamma}\tilde{\Gamma}_{\alpha\gamma}^{\beta}\right)\frac{\partial}{\partial
y^{\beta}},Je_{p}\right)g^{ap}.$

Now differentiate components of (3.3)

$\frac{\partial^{2}F^{\beta}}{\partial x^{i}\partial
x^{j}}=\frac{\partial\Psi^{\beta}}{\partial x^{j}\partial
x^{i}}+\frac{\partial^{3}\varphi}{\partial x^{j}\partial x^{i}\partial
x^{k}}\delta^{kl}\frac{\partial\Psi^{\beta}}{\partial
v^{l}}+\frac{\partial^{2}\varphi}{\partial x^{i}\partial
x^{k}}\delta^{kl}\frac{\partial^{2}\Psi^{\beta}}{\partial x^{j}\partial
v^{l}}.$

Plug in (3.3) to get

$\displaystyle h$ $\displaystyle\left(\frac{\partial^{2}F^{\beta}}{\partial
x^{i}\partial x^{j}}\frac{\partial}{\partial
y^{\beta}},Je_{p}\right)=\omega\left(e_{p},\frac{\partial^{2}F^{\beta}}{\partial
x^{i}\partial x^{j}}\frac{\partial}{\partial y^{\beta}}\right)$
$\displaystyle=$
$\displaystyle\omega\left(\frac{\partial\Psi^{\delta}}{\partial
x^{p}}\frac{\partial}{\partial y^{\delta}}+\frac{\partial^{2}\varphi}{\partial
x^{p}\partial x^{q}}\delta^{qm}\frac{\partial\Psi^{\delta}}{\partial
v^{m}}\frac{\partial}{\partial
y^{\delta}},\frac{\partial\Psi^{\beta}}{\partial x^{j}\partial
x^{i}}\frac{\partial}{\partial y^{\beta}}+\frac{\partial^{3}\varphi}{\partial
x^{j}\partial x^{i}\partial
x^{k}}\delta^{kl}\frac{\partial\Psi^{\beta}}{\partial
v^{l}}\frac{\partial}{\partial y^{\beta}}+\frac{\partial^{2}\varphi}{\partial
x^{i}\partial x^{k}}\delta^{kl}\frac{\partial^{2}\Psi^{\beta}}{\partial
x^{j}\partial v^{l}}\frac{\partial}{\partial y^{\beta}}\right)$
$\displaystyle=$ $\displaystyle\frac{\partial^{3}\varphi}{\partial
x^{j}\partial x^{i}\partial
x^{k}}\delta^{kl}\omega\left(\frac{\partial\Psi^{\delta}}{\partial
x^{p}}\frac{\partial}{\partial
y^{\delta}},\frac{\partial\Psi^{\beta}}{\partial
v^{l}}\frac{\partial}{\partial
y^{\beta}}\right)+\frac{\partial^{2}\varphi}{\partial x^{p}\partial
x^{q}}\delta^{qm}\frac{\partial^{3}\varphi}{\partial x^{j}\partial
x^{i}\partial
x^{k}}\delta^{kl}\omega\left(\frac{\partial\Psi^{\delta}}{\partial
v^{m}}\frac{\partial}{\partial
y^{\delta}},\frac{\partial\Psi^{\beta}}{\partial
v^{l}}\frac{\partial}{\partial y^{\beta}}\right)$
$\displaystyle+F_{p}^{\delta}\left(\frac{\partial\Psi^{\beta}}{\partial
x^{j}\partial x^{i}}+\frac{\partial^{2}\varphi}{\partial x^{i}\partial
x^{k}}\delta^{kl}\frac{\partial^{2}\Psi^{\beta}}{\partial x^{j}\partial
v^{l}}\right)\omega_{\delta\beta}\circ F$ $\displaystyle=$
$\displaystyle\frac{\partial^{3}\varphi}{\partial x^{j}\partial x^{i}\partial
x^{k}}\delta^{kl}\Psi^{\ast}\omega\left(\frac{\partial}{\partial
x^{p}},\frac{\partial}{\partial
v^{l}}\right)+\frac{\partial^{2}\varphi}{\partial x^{p}\partial
x^{q}}\delta^{qm}\frac{\partial^{3}\varphi}{\partial x^{j}\partial
x^{i}\partial x^{k}}\delta^{kl}\Psi^{\ast}\omega\left(\frac{\partial}{\partial
v^{m}},\frac{\partial}{\partial v^{l}}\right)$
$\displaystyle+F_{p}^{\delta}\left(\frac{\partial\Psi^{\beta}}{\partial
x^{j}\partial x^{i}}+\frac{\partial^{2}\varphi}{\partial x^{i}\partial
x^{k}}\delta^{kl}\frac{\partial^{2}\Psi^{\beta}}{\partial x^{j}\partial
v^{l}}\right)\omega_{\delta\beta}\circ F$ $\displaystyle=$
$\displaystyle\frac{\partial^{3}\varphi}{\partial x^{j}\partial x^{i}\partial
x^{k}}\delta^{kl}\delta_{pl}+F_{p}^{\delta}\left(\frac{\partial\Psi^{\beta}}{\partial
x^{j}\partial x^{i}}+\frac{\partial^{2}\varphi}{\partial x^{i}\partial
x^{k}}\delta^{kl}\frac{\partial^{2}\Psi^{\beta}}{\partial x^{j}\partial
v^{l}}\right)\omega_{\delta\beta}\circ F.$

Now also

$h\left(\frac{\partial}{\partial
y^{\beta}},Je_{p}\right)=\omega\left(F_{p}^{\delta}\frac{\partial}{\partial
y^{\delta}},\frac{\partial}{\partial
y^{\beta}}\right)=F_{p}^{\delta}\omega_{\delta\beta}\circ F.$

Combining (3.10) and the above

$\displaystyle H^{a}$
$\displaystyle=g^{ij}g^{ap}\left(\frac{\partial^{3}\varphi}{\partial
x^{j}\partial x^{i}\partial
x^{k}}\delta^{kl}\delta_{pl}+F_{p}^{\delta}\left(\frac{\partial\Psi^{\beta}}{\partial
x^{j}\partial x^{i}}+\frac{\partial^{2}\varphi}{\partial x^{i}\partial
x^{k}}\delta^{kl}\frac{\partial^{2}\Psi^{\beta}}{\partial x^{j}\partial
v^{l}}\right)\omega_{\delta\beta}\circ F\right)$
$\displaystyle+g^{ij}g^{ap}F_{i}^{\alpha}F_{j}^{\gamma}\tilde{\Gamma}_{\alpha\gamma}^{\beta}F_{p}^{\delta}\omega_{\delta\beta}\circ
F.$

Thus using the expression we derived in (3.9)

(3.11) $\displaystyle\operatorname*{div}$
$\displaystyle(JH)=-g^{ap}g^{ij}\frac{\partial^{4}\varphi}{\partial
x^{a}\partial x^{j}\partial x^{i}\partial
x^{p}}-\left(\frac{\partial}{\partial
x^{a}}\left(g^{ij}g^{ap}\right)\right)\frac{\partial^{3}\varphi}{\partial
x^{j}\partial x^{i}\partial x^{p}}$ $\displaystyle-\frac{\partial}{\partial
x^{a}}\left(g^{ij}g^{ap}\right)F_{p}^{\delta}\left[\left(\frac{\partial\Psi^{\beta}}{\partial
x^{j}\partial x^{i}}+\frac{\partial^{2}\varphi}{\partial x^{i}\partial
x^{k}}\delta^{kl}\frac{\partial^{2}\Psi^{\beta}}{\partial x^{j}\partial
v^{l}}\right)+F_{i}^{\alpha}F_{j}^{\gamma}\tilde{\Gamma}_{\alpha\gamma}^{\beta}\right]\omega_{\delta\beta}\circ
F-H^{m}\Gamma_{am}^{a}.$

Now recalling (3.4), we see the metric components $g_{ab}$ involve second
order derivatives in terms of $\varphi,$ thus $\Gamma_{ij}^{k}$ are third
order. So each term above after the first term is at most third order. ∎

###  3.2. Short time existence

######  Proposition 3.2.

Given an initial smooth immersion of a compact $L\rightarrow M,$ there exists
a solution to (1.1,1.2) for some short time.

###### Proof.

Choose a Weinstein neighborhood containing $L.$ Now suppose we have $\varphi$
which satisfies the fourth order equation

(3.12)
$\displaystyle\varphi_{t}=-g^{ap}g^{ij}\frac{\partial^{4}\varphi}{\partial
x^{a}\partial x^{j}\partial x^{i}\partial
x^{p}}+G(x,D\varphi,D^{2}\varphi,D^{3}\varphi)=\operatorname{div}(JH)$
$\displaystyle\varphi(\cdot,0)=0.$

Then the immersions $F$ generated from $\varphi(x,t)$ satisfy

$\displaystyle\left(\frac{\partial F}{\partial t}\right)^{\perp}$
$\displaystyle=J\nabla\varphi_{t}=J\nabla\operatorname{div}(JH).$

As the normal component satisfies the appropriate equation, we may compose
with diffeomorphisms to get a flow (see Claim 3.3 below) such that

(3.13) $\frac{\partial F}{\partial t}=J\nabla\operatorname{div}(JH).$

Now the equation (3.12) is precisely of the form of $2p$ order quasilinear
parabolic equation studied in [MM12]. By [MM12, Theorem 1.1] we have short
time existence for the solution to (3.12), thus we have short-time existence
for the flow (3.13). ∎

###  3.3. Uniqueness

We start with a standard observation.

######  Claim 3.3.

Suppose that $F:L\times[0,T)\rightarrow M$ is a family of immersions
satisfying

$\left(\frac{\partial F}{\partial t}\right)^{\perp}=N(x,t)$

for some vector field $N(x,t)$ which is normal to the immersed submanifold
$F(\cdot,t)(L)$. There exists a unique family of diffeomorphisms
$\chi_{t}:L\rightarrow L$ such that

$\frac{\partial}{\partial
t}F(\chi_{t}(x),t)=N(\chi_{t}(x),t)\,\,\,\,{and}\,\,\,\,\chi_{0}=Id_{|L}.$

###### Proof.

Given the flow exists, the given velocity field will decompose orthogonally
into normal and tangential components:

$\frac{\partial F}{\partial t}=N(x,t)+T\left(x,t\right).$

Consider the time-dependent vector field on $L$

$V(x,t)=-D_{L}F(x,t)^{-1}T\left(x,t\right)$

By the Fundamental Theorem on Flows, (cf. [Lee13, Theorem 9.48]) there is a
unique flow on $L$ starting at the identity and satisfying

$\frac{\partial}{\partial t}\chi_{t}(x)=V(\chi_{t}(x),t).$

Composing this flow with the original flow $F$ yields the result. ∎

######  Theorem 3.4.

The solution to the initial value problem (1.1) - (1.2) is unique. More
precisely, if $F_{1}$ and $F_{2}$ are two solutions of (1.1) such that
$F_{1}(x,t_{0})=F_{2}(x,t_{0}^{\prime})$ for some $t_{0},t_{0}^{\prime}$ and
all $x\in L$, then

$F_{1}(x,t_{0}+\tau)=F_{2}(x,t_{0}^{\prime}+\tau)$

for all $\tau$ in an open neighborhood of $0$ where both sides above are
defined.

###### Proof.

Without loss of generality, we take $t_{0}=t_{0}^{\prime}=0$. Let
$L=F_{1}(\cdot,0)=F_{2}(\cdot,0)$ and $\Psi:U\subset T^{\ast}L\to V\subset M$
be a Lagrangian neighborhood mapping.

First, we show that the normal flow of $F_{i}(\cdot,t)$ is given in the
neighborhood $\Psi$ by the graph of an exact section $d\varphi_{i}(\cdot,t)$
where $\varphi_{i}$ solves a problem of the form (3.12). To this end, note
that for $\tau$ in the domain, the path
$\left\\{F_{i}(\cdot,t),t\in[0,\tau]\text{ }\right\\}$ is a Hamiltonian
isotopy between $F_{i}(\cdot,0)$ and $F_{i}(\cdot,\tau)$. Being a Hamiltonian
isotopy is invariant under the symplectomorphism $\Psi$, so the sections
$\Psi^{-1}\left(F_{i}(\cdot,0)\right)$ and
$\Psi^{-1}\left(F_{i}(\cdot,\tau)\right)$ are Hamiltonian isotopic. By [Wei71,
Corollary 6.2], Lagrangian submanifolds that are near to the $0$-section are
given as graphs of closed sections of the cotangent bundle. As the flow is
smooth, for small times the Lagrangian submanifolds are near enough to be
described by closed sections. According to [MS17, Proposition 9.4.2], these
sections are exact, that is

$\Psi^{-1}\circ
F_{i}(\cdot,\tau)\left(L\right)=\left\\{d\varphi_{i}\left(x,\tau\right):x\in
L\right\\}.$

In other words,

$\Psi(\left\\{d\varphi_{i}\left(x,\tau\right):x\in
L_{i}\right\\})=F_{i}(\cdot,\tau)\left(L\right)$

meaning that for each $\tau$

$\Psi\circ d\varphi_{i}:L\rightarrow T^{\ast}L\rightarrow M$

is a Lagrangian immersion, which may have reparameterized the base. In
particular, the flow $F_{i}$ determines a flow of scalar functions
$\varphi_{i},$ which recovers the same family of submanifolds at $F_{i}$ (up
to reparameterization) as do $\Psi\circ d\varphi_{i}\left(\cdot,t\right)$. By
Proposition 3.1, the scalar equation (3.12) holds on $\varphi_{i}$ as does the
initial condition $d\varphi_{i}\equiv 0$. It follows that $\varphi_{1}$ and
$\varphi_{2}$ both satisfy the same equation (3.12) and have the same initial
condition, so $\varphi_{1}=\varphi_{2}+C$ for some constant $C$. Thus the
flows $F_{1}$ and $F_{2}$ are the same. ∎

Theorem 3.4 allows for seamless extension of the flow: While the Weinstein’s
Lagrangian neighborhood may only exist around $L_{0}$, if another Lagrangian
neighborhood of $L_{0}$ extends the flow, the two flows patch together
smoothly.

##  4\. Higher order estimates based on curvature bounds

The goal of this section is to show that a solution with uniformly bounded
second fundamental form over $[0,T)$ enjoy estimates of all orders and can be
extended.

######  Theorem 4.1.

Suppose that the flow (1.1) exists on $[0,T)$ and the second fundamental form
has a uniform bound on $[0,T)$. Then the flow converges smoothly as
$t\rightarrow T$ so can be extended to $[0,T+\varepsilon)$ for some
$\varepsilon>0$.

To prove this theorem, it is essential in our approach to establish a-priori
estimates from the integral estimates derived from the following differential
inequality:

######  Proposition 4.2.

Suppose that $F$ is a solution to (1.1) on $[0,T)$ for a compact Lagrangian
submanifold $L$ inside a compact $M$. Suppose the second fundamental form has
a uniform bound $K$. There exists $C$ depending on $K$, the ambient geometry
of $M$ and $\operatorname{Vol}(L_{0})$ such that for all $k\geq 2$

(4.1) $\frac{d}{dt}\int_{L}\left|\nabla^{k-1}A\right|^{2}dV_{g}(t)\leq
C\int_{L}\left|\nabla^{k-1}A\right|^{2}dV_{g}(t)+C\sum_{l=0}^{k-2}\int_{L}\left|\nabla^{l}A\right|^{2}dV_{g}(t).$

A Weinstein neighborhood map determines the equation the flow must satisfy,
and we could derive estimates of all orders based on this particular equation.
However, the flow is expected to leave a given neighborhood after some time,
and we will need to take a new neighborhood. We would need to know the speed
of the flow to patch estimates from one neighborhood to another, but this
requires knowing the size of the Weinstein neighborhoods around the Lagrangian
submanifolds at different times.

We require charts with uniform geometric estimates. To obtain these we appeal
to uniform local Darboux coordinates given in [JLS11]. These charts are local
but are given with uniform geometric bounds. The short-time existence of the
flow is already determined by the global Weinstein neighborhoods; we write the
flow in these Darboux charts as a scalar equation from which we derive
integral estimates for derivatives of any order.

###  4.1. Uniform Darboux charts.

We record [JLS11, Prop.3.2 and Prop.3.4] on existence of Darboux coordinates
with estimates on a symplectic manifold. Let $\pi:\mathcal{U}\rightarrow M$ be
the $U(n)$ frame bundle of $M$. A point in $\mathcal{U}$ is a pair $(p,v)$
with $\pi(p,v)=p\in M$ and $v:\mathbb{R}^{2n}\rightarrow T_{p}M$ an
isomorphism satisfying $v^{\ast}(\omega_{p})=\omega_{0}$ and
$v^{\ast}(h|_{p})=h_{0}$ (the standard metric on $\mathbb{C}^{n}$). The right
action of $U(n)$ on $\mathcal{U}$ is free: $\gamma(p,v)=(p,v\circ\gamma)$ for
any $\gamma\in U(n)$.

######  Proposition 4.3 (Joyce-Lee-Schoen).

Let $(M,\omega)$ be a real $2n$-dimensional symplectic manifold without
boundary, and a Riemannian metric $h$ compatible with $\omega$ and an almost
complex structure $J$. Let $\mathcal{U}$ be the $U(n)$ frame bundle of $M$.
Then for small $\varepsilon>0$ we can choose a family of embeddings
$\Upsilon_{p,v}:B^{2n}_{\varepsilon}\rightarrow M$ depending smoothly on
$\left(p,v\right)\in U$, where $B^{2n}_{\varepsilon}$ is the ball of radius
$\varepsilon$ about $0$ in $\mathbb{C}^{n},$ such that for all
$\left(p,v\right)\in U$ we have

  1. (1)

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

  2. (2)

$\Upsilon_{p,v\circ\gamma}(0)\equiv\Upsilon_{p,v}\circ\gamma$ for all
$\gamma\in U(n);$

  3. (3)

$\Upsilon_{p,v}^{\ast}(\omega)\equiv\omega_{0}=\frac{\sqrt{-1}}{2}\sum_{j=1}^{n}dz_{j}\wedge
d\bar{z}_{j};$ and

  4. (4)

$\Upsilon_{p,v}^{\ast}(h)=h_{0}+O(|z|)$.

Moreover, for a dilation map $\mathbf{t}:B^{2n}_{R}\to B^{2n}_{\varepsilon}$
given by $\mathbf{t}(z)=tz$ where $t\leq\varepsilon/R$, set
$h^{t}_{p,v}=t^{-2}(\Upsilon_{p,v}\circ\mathbf{t})^{*}h$. Then it holds

  1. (5)

$\|h^{t}_{p,v}-h_{0}\|_{C^{0}}\leq C_{0}t\ \ \ \ \mbox{and}\ \ \ \|\partial
h^{t}_{p,v}\|_{C^{0}}\leq C_{1}t$,

where norms are taken w.r.t. $h_{0}$ and $\partial$ is the Levi-Civita
connection of $h_{0}$.

######  Proposition 4.4.

Suppose that $M$ is a compact symplectic manifold with a compatible Riemannian
metric $h$. Suppose that $L$ is a compact Lagrangian submanifold of $M$ with
second fundamental form bounded above by $K$ and volume bounded above by
$V_{0}$. Given $c_{n}>0,$ there exists an $r_{0}=r_{0}(K,c_{n})>0$ and a
finite cover of $L$ by Darboux charts
$\Upsilon_{p_{i},v_{i}}$:$B_{r_{0}}^{2n}\rightarrow M$ centered at points
$p_{i}$ on $L$ such that

  1. (1)

The connected component of $L\cap B_{r_{0}}^{2n}$ containing $p_{j}$ is
represented by a graph $\left(x,d\varphi^{(j)}\right)$ over
$B_{r_{0}}^{2n}\cap\mathbb{R}^{n}\times\left\\{0\right\\}$ for some potential
$\varphi^{(j)}$.

  2. (2)

The tangent plane at each point of this connected component satisfies a
closeness condition with respect to the planes
$v_{i}(\mathbb{R}^{n}\times\\{0\\}):$

(4.2) $\max_{\begin{subarray}{c}\left|e\right|_{g}=1,\text{ }e\in T_{p}L\\\
\left|\nu\right|_{\delta_{0}}=1,\nu\in\left\\{0\right\\}\times\mathbb{R}^{n}\end{subarray}}e\cdot\nu<c_{n}$

where the dot product is in the euclidean metric $\delta_{0}$, and $c_{n}$ is
a small universal constant (say $c_{n}=\frac{1}{10\sqrt{n}}$) chosen so that
quantities such as the volume element and coordinate expression for $h$ are
bounded by universal constants.

  3. (3)

The ambient metric $h$ is very close to the euclidean metric, that is
$\|h-\delta_{0}\|<c_{n}$ for some $c_{n}$ (can be the same $c_{n}$ as in (2)
above).

  4. (4)

The submanifold $L$ is covered by the charts obtained by restricting these
charts to $B_{r_{0}}^{2n}(p_{j})$

  5. (5)

The number of such points $\left\\{p_{j}\right\\}$ satisfies

(4.3) ${N}(K,V_{0})\leq\frac{C(n)V_{0}}{r_{0}^{n}(K,c_{n})}.$

###### Proof.

At each point $p\in L$ we take a Darboux chart $\Upsilon_{p,v}$ as described
above with that $T_{p}L=\mathbb{R}^{n}\times\left\\{0\right\\}$ in the given
chart. Note that after some fixed re-scalings, we can assert via Proposition
4.3 that $\Upsilon_{p,v}$ exists on $B_{\varepsilon_{0}}^{2n}$ and satisfies
any near euclidean metric conditions we choose to prescribe, including the
closeness condition: $\left|h-\delta_{0}\right|<c$. Now we may apply
Proposition 5.1 which asserts existence of a ball
$B_{r_{0}}^{n}(p)\subset\mathbb{R}^{n}\times\\{0\\}$ over which $L$ is
representable as a graph, with (4.2) holding. The quantity $r_{0}$ will depend
on $K$.

Consider the compact immersed submanifold $L$ as a metric space $(L,d)$.
Taking a finite cover of metric balls $B_{r_{0}/4}(p)$ for $p\in L$ and
applying Vitalli’s covering Lemma, we conclude that there is a subset of these
points $\\{p_{j}\\}$ so that $L=\cup_{i}B_{3_{0}/4}(p_{i})$ and
$B_{r_{0}/4}(p_{i})$ are mutually disjoint. By (4.2), $L\cap
B_{3r_{0}/4}(p_{i})$ is in the image of a graph given by Proposition 5.1. In
particular, the disjoint $B_{r_{0}/4}(p_{i})$’s have a minimum total volume
$\omega_{n}c^{n}r_{0}^{n}$. The bound (4.3) on the number of balls follows. As
$\\{B_{3r_{0}/4}(p_{i})\\}$ covers $L$ and each of these balls is contained in
a graph over $B_{r_{0}}^{n},$ we take the set of the graphs as the cover. ∎

The scalar functions from the the exact sections of $T^{*}L$ are globally
defined on $L$ via the abstract Weinstein map $\Psi$. We have utilized them to
establish short-time existence and uniqueness for our geometric flow of $F$.
However, for higher order a-priori estimates, we need to set up the flow
equation in a Darboux chart with estimates on the metric as described above.
Fortunately, each $\Upsilon_{p,v}$ is a symplectomorphism, which takes
gradient graphs $(x,d\varphi)$ to Lagrangian submanifolds, so the computations
in section 3.1 can be repeated verbatim, with $\Upsilon_{p,v}$ in place of
$\Psi$. In particular, in each chart, the flow is determined by an equation

(4.4) $\varphi_{t}=-g^{ap}g^{ij}\frac{\partial^{4}\varphi}{\partial
x^{a}\partial x^{j}\partial x^{i}\partial
x^{p}}+G(x,D\varphi,D^{2}\varphi,D^{3}\varphi).$

######  Remark 4.5.

A precise computation in Darboux coordinates of the expression (3.10) gives

$\displaystyle h\left(H,Je_{p}\right)$
$\displaystyle=g^{ij}\varphi_{pij}+g^{ij}\tilde{\Gamma}_{ij}^{p+n}+g^{ij}\varphi_{kj}\delta^{km}\tilde{\Gamma}_{i,m+n}^{p+n}+g^{ij}\varphi_{ki}\delta^{km}\tilde{\Gamma}_{m+n,j}^{p+n}$
$\displaystyle+g^{ij}\varphi_{ki}\varphi_{lj}\delta^{km}\delta^{lr}\tilde{\Gamma}_{m+n,r+n}^{p+n}-g^{ij}\tilde{\Gamma}_{ij}^{q}\varphi_{pq}-g^{ij}\varphi_{kj}\delta^{km}\tilde{\Gamma}_{i,m+n}^{q}\varphi_{pq}$
$\displaystyle-g^{ij}\varphi_{ki}\delta^{km}\tilde{\Gamma}_{m+n,j}^{q}\varphi_{pq}-g^{ij}\varphi_{ki}\varphi_{lj}\delta^{km}\delta^{lr}\tilde{\Gamma}_{m+n,r+n}^{q}\varphi_{pq}$

where $\tilde{\Gamma}_{ij}^{q}$ are Christoffel symbols in the ambient metric
$\left(M,h\right)$. Considering that each expression of the form $g^{ab}$ is a
smooth function in terms of $D^{2}\varphi$ with dependence on zero order of
$h$ and each $\tilde{\Gamma}_{ij}^{\beta}$ expression depends on $Dh$ and $h$,
one may conclude (after computing $\operatorname{div}(JH)$ as in (3.11)) that
$G$ can be written as a sum of expressions that are

  1. (1)

quadratic in $D^{3}\varphi$ and smooth in $D^{2}\varphi,h$ in a predetermined
way

  2. (2)

linear in $D^{3}\varphi$, smooth in $D^{2}\varphi,h,Dh$ in a predetermined way

  3. (3)

smooth in $D^{2}\varphi,h$ and linear in $D^{2}h$ in a predetermined way

  4. (4)

smooth in $D^{2}\varphi,h,Dh$ in a predetermined way.

This allows us to make a claim that there is uniform control on the important
quantities involved in the equation we are solving.

######  Proposition 4.6.

Suppose that $L$ is a compact Lagrangian manifold with volume $V_{0}$ and
evolves by (1.1) on $[0,T)$. If the norm of second fundamental $A$ of $L_{t}$
satisfies $|A|_{g(t)}\leq K$ for $t\in[0,T)$, then after a fixed rescaling on
$M$ there is a finite set of Darboux charts such that

  1. (1)

The submanifold is covered by graphs over
$B_{1}^{2n}\cap\mathbb{R}^{n}\times\left\\{0\right\\}$.

  2. (2)

The submanifold is graphical over
$B_{5}^{2n}\cap\mathbb{R}^{n}\times\left\\{0\right\\}$ in each chart.

  3. (3)

The slope bound (4.2) holds over $B_{5}^{n}(0).$

  4. (4)

The flow (1.1) is governed by (4.4) locally in these charts.

  5. (5)

For each chart, the $G$ from (4.4) satisfies a uniform bound on any fixed
order derivatives of $G$ (in terms of all four arguments, not with respect to
$x$ coordinate before embedding.)

  6. (6)

The number of charts is controlled

(4.5) ${N}(K,V_{0})\leq C(K)\frac{V_{0}}{r_{0}^{n}(K)}.$

###### Proof.

Rescale $M$ so that $r_{0}=5.$ Then the expression $G$ becomes predictably
controlled by Remark 4.5. Choosing a cover with interior balls, as in the
proof of Proposition 4.4, determines the necessary number of balls. ∎

###  4.2. Localization

Let $L_{t}$ evolve by (1.1) with time $t\in[0,T),$ and assume
$\left|A\right|_{g(t)}\leq K$ for all $L_{t}.$ Our goal is to establish
integral bounds for $\left|\nabla^{l}A\right|_{g(t)}^{2},$ which only depend
on $k,K,M$ and the initial volume $V_{0}$ of $L_{0}.$ To derive the
differential inequality (4.1) at any time $t_{0}$, we use Proposition 4.6 and
express geometric quantities $g,A,\nabla^{l}A$, etc., in the (no more than
$N)$ Darboux charts in terms of $\varphi(x,t)$ for $x\in B_{5}^{n}.$ By
compactness of $L$ and smoothness of the flow, the flow will continue to be
described by graphs of $d\varphi(x,t)$ in this open union of $N$ charts for
$t\in[t_{0},t_{1})$ for some $t_{1}>t_{0}.$

To be precise, each of the Darboux charts in Lemma 4.6 has a product
structure; we may assume that each chart contains coordinates
$B_{4}^{n}(0)\times B_{2}^{n}(0)$ so that $L$ is graphical over $B_{5}^{n}(0)$
and further that the collection of $B_{1}^{n}(0)\times B_{1}^{n}(0)$ covers
$L$. Now we may fix once and for all a function $\eta$ which is equal to $1$
on $B_{1}^{n}(0)\times B_{1}^{n}(0)$ and vanishes within $B_{2}^{n}(0)\times
B_{2}^{n}(0)$. For a given chart $\Upsilon^{\alpha}$ (here
$\alpha\in\left\\{1,..N\right\\}$ indexes our choice of charts) we call the
function $\eta_{\alpha}$. This function will have uniformly bounded dependence
on the variables $x$ and $y$ in the chart.

Now once these $\eta_{\alpha}$ are chosen, we may then define a partition of
unity for the union of charts which form a tubular neighborhood of $L$, which
will restrict to a partition of unity for small variations of $L$:

(4.6) $\rho_{\alpha}^{2}:=\frac{\eta_{\alpha}^{2}}{\sum\eta_{\alpha}^{2}}.$

By compactness of the unit frame bundle and the smoothness of the family of
charts defined in Proposition 4.3, the transition functions between charts
will have bounded derivatives to any order. Thus, in a fixed chart, where a
piece of $L$ is represented as $\left\\{\left(x,d\varphi(x)\right):x\in
B_{1}(0)\right\\}$, the dependence of $\rho_{\alpha}^{2}$ will be uniformly
controlled in terms of these variables, so there is a uniform pointwise bound

(4.7) $\left|D_{x}^{2}\rho_{\alpha}^{2}\right|\leq
C(D^{3}\varphi,D^{2}\varphi,D\varphi,x)$

were this dependence is at most linear on $D^{3}\varphi$. We will be using the
$x$ coordinates as charts for $L.$

Note also that, if we have a uniform bound on $\frac{d}{dt}D\varphi$ and
$\frac{d}{dt}D^{2}\varphi$ we can conclude a positive lower bound on
$t_{1}-t_{0};$ the flow will be described by graphs of $\varphi(x,t)$ in these
$N$ charts, and the condition (4.2) will be satisfied for a slightly larger
$c_{n}^{\prime}$ (say $c_{n}^{\prime}=\frac{1}{5\sqrt{n}}$ instead of
$c_{n}=\frac{1}{10\sqrt{n}})$.

####  4.2.1. Expression for metric and second fundamental form

In the Darboux charts for $M$, the manifold $L$ is expressed graphically over
the $x$ coordinate via

$x\mapsto F(x)=\left(x,d\varphi(x)\right).$

Thus we have a tangential frame:

(4.8) $e_{i}=\partial_{x^{i}}F=E_{i}+\varphi_{ik}\delta^{km}E_{m+n}$

with

$g_{ij}=h_{ij}+\varphi_{ik}\delta^{km}\varphi_{jl}\delta^{lr}h_{\left(m+n\right)\left(l+n\right)}+\varphi_{jl}\delta^{lr}h_{\left(i\right)\left(l+n\right)}+\varphi_{ik}\delta^{km}h_{\left(m+n\right)\left(j\right)}.$

Recalling (2) and (3) in Proposition 4.4 we may assume that the expression of
$h$ in these coordinates is very close to $\delta_{ij}$ and that
$D^{2}\varphi$ is not large.

Differentiating the components of the induced metric gives

(4.9) $\partial_{x^{p}}g_{ij}=\text{function of }\left(x,D\varphi\right)\\\
+\text{Terms involving up to three factors of }D^{2}\varphi\text{ but no
higher}\\\ +\text{Terms involving up to two factors of }D^{2}\varphi\text{ and
one factor of }D^{3}\varphi.$

######  Lemma 4.7.

In a Darboux chart, using the coordinate basis (4.8) for the tangent space and
$\left\\{Je_{l}\right\\}$ for the normal space, the covariant derivatives of
the second fundamental form and of the potential $\varphi$ are related by

(4.10) $\nabla^{k-1}A=D^{k+2}\varphi+S_{k}$

where $S_{1}$ is a smooth controlled function depending on the chart, $h$ and
$D^{2}\varphi$, $S_{2}$ depends also on $D^{3}\varphi$ and for $k\geq 3:$

  1. (1)

Each $S_{k}$ is a sum of of multilinear forms of
$D^{4}\varphi,...,D^{k+1}\varphi$

  2. (2)

The coefficients of these forms are functions of
$\left(x,D\varphi,D^{2}\varphi,D^{3}\varphi\right)$

  3. (3)

The total sum of the derivatives of $D^{3}\varphi$ that occur in a given term
is no more than $k-2.$

(Note that (4.10) is interpreted as literal equality of the symbols in the
choice of basis, not simply “up to a smooth function”)

###### Proof.

Starting with $k=1$, differentiate in the ambient space

$\tilde{\nabla}_{e_{i}}e_{j}=\tilde{\Gamma}_{ji}^{\beta}E_{\beta}+\varphi_{jmi}\delta^{mk}E_{n+k}+\varphi_{jm}\delta^{mk}\tilde{\Gamma}_{n+k,i}^{\beta}E_{\beta}+\varphi_{jm}\varphi_{ri}\delta^{mk}\delta^{rl}\tilde{\Gamma}_{n+k,n+l}^{\beta}E_{\beta}.$

Using $e_{j}$ and $Je_{k}$ as frame and normal frame,

$\displaystyle A_{ijl}$
$\displaystyle=\langle\tilde{\nabla}_{e_{i}}e_{j},Je_{l}\rangle$
$\displaystyle=\omega(e_{l},\varphi_{jmi}\delta^{mk}E_{n+k})+\omega(e_{l},\tilde{\Gamma}_{ji}^{\beta}E_{\beta}+\varphi_{jm}\delta^{mk}\tilde{\Gamma}_{n+k,i}^{\beta}E_{\beta}+\varphi_{jm}\varphi_{ri}\delta^{mk}\delta^{rl}\tilde{\Gamma}_{n+k,n+l}^{\beta}E_{\beta})$
$\displaystyle=\varphi_{jli}+S_{1}$

where $S_{1}$ is a smooth function involving $D^{2}\varphi$ and the ambient
Christoffel symbols at $\left(x,D\varphi\right)$ and recalling

$\omega(e_{l},E_{\beta})=\omega(E_{l}+\varphi_{lk}\delta^{jk}E_{j+n},E_{\beta})=\left\\{\begin{array}[c]{cl}\delta_{sl},&\
\ \ \text{ if }\beta=s+n\text{ for }s\in\left\\{1,...,n\right\\}\\\
-\varphi_{sl},&\,\,\,\,\text{ if
}\beta=s\in\left\\{1,...,n\right\\}.\end{array}\right.$

Now for $k=2$ ($\nabla$ denotes covariant derivatives on the submanifold):

$\left(\nabla
A\right)_{pijl}=\partial_{p}A_{ijl}-A\left(\nabla_{e_{p}}e_{i},e_{j},e_{l}\right)-A\left(e_{i},\nabla_{e_{p}}e_{j},e_{l}\right)-A\left(e_{i},e_{j},\nabla_{e_{p}}e_{l}\right).$

Now we can compute the Christoffel symbols with respect to the induced metric
$g$:

$\nabla_{e_{p}}e_{i}=1\ast D^{3}\varphi\text{ + lower order }$

thus

$\displaystyle\left(\nabla A\right)_{pijl}$
$\displaystyle=\varphi_{jlip}+\partial_{x^{p}}S_{1}-A\ast\left(D^{3}\varphi\right)+\text{lower
order}$ $\displaystyle=\varphi_{jlip}+D^{3}\varphi\ast D^{3}\varphi+1\ast
D^{3}\varphi+\text{ smooth in other arguments.}$

Here and in sequel, we use $A\ast B$ to denote a predictable linear
combination of terms from tensors $A$ and $B$, and $1\ast T$ to be a
predictable linear combination of $T.$ Now

$\displaystyle\nabla^{2}A$ $\displaystyle=D^{5}\varphi+D^{4}\varphi\ast
D^{3}\varphi+\text{lower order}$ $\displaystyle\nabla^{3}A$
$\displaystyle=D^{6}\varphi+D^{5}\varphi\ast D^{3}\varphi+D^{4}\varphi\ast
D^{4}\varphi\text{ + lower order}$

and so forth. The result follows by inductively applying the product rule and
noting

$\nabla^{k-1}A=D(\nabla^{k-2}A)+D^{3}\varphi\ast\nabla^{k-2}A\text{ + lower
order}$

by the formula for covariant derivative. ∎

###  4.3. Integral inequalities

We will use $\|\cdot\|_{\infty}$ for the supremum norm in the euclidean metric
$\delta_{0}$ and

$\left|D^{m}\varphi\right|_{g}^{2}=g^{i_{1}j_{1}}g^{i_{2}j_{2}}...g^{i_{m}j_{m}}\varphi_{i_{1}...i_{m}}\varphi_{j_{1}...j_{m}}$

to denote the norm squared with respect to $g$ for the locally defined
$m$-tensor $D^{m}\varphi$ instead of the higher covariant derivative tensor
$\nabla^{m}\varphi$. We find that this makes computations on the chosen
Darboux chart more transparent. Note that since $g$ is close to $\delta_{0}$
on the chart (with estimates on errors)

$\frac{\left|D^{m}\varphi\right|_{g}^{2}}{\left|D^{m}\varphi\right|_{\delta_{0}}^{2}}\in(1-c_{n},1+c_{n})$

and

$(1-c_{n})dx\leq dV_{g}\leq\left(1+c_{n}\right)dx$

for some small $c_{n}$. Thus we may regard as equivalent estimates on
integrals against $dx$ and integrals against $dV_{g}$, provided the quantities
we are integrating are nonnegative. However, if the quantity being integrated
is not known to be non-negative, we have to be precise in performing
estimates.

All estimates below implicitly depend on $c_{n},$ but $c_{n}$ need not be
tracked closely: it need not be close to zero.

####  4.3.1. Interpolation inequalities

We use Gagliardo-Nirenberg interpolation to derive integral inequalities that
allow us to integrate multilinear combinations of higher derivatives of
$\varphi$. For simplicity of notation, we will use $C$ for uniform constants
with dependence indicated in its arguments. For our application, we give
interpolations for different range of indices.

######  Lemma 4.8.

Let $\xi$ be a smooth compactly supported vector-valued function on
$\mathbb{R}^{n}$.

  1. (1)

If $j_{1}+j_{2}+j_{3}+...+j_{q}=m,$ then

$\int\left|D^{j_{1}}\xi\ast D^{j_{2}}\xi...\ast D^{j_{q}}\xi\right|^{2}\leq
C\left\|\xi\right\|_{\infty}^{2q-2}\int\left|D^{m}\xi\right|^{2}.$

  2. (2)

If $j_{1}+j_{2}+j_{3}+...+j_{q}+j^{\ast}=2\tilde{m}$ where $j_{q}<\tilde{m}$
and $j^{\ast}\geq 0$, then

$\int\left|D^{j_{1}}\xi\ast D^{j_{2}}\xi...\ast D^{j_{q}}\xi\right|\leq
C\left\|\xi\right\|_{\infty}^{q-\left(2-j^{\ast}/2\tilde{m}\right)}\left(\frac{2\tilde{m}-j^{\ast}}{2\tilde{m}}\int\left|D^{\tilde{m}}\xi\right|^{2}+\frac{j^{\ast}}{2\tilde{m}}\left\|\chi_{\text{supp}\left(\xi\right)}\right\|_{p^{\ast}}^{2\tilde{m}/j^{\ast}}\right).$

  3. (3)

If $j_{1}+...+j_{r}=2\bar{m}+1$ and all $j_{i}\leq\bar{m},$ then for
$\varepsilon>0$

$\int\left|D^{j_{1}}\xi\ast D^{j_{2}}\xi...\ast
D^{j_{r}}\xi\right|\leq\varepsilon\int\left|D^{\bar{m}+1}\xi\right|^{2}+C(\varepsilon,\bar{m},\|\xi\|_{\infty})\,\left(\int\left|D^{\bar{m}}\xi\right|^{2}+\int\chi_{\text{supp}\left(f\right)}\right).$

###### Proof.

For (1), use $p_{i}=\frac{m}{j_{i}}$ and apply the generalized Hölder’s
inequality

$\int\left|D^{j_{1}}\xi\ast D^{j_{2}}\xi...\ast
D^{j_{q}}\xi\right|^{2}\leq\left\|D^{j_{1}}\xi\right\|_{2p_{1}}^{2}...\left\|D^{j_{q}}\xi\right\|_{2p_{q}}^{2}$

and then use the Gagliardo-Nirenberg interpolation inequality (cf. [FFRS21,
Theorem 1.1]) with $\theta_{i}=\frac{j_{i}}{m}$.

For (2), taking $p_{i}=\frac{2\tilde{m}}{j_{i}}$ and
$p^{\ast}=\frac{2\tilde{m}}{j^{\ast}}$ if $j^{\ast}>0$, then

$\int\left|D^{j_{1}}\xi\ast D^{j_{2}}\xi...\ast
D^{j_{q}}\xi\right|\leq\left\|D^{j_{1}}\xi\right\|_{p_{1}}...\left\|D^{j_{q}}\xi\right\|_{p_{q}}\left\|\chi_{\text{supp}\left(\xi\right)}\right\|_{p^{\ast}}.$

Now apply the Gagliardo-Nirenberg interpolation inequality with
$\theta_{i}=\frac{j_{i}}{\tilde{m}}$, applying Young’s inequality if
$j^{\ast}>0$.

For (3), we may split, with $a\leq\bar{m}\leq\bar{m}+1\leq b$

$\displaystyle j_{1}+...+j_{s}$ $\displaystyle=a$ $\displaystyle
j_{s+1}+...+j_{r}$ $\displaystyle=b.$

Now for some $p,q$ conjugates to be determined, let

$p_{i}=\left\\{\begin{array}[c]{cl}\frac{ap}{j_{i}},&\ \ \ \text{ if
}i\in\left\\{1,...,s\right\\}\\\ \frac{bq}{j_{i}},&\,\,\,\,\text{ if
}i\in\left\\{s+1,...,r\right\\}.\end{array}\right.$

Apply the generalized Hölder’s inequality

(4.11) $\int\left|D^{j_{1}}\xi\ast D^{j_{2}}\xi...\ast
D^{j_{r}}\xi\right|\leq\left\|D^{j_{1}}\xi\right\|_{p_{1}}...\left\|D^{j_{q}}\xi\right\|_{p_{r}}.$

We have from the Gagliardo-Nirenberg interpolation inequality

$\left\|D^{j_{i}}\xi\right\|_{p_{i}}\leq\left\\{\begin{array}[c]{cl}\left\|D^{\bar{m}}\xi\right\|_{\frac{ap}{\bar{m}}}^{\frac{j_{i}}{\bar{m}}}\left\|\xi\right\|_{\infty}^{1-\frac{j_{i}}{\bar{m}}}&\
\ \ \text{ if }i\in\left\\{1,...,s\right\\}\text{ with
}\theta_{i}=\frac{j_{i}}{\bar{m}}\\\
\left\|D^{\bar{m}+1}\xi\right\|_{\frac{bq}{\bar{m}+1}}^{\frac{j_{i}}{\bar{m}+1}}\left\|\xi\right\|_{\infty}^{1-\frac{j_{i}}{\bar{m}+1}},&\,\,\,\,\text{
if }i\in\left\\{s+1,...,r\right\\}\text{ with
}\theta_{i}=\frac{j_{i}}{\bar{m}+1}.\end{array}\right.$

Taking the product and then applying Young’s inequality (for the same $p,q$)

$\displaystyle\left\|D^{j_{1}}\xi\right\|_{p_{1}}...\left\|D^{j_{q}}\xi\right\|_{p_{r}}$
$\displaystyle\leq\left\|D^{\bar{m}}\xi\right\|_{\frac{ap}{\bar{m}}}^{\frac{a}{\bar{m}}}\left\|D^{\bar{m}+1}\xi\right\|_{\frac{bq}{\bar{m}+1}}^{\frac{b}{\bar{m}+1}}\left\|\xi\right\|_{\infty}^{r-\frac{a}{m}-\frac{b}{m+1}}$
$\displaystyle\leq
C(\varepsilon,p,q,r,\left\|\xi\right\|_{\infty})\left\|D^{\bar{m}}\xi\right\|_{\frac{ap}{\bar{m}}}^{\frac{ap}{\bar{m}}}+\varepsilon\left\|D^{\bar{m}+1}\xi\right\|_{\frac{bq}{\bar{m}+1}}^{\frac{bq}{\bar{m}+1}}$
(4.12)
$\displaystyle=C(\varepsilon,p,q,r,\left\|\xi\right\|_{\infty})\left\|D^{\bar{m}}\xi\right\|_{2\frac{a(\bar{m}+1)}{\bar{m}(a+1)}}^{2\frac{a(\bar{m}+1)}{\bar{m}(a+1)}}+\varepsilon\left\|D^{\bar{m}+1}\xi\right\|_{2}^{2}$

where in the last line we have made the choices

$q=\frac{2(\bar{m}+1)}{b},\text{ \ }p=\frac{2(\bar{m}+1)}{a+1}.$

Since $1\leq a\leq\bar{m}$ we have

$\frac{a(\bar{m}+1)}{\bar{m}(a+1)}\leq 1$

and can use Hölder’s and Young’s inequalities to get

(4.13)
$C(\varepsilon,p,q)\left\|D^{\bar{m}}\xi\right\|_{2\frac{a(\bar{m}+1)}{\bar{m}(a+1)}}^{2\frac{a(\bar{m}+1)}{\bar{m}(a+1)}}\leq
C(a,\bar{m})\left(\int\left|D^{\bar{m}}\xi\right|^{2}+\int\chi_{\text{supp}\left(f\right)}\right)$

omitting the last term in the case $a=\bar{m}$. Chaining together (4.11, 4.12,
4.13) gives the result.

∎

######  Lemma 4.9.

Let $f\in C^{\infty}(B_{4})$ and $r_{1}<r_{2}\leq 4$.

  1. (1)

If $j_{1}+...+j_{s}=m,$ then

$\int_{B_{r_{1}}}\left|D^{j_{i}}f\cdots D^{j_{s}}f\right|^{2}\leq
C(m,r\,_{1},r_{2})\,\|f\|_{\infty}^{2s-2}\sum_{j=0}^{m}\int_{B_{r_{2}}}\left|D^{j}f\right|^{2}.$

  2. (2)

If $j_{1}+j_{2}+j_{3}+...+j_{s}+j^{\ast}=2\tilde{m}$ where $j_{q}<\tilde{m}$
and $j^{\ast}\geq 0,$ then

$\int_{B_{r_{1}}}\left|D^{j_{i}}f\cdots D^{j_{s}}f\right|\leq
C(\tilde{m},r_{1},r_{2})\,\left\|f\right\|_{\infty}^{s-\left(2-j^{\ast}/2\tilde{m}\right)}\left(\frac{2\tilde{m}-j^{\ast}}{2\tilde{m}}\sum_{j=0}^{\tilde{m}}\int_{B_{r_{2}}}\left|D^{j}f\right|^{2}+\frac{j^{\ast}}{2\tilde{m}}\left\|\chi_{\text{supp}\left(f\right)}\right\|_{p^{\ast}}^{2\tilde{m}/j^{\ast}}\right).$

  3. (3)

If $j_{1}+...+j_{r}=2\bar{m}+1$ and all $j_{i}\leq\bar{m},$ then for
$\varepsilon>0$

$\int_{B_{r_{1}}}\left|D^{j_{1}}f\ast D^{j_{2}}f...\ast
D^{j_{r}}f\right|\leq\varepsilon\int_{B_{r_{2}}}\left|D^{\bar{m}+1}\xi\right|^{2}+C(\varepsilon,r_{1},r_{2},\bar{m},\|f\|_{\infty})\,\left(\sum_{j=0}^{\tilde{m}}\int_{B_{r_{2}}}\left|D^{j}\xi\right|^{2}+1\right).$

###### Proof.

Set $\tilde{\eta}\in C_{0}^{\infty}(B_{3})$ that is 1 on $B_{r_{1}}$, 0 on
$B_{3}(0)\backslash B_{r_{2}}(0)$, $0\leq\tilde{\eta}\leq 1$ and
$\|\tilde{\eta}\|_{C^{m}}\leq C(m)$. By Lemma 4.8 (second line below)

$\displaystyle\int_{B_{r_{1}}}\left|D^{i_{1}}f\cdots D^{i_{s}}f\right|^{2}$
$\displaystyle\leq\int_{B_{r_{2}}}\left|D^{i_{1}}(\tilde{\eta}f)\cdots
D^{i_{s}}(\tilde{\eta}f)\right|^{2}$
$\displaystyle\leq\|\tilde{\eta}f\|_{\infty}^{2s-2}\int_{B_{r_{2}}}\left|D^{m}(\tilde{\eta}f)\right|^{2}$
$\displaystyle\leq
C(m,r_{1},r_{2},\|\tilde{\eta}\|_{C^{m}})\,\|f\|_{\infty}^{2s-2}\sum_{j=0}^{m}\int_{B_{r_{2}}}\left|D^{j}f\right|^{2}.$

The second and third inequalities in the statement of the Lemma follows by
applying the previous Lemma in a similar way. ∎

The following is simple but will be used repeatedly, so we explicitly note it.

######  Lemma 4.10.

Suppose that $r_{1}<r_{2}.$ Then for $\tilde{\varepsilon}>0$

$\int_{B_{r_{1}}}\left|D^{k+3}\varphi\right|^{2}\leq\tilde{\varepsilon}\int_{B_{r_{2}}}\left|D^{k+4}\varphi\right|^{2}+C(\tilde{\varepsilon},r_{1},r_{2})\int_{B_{r_{2}}}\left|D^{k+2}\varphi\right|^{2}.$

###### Proof.

For some $\tilde{\eta}=1$ on $B_{r_{1}}$ supported inside $B_{r_{2}}$

$\displaystyle\int_{B_{r_{2}}}\left|D^{k+3}\varphi\right|^{2}\tilde{\eta}^{2}$
$\displaystyle=-\int_{B_{r_{2}}}\
D^{k+2}\varphi\ast\left(\tilde{\eta}^{2}D^{k+4}\varphi+2\tilde{\eta}D\tilde{\eta}D^{k+3}\varphi\right)$
$\displaystyle\leq\tilde{\varepsilon}\int_{B_{r_{2}}}\tilde{\eta}^{2}\left|D^{k+4}\varphi\right|^{2}+\frac{1}{\tilde{\varepsilon}}C\int_{B_{r_{2}}}\tilde{\eta}^{2}\left|D^{k+2}\varphi\right|^{2}$
$\displaystyle+\frac{1}{2}\int_{B_{r_{2}}}\tilde{\eta}^{2}\left|D^{k+3}\varphi\right|^{2}+C\int_{B_{r_{2}}}\left|D\tilde{\eta}\right|^{2}\left|D^{k+2}\varphi\right|^{2}.$

Thus

$\int_{B_{r_{1}}}\left|D^{k+3}\varphi\right|^{2}\leq
2\tilde{\varepsilon}\int_{B_{r_{2}}}\tilde{\eta}^{2}\left|D^{k+4}\varphi\right|^{2}+\frac{2}{\tilde{\varepsilon}}C\int_{B_{r_{2}}}\left(\tilde{\eta}^{2}+\left|D\tilde{\eta}\right|^{2}\right)\left|D^{k+2}\varphi\right|^{2}.$

∎

####  4.3.2. Evolution inequalities

######  Proposition 4.11.

Let $\rho_{\alpha}^{2}\in C_{0}^{\infty}(B_{2}(0))$ defined by (4.6). Working
in Darboux charts, for $\varepsilon>0$ we have

(4.14)
$\displaystyle\int_{B_{2}}\frac{d}{dt}\left(\left|D^{k+2}\varphi\right|_{g}^{2}dV_{g}\right)\rho_{\alpha}^{2}$
$\displaystyle\leq-2\int_{B_{2}}\left|D^{k+4}\varphi\right|_{g}^{2}\rho_{\alpha}^{2}dV_{g}+\varepsilon\int_{B_{3}}|D^{k+4}\varphi|^{2}dV_{g}$
$\displaystyle+C(k,\varepsilon,\left\|\varphi\right\|_{C^{3}})\left(\sum_{m=3}^{k+2}\int_{B_{3}}\left|D^{m}\varphi\right|_{g}^{2}dV_{g}+1\right).$

###### Proof.

In a Darboux chart, express $dV_{g}=V_{g}dx$. We have

$\displaystyle\frac{d}{dt}\left(\left|D^{k+2}\varphi\right|_{g}^{2}V_{g}\right)=\,$
$\displaystyle
2\left(\partial_{t}\varphi_{i_{1}...i_{k+2}}\right)\left(\varphi_{j_{1}...j_{k+2}}g^{i_{1}j_{1}}g^{i_{2}j_{2}}...g^{i_{k+2}j_{k+2}}V_{g}\right)$
$\displaystyle+\varphi_{i_{1}...i_{k+2}}\varphi_{j_{1}...j_{k+2}}\partial_{t}\left(g^{i_{1}j_{1}}g^{i_{2}j_{2}}...g^{i_{k+2}j_{k+2}}V_{g}\right)$
(4.15) $\displaystyle=\,$
$\displaystyle-2(g^{kl}g^{pq}\varphi_{klpq}+G)_{i_{1}...i_{k+2}}\left(\varphi_{j_{1}...j_{k+2}}g^{i_{1}j_{1}}...g^{i_{k+2}j_{k+2}}V_{g}\right)$
$\displaystyle+\varphi_{i_{1}...i_{k+2}}\varphi_{j_{1}...j_{k+2}}\partial_{t}\left(g^{i_{1}j_{1}}g^{i_{2}j_{2}}...g^{i_{k+2}j_{k+2}}V_{g}\right).$

We count the highest order of derivatives of $\varphi$ in $x^{1},...,x^{n}$
for each term below:

  1. (1)

$g,g^{-1},V_{g}$ are of 2nd order

  2. (2)

$\partial_{t}g,\partial_{t}g^{-1}$ and
$\partial_{t}V_{g}=V_{g}g^{ij}\partial_{t}g_{ij}$ are of 6th order

  3. (3)

$(g^{kl}g^{pq}\varphi_{klpq})_{i_{1}...i_{k+2}}$ is of $(k+6)$th order and
$G_{i_{1}...i_{k+2}}$ is of $(k+5)$th order.

For the sake of notation, we will use

  1. (1)

$P=P(x,D\varphi,D^{2}\varphi,D^{3}\varphi)$,

  2. (2)

$Q=P(x,D\varphi,...,D^{k+3}\varphi)\ $to be described in (4.19) and below.

  3. (3)

Bounded second order quantities are absorbed and not explicitly stated unless
necessary (in particular, $dV_{g}$ will be dropped when not being
differentiated.)

Multiplying $\rho_{\alpha}^{2}$ to localize in a chart then integrate on $L$,
we may then perform:

(A) Integration by parts twice the first term in (4.15) leads to

(4.16)
$\int_{B_{2}}(-2g^{kl}g^{pq}\varphi_{klpq})_{i_{1}...i_{k+2}}\left(\varphi_{j_{1}...j_{k+2}}g^{i_{1}j_{1}}...g^{i_{k+2}j_{k+2}}V_{g}\right)\rho_{\alpha}^{2}=-2\int_{B_{2}}\left|D^{k+4}\varphi\right|_{g}^{2}\rho_{\alpha}^{2}\,dV_{g}+I$

where

$I=\int_{B_{2}}D^{k+4}\varphi\ast\left(D^{4}\varphi+P^{2})\ast
D^{k+2}\varphi\rho_{\alpha}^{2}+D^{k+3}\varphi\ast\left(P\ast\rho_{\alpha}^{2}+D\rho_{\alpha}^{2}\right)+D^{k+2}\varphi\ast\left(P\ast
D\rho_{\alpha}^{2}+D^{2}\rho_{\alpha}^{2}\right)\right).$

To deal with the first term in $I$:

(4.17) $\int_{B_{2}}D^{k+4}\varphi\ast\left(D^{4}\varphi+P^{2}\right)\ast
D^{k+2}\varphi\leq\varepsilon\int_{B_{2}}\rho_{\alpha}^{2}|D^{k+4}\varphi|^{2}+C(\varepsilon)\int_{B_{2}}\rho_{\alpha}^{2}\left(\left|D^{k+2}\varphi\ast\left(D^{4}\varphi+P^{2}\right)\right|^{2}\right).$

Lemma 4.9 with $m=k$, $f=$ $D^{3}\varphi$ and $r_{2}=5/2$ yields

$\int_{B_{2}}\left|D^{k+2}\varphi\ast D^{4}\varphi\right|^{2}\leq
C(D^{3}\varphi)\sum_{j=0}^{m}\int_{B_{5/2}}\left|D^{j+3}\varphi\right|^{2}.$

Applying Lemma (4.10) to the highest order term provides a bound of (4.17) by
the positive terms in (4.14) noting also that

$\int_{B_{2}}\left|D^{k+2}\varphi\ast P^{2}\right|^{2}\leq
C(D^{3}\varphi)\int_{B_{2}}\left|D^{k+2}\varphi\right|^{2}.$

Next

(4.18)
$\int_{B_{2}}\left|(P\ast\rho_{\alpha}^{2}+D\rho_{\alpha}^{2})D^{k+4}\varphi\ast
D^{k+3}\varphi\right|\leq\varepsilon\int_{B_{2}}\rho_{\alpha}^{2}|D^{k+4}\varphi|^{2}+\frac{1}{\varepsilon}C(P)\int_{B_{2}}\left|D^{k+3}\varphi\right|^{2}$

recalling that $D\rho_{\alpha}$ is bounded by uniform constants and
$D^{2}\varphi.$ By Lemma 4.10 (choosing $\tilde{\varepsilon}\approx
c\varepsilon^{2}$), (4.18) is bounded by

$\varepsilon\int_{B_{2}}\rho_{\alpha}^{2}|D^{k+4}\varphi|^{2}+\varepsilon\int_{B_{3}}\left|D^{k+4}\varphi\right|^{2}+\frac{1}{\varepsilon^{3}}C\int_{B_{3}}\left|D^{k+2}\varphi\right|^{2}$

which is of the correct form. Finally for $I$, using (4.7),

$\int_{B_{2}}\left|D^{k+4}\varphi\ast D^{k+2}\varphi\ast\left(P\ast
D\rho_{\alpha}^{2}+D^{2}\rho_{\alpha}^{2}\right)\right|\leq\varepsilon\int_{B_{2}}\rho_{\alpha}^{2}|D^{k+4}\varphi|^{2}+C(\varepsilon,P)\int_{B_{2}}\left|D^{k+2}\varphi\right|^{2}.$

(B) Note that when applying the product rule successively to $G$, we will get

  1. (1)

A single highest order term which is linear in the highest order with
coefficients involving at most order $D^{3}\varphi.$

  2. (2)

Second to highest order terms that are linear in the second highest order, may
have a factor of $D^{4}\varphi$, all other dependence of lower order.

  3. (3)

Terms of lower order, which could be multilinearly dependent on various lower
orders.

Thus

(4.19) $G_{i_{1}...i_{k+1}}=1\ast D^{k+4}\varphi+Q.$

with $Q$ having highest order $D^{k+3}\varphi.$ This is observed by iterating
the following expansion: Using $DG$ to denote a derivative in $x$ of the
composition

$x\mapsto G(x,D\varphi(x),D^{2}\varphi(x),D^{3}\varphi(x))$

and $\bar{D}G$ to denote derivatives in all 4 arguments of $G,$ we have

$DG=\bar{D}G\ast\left(D^{4}\varphi+D^{3}\varphi+D^{2}\varphi+\phi\right)$

where $\phi$ is the term generated by $\bar{D}G/Dx.$ Continuing

$\displaystyle D^{2}G$
$\displaystyle=\bar{D}G\ast\left(D^{5}\varphi+D^{4}\varphi+D^{3}\varphi+D\phi\ast\left(D^{4}\varphi+D^{3}\varphi+D^{2}\varphi\right)\right)$
$\displaystyle+\bar{D}^{2}G\ast\left(D^{4}\varphi+D^{3}\varphi+D^{2}\varphi+\phi\right)\ast\left(D^{4}\varphi+D^{3}\varphi+D^{2}\varphi+\phi\right)$
$\displaystyle...$ (4.20) $\displaystyle D^{k+1}G$
$\displaystyle=\bar{D}G\ast\left(D^{k+4}\varphi+D^{k+3}\varphi+D^{k+2}\varphi+...\right)$
$\displaystyle+\bar{D}^{2}G\ast\left(D^{k+3}\varphi+D^{k+2}\varphi+D^{k+1}\varphi+...\right)\ast\left(D^{4}\varphi+D^{3}\varphi+D^{2}\varphi+\phi\right)$
$\displaystyle+\bar{D}^{3}G\ast\left(D^{k+2}\varphi+...\right)\ast\left\\{\left(D^{5}\varphi+...\right)+\left(D^{4}\varphi+...\right)\ast\left(D^{4}\varphi+...\right)\right\\}$
$\displaystyle...$

Now integrate by parts:

$\displaystyle\int_{B_{2}}$ $\displaystyle
G_{i_{1}...i_{k+2}}\left(\varphi_{j_{1}...j_{k+2}}g^{i_{1}j_{1}}...g^{i_{k+2}j_{k+2}}V_{g}\right)\rho_{\alpha}^{2}=-\int_{B_{2}}G_{i_{1}...i_{k+1}}\partial_{{i_{k+2}}}\left[\rho_{\alpha}^{2}\left(\varphi_{j_{1}...j_{k+2}}g^{i_{1}j_{1}}...g^{i_{k+2}j_{k+2}}V_{g}\right)\right]$
$\displaystyle=$ $\displaystyle\int_{B_{2}}\left(D^{k+4}\varphi\ast
D^{k+3}\varphi\right)\rho_{\alpha}^{2}+\int_{B_{2}}(D\rho_{\alpha}^{2}+\rho_{\alpha}^{2}P)D^{k+4}\varphi\ast
D^{k+2}\varphi$ $\displaystyle+\int_{B_{2}}\left(Q\ast
D^{k+3}\varphi\right)\rho_{\alpha}^{2}+\int_{B_{2}}(D\rho_{\alpha}^{2}+\rho_{\alpha}^{2}P)Q\ast
D^{k+2}\varphi.$

We use Peter-Paul’s inequality we split into two types of terms:

$\varepsilon\int_{B_{2}}\left|D^{k+4}\varphi\right|^{2}\,\rho_{\alpha}^{2}+\varepsilon\int_{B_{2}}Q^{2}\rho_{\alpha}^{2}$

and

$C(\varepsilon,D\rho_{\alpha})\int_{B_{2}}\left(\left|D^{k+3}\varphi\right|^{2}+\left|D^{k+2}\varphi\right|^{2}\right)(P^{2}+P+1)\rho_{\alpha}^{2}.$

First,

$\int_{B_{2}}\left|D^{k+3}\varphi\right|^{2}\left(P^{2}+P+1\right)\rho_{\alpha}^{2}\leq
C\left(D^{3}\varphi\right)\int_{B_{2}}\left|D^{k+3}\varphi\right|^{2}$

is bounded by the argument in Lemma (4.10).

One can prove by induction, observing (4.19) and (4.20), that for each term in
$Q,$ the total number of derivatives of $D^{3}\varphi$ that arise will sum up
to no more than $k+1$ (i.e. $D^{k-1}\varphi\ast D^{6}\varphi\ast
D^{5}\varphi=D^{3+k-4}\varphi\ast D^{3+3}\varphi\ast D^{3+2}\varphi$, here
$k-4+3+2=k+1.$) Applying Lemma 4.9 for $f=D^{3}\varphi$ and $m=k+1$ to each of
the squared terms gives

$\int_{B_{2}}Q^{2}dV_{g}\leq
C\left(\|D^{3}\varphi\|\right)\sum_{i=0}^{k+1}\int_{B_{3}}|D^{i+3}\varphi|^{2}$

thus $\varepsilon\int_{B_{2}}Q^{2}dV_{g}$ has the correct bound, by Lemma
4.10.

Finally we finish bounding the last term in (4.15)

$\int\varphi_{i_{1}...i_{k+2}}\varphi_{j_{1}...j_{k+2}}\partial_{t}\left(g^{i_{1}j_{1}}g^{i_{2}j_{2}}...g^{i_{k+2}j_{k+2}}V_{g}\right)\rho_{\alpha}^{2}=\int\left(D^{k+2}\varphi\ast
D^{k+2}\varphi\ast D^{6}\varphi\right)\rho_{\alpha}^{2}.$

Apply Lemma 4.9 for $f=D^{3}\varphi$ and $\tilde{m}=k$

$\int_{B_{2}}\left(D^{k+2}\varphi\ast D^{k+2}\varphi\ast
D^{6}\varphi\right)\rho_{\alpha}^{2}\leq\varepsilon\int_{B_{3}}\left|D^{k+4}\varphi\right|^{2}+C\left(\varepsilon,k,D^{3}\varphi\right)\sum_{j=3}^{k+3}\int_{B_{3}}\left|D^{j}\varphi\right|^{2}$

We may then sweep away the $\int_{B_{2}}\left|D^{k+3}\varphi\right|^{2}$ term
Lemma 4.10 (choosing $\tilde{\varepsilon}\approx c\varepsilon^{2}$) to
conclude the proof. ∎

######  Proposition 4.12.

Let $\rho_{\alpha}^{2}\in C_{0}^{\infty}(B_{2}(0))$. Considering the
decomposition in Lemma 4.7, for $\varepsilon>0$ we have

$\displaystyle\int_{B_{2}}\frac{d}{dt}\left(\left(\left|S_{k}\right|_{g}^{2}+2\langle
D^{k+2}\varphi,S_{k}\rangle_{g}\right)dV_{g}\right)\rho_{\alpha}^{2}$
$\displaystyle\leq
C(k,\varepsilon,D^{3}\varphi)\,\left(\sum_{j=3}^{k+2}\int_{B_{3}}\left|D^{j}\varphi\right|^{2}+1\right)$
$\displaystyle+\varepsilon\int_{B_{3}}\left|D^{k+4}\varphi\right|^{2}.$

###### Proof.

Recall that

(4.21) $S_{k}=\left(1+D^{3}\varphi\right)\ast(D^{k+1}\varphi+D^{k}\varphi\ast
D^{4}\varphi+D^{k-1}\varphi\ast D^{4}\varphi\ast
D^{4}\varphi+D^{k-1}\varphi\ast D^{5}\varphi+...)$

Differentiating with respect to $t$ generates product rule expansions with 4
orders of derivatives added to a factor in each term, that is (modulo lower
order geometrically controlled values like $V_{g})$

(4.22) $\displaystyle\frac{d}{dt}\left|S_{k}\right|_{g}^{2}dV_{g}$
$\displaystyle=\left(1+D^{3}\varphi\right)\ast\left(D^{k+5}\varphi+D^{k+4}\varphi\ast
D^{4}\varphi+D^{k}\varphi\ast D^{8}\varphi+...\right)\ast S_{k}$
$\displaystyle\left(D^{6}\varphi+D^{7}\varphi\right)\ast\left(D^{k+1}\varphi+D^{k}\varphi\ast
D^{4}\varphi+D^{k-1}\varphi\ast D^{4}\varphi\ast D^{4}\varphi+...\right)\ast
S_{k}.$

Integrate by parts:

$\displaystyle\int_{B_{2}}$
$\displaystyle\left(D^{k+5}\varphi\ast\left(1+D^{3}\varphi\right)\ast
S_{k}\right)\rho_{\alpha}^{2}=\int_{B_{2}}D^{k+4}\varphi\ast\left(D^{4}\varphi\ast
S_{k}+\left(1+D^{3}\varphi\right)\ast DS_{k}\right)\rho_{\alpha}^{2}$
$\displaystyle+\int_{B_{2}}\left(D^{k+4}\varphi\ast\left(1+D^{3}\varphi\right)\ast
S_{k}\right)\ast D\rho_{\alpha}^{2}$
$\displaystyle\leq\varepsilon\int_{B_{2}}\left|D^{k+4}\varphi\right|^{2}\rho_{\alpha}^{2}+C(\varepsilon)\int_{B_{2}}\left(\left|D^{4}\varphi\ast
S_{k}\right|^{2}+\left|D^{3}\varphi\ast
DS_{k}\right|^{2}\right)\rho_{\alpha}^{2}$
$\displaystyle+C(\varepsilon)\int_{B_{2}}\left|\left(1+D^{3}\varphi\right)\ast
S_{k}\right|^{2}\left|D\rho_{\alpha}\right|^{2}.$

Now apply Lemma 4.9 with $f=D^{3}\varphi$ and $m=k-1$

(4.23) $\int_{B_{2}}\left(\left|D^{4}\varphi\ast
S_{k}\right|^{2}+\left|D^{3}\varphi\ast
DS_{k}\right|^{2}\right)\rho_{\alpha}^{2}\leq
C(k,D^{3}\varphi)\,\sum_{j=3}^{k+2}\int_{B_{3}}\left|D^{j}\varphi\right|^{2}.$

Similarly using $\tilde{\eta}$ as in the proof of Lemma 4.9, we have

$\int_{B_{2}}\left|\left(1+D^{3}\varphi\right)\ast
S_{k}\right|^{2}\left|D\rho_{\alpha}\right|^{2}\leq
C\left(D\rho_{\alpha},D^{3}\varphi\right)\sum_{j=3}^{k+1}\int_{B_{3}}\left|D^{j}\varphi\right|^{2}.$

Continuing with the terms in (4.22)

$\int_{B_{2}}\left(1+D^{3}\varphi\right)\ast\left(D^{k+4}\varphi\ast
D^{4}\varphi\ast
S_{k}\right)\rho_{\alpha}^{2}\leq\varepsilon\int_{B_{2}}\left|D^{k+4}\varphi\right|^{2}\rho_{\alpha}^{2}+C(\varepsilon)\int_{B_{2}}\left|D^{3}\varphi\ast
D^{4}\varphi\ast S_{k}\right|^{2}\rho_{\alpha}^{2}$

with the latter term enjoying the same bound as (4.23). The remaining terms
are of the form

$\int\left(D^{3+j_{1}}\varphi\ast D^{3+j_{2}}\varphi\ast..\ast
D^{3+j_{q}}\varphi\right)\rho_{\alpha}^{2}$

with $j_{1}+...+j_{q}\leq 2k$ so

$\int_{B_{2}}\left(D^{3+j_{1}}\varphi\ast D^{3+j_{2}}\varphi\ast..\ast
D^{3+j_{q}}\varphi\right)\rho_{\alpha}^{2}\leq
C(m,D^{3}\varphi)\,\left(\sum_{j=3}^{k+3}\int_{B_{3}}\left|D^{j}\varphi\right|^{2}+1\right)$

by Lemma 4.9 again. Applying Lemma 4.10 to
$\int_{B_{3}}\left|D^{k+3}\varphi\right|^{2}$ completes the desired bound for
the integral of the (4.22) terms.

Next

(4.24) $\int_{B_{2}}\frac{d}{dt}\left(2\langle
D^{k+2}\varphi,S_{k}\rangle_{g}dV_{g}\right)\rho_{\alpha}^{2}dV_{g}=\int_{B_{2}}\left(D^{k+6}\varphi\ast
S_{k}\right)\rho_{\alpha}^{2}dV_{g}+\int_{B_{2}}D^{k+2}\varphi\ast\frac{d}{dt}\left(S_{k}\ast
V\right)\rho_{\alpha}^{2}.$

Integrating the first term by parts twice yields

$\displaystyle\int_{B_{2}}$ $\displaystyle D^{k+6}\varphi\ast
S_{k}\rho_{\alpha}^{2}=\int_{B_{2}}D^{k+4}\varphi\ast D^{2}\left(S_{k}\ast
V\right)\rho_{\alpha}^{2}+D\rho_{\alpha}^{2}\ast D\left(S_{k}\ast
V\right)+D^{2}\rho_{\alpha}^{2}\ast\left(S_{k}\ast V\right)$
$\displaystyle\leq\varepsilon\int_{B_{2}}\left|D^{k+4}\varphi\right|^{2}\rho_{\alpha}^{2}+C(\varepsilon)\int_{B_{2}}\left|D^{2}S_{k}\right|^{2}\rho_{\alpha}^{2}+C\left(\varepsilon,D^{2}\rho_{\alpha}^{2}\right)\int_{B_{2}}\left(\left|DS_{k}\right|^{2}+\left|S_{k}\right|^{2}\right).$

Again Lemma 4.9 with $f=D^{3}\varphi$ and $\tilde{m}=k$ and $r_{2}=5/2$

(4.25) $\displaystyle\int_{B_{2}}\left|D^{2}S_{k}\right|^{2}\rho_{\alpha}^{2}$
$\displaystyle\leq
C(k,D^{3}\varphi)\,\sum_{j=3}^{k+3}\int_{B_{5/2}}\left|D^{j}\varphi\right|^{2}.$
(4.26)
$\displaystyle\leq\varepsilon\int_{B_{3}}\left|D^{k+4}\varphi\right|^{2}+C\left(\varepsilon,k,D^{3}\varphi\right)\sum_{j=3}^{k+2}\int_{B_{3}}\left|D^{j}\varphi\right|^{2}$

using Lemma 4.10.

Now look at second term in (4.24). Note that

$D^{k+2}\varphi\ast\frac{d}{dt}\left(S_{k}\ast V\right)=D^{k+2}\varphi\ast
D^{k+5}\varphi\ast D^{3}\varphi+D^{k+2}\varphi\ast\left(D^{k+4}\varphi\ast
D^{4}\varphi+...\right).$

The highest order term can be dealt with via integration by parts away from
$D^{k+5}\varphi$ and then an iterated Peter-Paul, carefully choosing smaller
$\varepsilon$ and using Lemma 4.9. For the remaining terms, we need the third
statement in Lemma 4.9 which gives

$\displaystyle\int_{B_{2}}\left(D^{3+j_{1}}\varphi\ast
D^{3+j_{2}}\varphi\ast..\ast D^{3+j_{q}}\varphi\right)\rho_{\alpha}^{2}$
$\displaystyle\leq\varepsilon\int_{B_{5/2}}\left|D^{k+4}\varphi\right|^{2}$
$\displaystyle+C(\varepsilon,k,\frac{5}{2},\|f\|_{\infty})\,\left(\sum_{j=0}^{k}\int_{B_{5/2}}\left|D^{3+j}\varphi\right|^{2}+1\right)$

as $j_{1}+...+j_{q}=2k+1$. A final application of Lemma 4.10 to
$\int_{B_{5/2}}\left|D^{k+3}\varphi\right|^{2}$ completes the proof. ∎

In Proposition 4.11 we have isolated ‘good’ terms
$-2\int_{B_{2}}\left|D^{k+4}\varphi\right|^{2}\rho_{\alpha}^{2}.$ We would
like to use them to offset the ‘bad’ terms of the form
$\varepsilon\int_{B_{3}}\left|D^{k+4}\varphi\right|^{2}dV_{g}$ that occur in
Propositions 4.11 and 4.12. Because the expressions for $D^{k+4}\varphi$ are
different in each chart in the cover, the difficulty arises that we cannot
directly beat the terms occurring on a larger ball by terms on a smaller ball
of different charts, even when the smaller balls cover the larger ball. To
make an argument that the bad terms in a larger ball of one chart are offset
by the good terms in a smaller ball in a different chart requires bounding the
bad terms by a global, well-defined geometric quantity involving derivatives
of the second fundamental form, modulo a lower order difference. This is the
point of the following lemma.

######  Lemma 4.13.

Take a finite cover of charts $\Upsilon^{\alpha}$, each over $B_{4}(0)$ and
partition of unity $\rho_{\alpha}^{2}$ in $B_{2}(0)$ in each respective chart
as described by (4.6). Then

$\sum_{\alpha}\int_{B_{3}}\left|D^{k+4}\varphi\right|^{2}dV_{g}\leq
2N\sum_{m=0}^{k+1}\int_{L}\left|\nabla^{k+1-m}A\right|^{2}dV_{g}+C.$

###### Proof.

Note that from Lemma 4.7

$\left|D^{k+4}\varphi\right|_{g}^{2}\leq
2\left|\nabla^{k+1}A\right|_{g}^{2}+2\left|S_{k+2}\right|_{g}^{2}.$

Let $\iota=\frac{1}{k+2}.$ Then we have, taking $\tilde{\eta}=1$ on each
$B_{3}(0),$ with $\tilde{\eta}\in C_{c}^{\infty}(B_{3+\iota}(0))$

$\displaystyle\int_{B_{3}}\left|D^{k+4}\varphi\right|^{2}dV_{g}$
$\displaystyle\leq
2\int_{B_{3}}\left(\left|\nabla^{k+1}A\right|^{2}+\left|S_{k+2}\right|_{g}^{2}\right)dV_{g}$
$\displaystyle\leq
2\int_{B_{3}}\left|\nabla^{k+1}A\right|^{2}dV_{g}+2\int_{B_{3+\iota}}\left|S_{k+2}\right|_{g}^{2}\tilde{\eta}^{2}dV_{g}$
$\displaystyle\leq
2\int_{B_{3}}\left|\nabla^{k+1}A\right|^{2}dV_{g}+C\left(\sum_{m=3}^{k+3}\int_{B_{3+\iota}}\left|D^{m}\varphi\right|^{2}dV_{g}+1\right)$

by Lemma 4.9. Iterating this argument, using

$\int_{B_{3+\iota}}\left|D^{k+3}\varphi\right|^{2}dV_{g}\leq
2\int_{B_{3+\iota}}\left|\nabla^{k}A\right|^{2}dV_{g}+C\left(\sum_{m=3}^{k+2}\int_{B_{3+2\iota}}\left|D^{m}\varphi\right|^{2}dV_{g}+1\right)$

and so forth, for a total of $k+1$ steps, we have by using
$\left|D^{3}\varphi\right|\leq$ $\left|A\right|+C$ that

$\int_{B_{3}}\left|D^{k+4}\varphi\right|^{2}dV_{g}\leq
2\sum_{m=0}^{k+1}\int_{B_{3+\frac{k+1}{k+2}}}\left|\nabla^{k+1-m}A\right|^{2}\tilde{\eta}^{2}dV_{g}+C.$

Now for any set of functions $\tilde{\eta}_{\alpha}$ who are $1$ on
$B_{r}\subset B_{4}$ on each chart $\Upsilon^{\alpha}$, we can bound

$\displaystyle\sum_{\alpha}\int_{B_{4}}\left|\nabla^{m}A\right|^{2}\tilde{\eta}_{\alpha}^{2}dV_{g}$
$\displaystyle\leq\max_{x\in
L}\left(\sum_{\alpha}\tilde{\eta}_{\alpha}^{2}(x)\right)\int_{L}\left|\nabla^{m}A\right|^{2}dV_{g}\leq
N\int_{L}\left|\nabla^{m}A\right|^{2}dV_{g}.$

It follows that

$\sum_{\alpha}\int_{B_{3}}\left|D^{k+4}\varphi\right|^{2}dV_{g}\leq
2N\sum_{m=0}^{k+1}\int_{L}\left|\nabla^{k+1-m}A\right|^{2}dV_{g}+C.$

∎

###  4.4. Proof of the main theorem

###### Proof of Proposition 4.2.

At a fixed time $t_{0}$ we may take the ambient charts
$\left\\{\Upsilon^{\alpha}\right\\}$ for a tubular neighborhood of $L$ and
subordinate partition of unity $\left\\{\rho_{\alpha}^{2}\right\\}$ which
restrict to charts (via the $x$ coordinate) for $L$ with the same partition of
unity.

Differentiate

$\displaystyle\frac{d}{dt}\int_{L}\left|\nabla^{k-1}A\right|_{g}^{2}dV_{g}$
$\displaystyle=\int_{L}\frac{d}{dt}\left(\left|\nabla^{k-1}A\right|_{g}^{2}dV_{g}\right)$
$\displaystyle=\int_{B_{2}}\left(\sum_{\alpha}\rho_{\alpha}^{2}\right)\frac{d}{dt}\left[\left(\left|D^{k+2}\varphi\right|_{g}^{2}+\left|S_{k}\right|_{g}^{2}+2\langle
D^{k+2}\varphi,S_{k}\rangle_{g}\right)dV_{g}\right]$
$\displaystyle=\sum_{\alpha}\int_{B_{2}}\frac{d}{dt}\left(\left|D^{k+2}\varphi\right|_{g}^{2}dV_{g}\right)\rho_{\alpha}^{2}$
$\displaystyle+\sum_{\alpha}\int_{B_{2}}\frac{d}{dt}\left[\left(\left|S_{k}\right|_{g}^{2}+2\langle
D^{k+2}\varphi,S_{k}\rangle_{g}\right)dV_{g}\right]\rho_{\alpha}^{2}.$

Thus

(4.27)
$\displaystyle\frac{d}{dt}\int_{L}\left|\nabla^{k-1}A\right|_{g}^{2}dV_{g}$
$\displaystyle\leq-2\sum_{\alpha}\int_{B_{2}}\left|D^{k+4}\varphi\right|_{g}^{2}\rho_{\alpha}^{2}dV_{g}+\varepsilon\sum_{\alpha}\int_{B_{3}}|D^{k+4}\varphi|^{2}dV_{g}$
$\displaystyle+\sum_{\alpha}C(k,\varepsilon,\left\|\varphi\right\|_{C^{3}})\left(\sum_{m=3}^{k+2}\int_{B_{3}}\left|D^{m}\varphi\right|_{g}^{2}dV_{g}+1\right)$

by Propositions 4.11 and 4.12. Now apply Lemma 4.13

$\displaystyle\sum_{\alpha}\int_{B_{3}}|D^{k+4}\varphi|^{2}dV_{g}$
$\displaystyle\leq\left(NC\sum_{m=0}^{k+1}\int_{L}|\nabla^{m}A|^{2}dV_{g}+C\right)$
$\displaystyle=NC\sum_{m=0}^{k+1}\int_{L}|\nabla^{m}A|^{2}\left(\sum_{\alpha}\rho_{\alpha}^{2}\right)dV_{g}+C$
$\displaystyle=NC\sum_{m=0}^{k+1}\sum_{\alpha}\int_{B_{2}}|\nabla^{m}A|^{2}\rho_{\alpha}^{2}dV_{g}$
$\displaystyle\leq
NC\sum_{m=0}^{k+1}\sum_{\alpha}\int_{B_{2}}2\left(\left|D^{m+3}\varphi\right|^{2}+\left|S_{m+1}\right|_{g}^{2}\right)\rho_{\alpha}^{2}dV_{g}$
$\displaystyle=2NC\sum_{\alpha}\int_{B_{2}}\left(\left|D^{k+4}\varphi\right|^{2}+\left|S_{k+2}\right|_{g}^{2}\right)\rho_{\alpha}^{2}dV_{g}$
$\displaystyle+2NC\sum_{m=0}^{k}\sum_{\alpha}\int_{B_{2}}2\left(\left|D^{m+3}\varphi\right|^{2}+\left|S_{m+1}\right|_{g}^{2}\right)\rho_{\alpha}^{2}dV_{g}.$

Note that from Lemma 4.10

$2\int_{B_{2}}\left|D^{k+3}\varphi\right|^{2}\rho_{\alpha}^{2}dV_{g}\leq\frac{1}{4NC}\int_{B_{3}}|D^{k+4}\varphi|^{2}dV_{g}+C\left(N,\left|D\rho_{\alpha}^{2}\right|\right)\int_{B_{3}}|D^{k+2}\varphi|^{2}dV_{g}.$

Note also Lemma 4.9, recalling (4.21), then Lemma 4.10 on the highest order
resulting term gives

$\displaystyle
2\int_{B_{2}}\left|S_{k+2}\right|_{g}^{2}\rho_{\alpha}^{2}dV_{g}$
$\displaystyle\leq
C\sum_{m=3}^{k+3}\int_{B_{5/2}}\left|D^{m}\varphi\right|_{g}^{2}+C$
$\displaystyle\leq\frac{1}{4NC}\int_{B_{3}}|D^{k+4}\varphi|^{2}dV_{g}+C\left(N,\left|D\rho_{\alpha}^{2}\right|\right)\sum_{m=3}^{k+2}\int_{B_{3}}\left|D^{m}\varphi\right|_{g}^{2}+C.$

Thus

$\displaystyle\sum_{\alpha}\int_{B_{3}}|D^{k+4}\varphi|^{2}dV_{g}$
$\displaystyle\leq
4NC\sum_{\alpha}\int_{B_{2}}\left|D^{k+4}\varphi\right|^{2}\rho_{\alpha}^{2}dV_{g}+8NC\sum_{m=3}^{k+2}\sum_{\alpha}\int_{B_{2}}\left|D^{m}\varphi\right|^{2}\rho_{\alpha}^{2}dV_{g}$
(4.28)
$\displaystyle+8NC\sum_{m=1}^{k+1}\sum_{\alpha}\int_{B_{2}}\left|S_{m}\right|_{g}^{2}\rho_{\alpha}^{2}dV_{g}.$

Choosing $\varepsilon<(2NC)^{-1}$ in (4.27) in light of (4.4) we have

$\frac{d}{dt}\int_{L}\left|\nabla^{k-1}A\right|_{g}^{2}dV_{g}\leq
C(N,k,\left\|\varphi\right\|_{C^{3}})\left(\sum_{m=3}^{k+2}\int_{B_{3}}\left|D^{m}\varphi\right|_{g}^{2}+\sum_{m=1}^{k+1}\int_{B_{3}}\left|S_{m}\right|_{g}^{2}+1\right).$

Applying Lemma 4.9 to the $\int\left|S_{m}\right|_{g}^{2}$ terms and then
Lemma 4.13 to the $\left|D^{m}\varphi\right|_{g}^{2}$ terms yields the result.
∎

###### Proof of Theorem 4.1.

Suppose now that $F$ is a solution to (1.1) with $\left|A\right|\leq K$ on
$[0,T)$. Starting with

$\int_{L}\left|A\right|^{2}dV_{g}(t)\leq K\operatorname{Vol}\left(L\right)\leq
C$

we may apply Proposition 4.2 and apply differential inequalities: continuing
with

$\frac{d}{dt}\int_{L}\left|\nabla A\right|^{2}dV_{g}(t)\leq
C\int_{L}\left|\nabla
A\right|^{2}dV_{g}(t)+C\int_{L}\left|A\right|^{2}dV_{g}(t)$

and so forth, obtaining bounds of the form

(4.29) $\int_{L}\left|\nabla^{k-1}A\right|^{2}dV_{g}(t)\leq C(k,K,F_{0},T)$

for arbitrary $k$.

Now at any $t_{0}\in[0,T)$ we may take a cover $\Upsilon^{\alpha}$ as
described in Proposition 4.6. By Lemma 4.13 and (4.29) we have

(4.30) $\left\|D^{k}\varphi\right\|_{L^{2}(B_{3})}\leq C(k,K,F_{0},T)$

for all $k,$ in every chart. By Sobolev embedding theorems, we have Hölder
bounds on $D^{k}\varphi$ over $B_{2}$ for each chart. In particular, there
will be uniform bounds on $\frac{d}{dt}D\varphi$ and
$\frac{d}{dt}D^{2}\varphi$ which control the speed of the flow in the chart
and the rate of change of the slope the manifold $L_{t}$ makes with respect to
the tangent plane at the origin in the chart. We conclude then the manifolds
$L_{t}$ will continue to be described by the set of charts taken at $t_{0}$
for $t<\max\left\\{T,t_{0}+\tau\right\\}$ for some positive $\tau$ with an
apriori lower bound. (Perhaps we take $c_{n}$ slightly larger in (4.2)). By
choosing $t_{0}$ near $T$ we are assured that these fixed charts describe the
flow for all values $t\in[t_{0},T)$.

Now observe that with fixed speed bounds, the paths $x\mapsto F(x,t)$ of the
normal flow are Lipschitz and hence the normal flow extends to a well-defined
continuous map

(4.31) $F:L\times[0,T]\rightarrow M.$

We claim that $F\left(\cdot,T\right)$ is a smooth immersion. While within a
chart, the vertical maps

(4.32) $\bar{F}(x):=(x,d\varphi(x,t))$

converge in every Hölder norm to a smooth map at $T,$ we still must argue that
the charts given by the $x$ coordinates do not collapse as $t\rightarrow T$.
This can be argued locally, using coordinates on $L_{t_{0}}$. For any given
$x\in L_{t_{0}}$ we may choose a chart such that $x\in B_{1}(0)\subset
B_{3}(0).$ We are already assuming $F$ is an immersion at $t_{0}$ so this
coordinate chart gives us a coordinate chart for the abstract smooth manifold
$L.$ For $t>t_{0}$ the normal flow $F$ is given by

(4.33) $F(x,t)=(\chi_{t}(x),d\varphi(\chi_{t}(x),t))$

for some local diffeomorphism $\chi_{t}(x):B_{1}(0)\rightarrow B_{2}(0)$ from
Claim 3.3, provided that $t_{0}$ is chosen close enough to $T$ such that

${\chi}_{t}(x)\in B_{2}(0)\text{ for all }x\in B_{1}(0)\text{ and
}t\in[t_{0},T).\text{ }$

This choice of $t_{0}$ is possible given that $\chi_{t}(x)$ is controlled by
the normal projection of $\frac{d\bar{F}}{dt}$ and the inverse
$(d\bar{F})^{-1}$, for $\bar{F}$ defined by (4.32), both of which are
universally controlled given (4.2) and (4.30).

Now because (4.32) is uniformly smooth, it can be extended smoothly to
$[t_{0},T+\delta),$ as well as the normal flow associated to this extension.
Applying Claim 3.3 (note that we may extend the flow outside $B_{3}$ in a nice
way which doesn’t affect the behavior in $B_{2}(0))$ we get a smooth
diffeomorphism $\chi_{T}.$ For $x\in B_{1}(0)$ we can compute the normal flow
$F$,

$F(x,T)=(\chi_{T}(x),d\varphi(\chi_{T}(x),T))$

which is a smooth extension of (4.33) to $T$, by the uniform estimates on
$\varphi$. Now $F(x,T)$ is a smooth immersion from $B_{1}(0)$ because
$\chi_{T}$ is a diffeomorphism. As $x$ was chosen arbitrarily, we conclude the
continuous extension of $F$ defined in (4.31) must be a smooth immersion from
$L$ at $T$.

We may now restart the flow by Proposition 3.2 with initial immersion
$F(x,T)$. The time derivatives of the new flow and $F$ agree to any order at
$T$. Therefore the new flow is a smooth extension of $F$ to
$[0,T+\varepsilon)$ for some $\varepsilon>0$. Moreover, Theorem 3.4 asserts
that this is the only smooth extension. ∎

##  5\. Appendix

###  5.1. Submanifold with bounded second fundamental form $A$

It is a known and frequently used fact that when $|A|$ is bounded then the
submanifold can be written as a graph over a controlled region in its tangent
space. We provide a proof below for any dimension and codimension.

######  Proposition 5.1.

Let $L^{k}$ be a compact manifold embedded in a compact Riemannian manifold
$(M^{k+l},g)$. Suppose that the second fundamental form of $L$ satisfies
$|A|\leq K$ for some constant $K>0$. Then $L$ is locally a graph of a vector-
valued function over a ball $B_{r}(0)\subset T_{p}L$ in a normal neighbourhood
of $p\in L$ in $M$ and $r>C(M,g)(K+1)^{-1}$ for some constant $C(M,g)>0$.

###### Proof.

Step 1. Bound the injectivity radius of $L$ from below in terms of $K$. Assume
$M$ is isometrically embedded in some euclidean space. For the embedding
$F:L^{k}\overset{f}{\to}M^{k+l}\overset{\varphi}{\to}\mathbb{R}^{k+n}$, denote
its second fundamental form by $\tilde{A}$ and note that

$|\tilde{A}|\leq C(|A|+1)\leq C(K+1)$

where $C$ only depends on the isometric embedding $\varphi$. Let
$\gamma:\mathbb{S}^{1}\to L$ be a shortest geodesic loop based at a point
$p\in L$ which is parametrized by arc-length $s$. Suppose
$\gamma(0)=\gamma(a),\gamma^{\prime}(0)=\gamma^{\prime}(a)$. Take a hyperplane
$P$ in $\mathbb{R}^{k+n}$ such that $P$ intersects $\gamma$ at a point $p$
orthogonally. There is a point $q\in\gamma$ where $\gamma$ meets $P$ again at
first time. The angle between the unit vectors $\gamma^{\prime}(p)$ and
$\gamma^{\prime}(q)$ in $\mathbb{R}^{k+n}$ is at least $\frac{\pi}{2}$.
Therefore

$\left|\gamma^{\prime}(p)-\gamma(^{\prime}q)\right|\geq\sqrt{2}.$

Since
$F\circ\gamma:\mathbb{S}^{1}\overset{\gamma}{\to}{L}\overset{F}{\to}\mathbb{R}^{k+l}$
factors through $L$ where $\gamma$ is a geodesic, we have (cf. [ES64], [EL78]
for the notation of the second fundamental form $\nabla d\phi$ of a mapping
$\phi$ between Riemannian manifolds),

$\nabla d(F\circ\gamma)=dF\circ\nabla(d\gamma)+\nabla
d(F)(d\gamma,d\gamma)=\nabla d(F)(d\gamma,d\gamma).$

Since the Christoffel symbols of $\mathbb{S}^{1}$ and of $\mathbb{R}^{n+k}$
are 0 we have

$\nabla d(F\circ\gamma)=(F\circ\gamma)^{\prime\prime}.$

Therefore

$(F\circ\gamma)^{\prime\prime}=\tilde{A}(F)(\gamma^{\prime},\gamma^{\prime}).$

Integrating along the portion of $\gamma$ from $p$ to $q$, we get

$\sqrt{2}\leq\left|\gamma^{\prime}(p)-\gamma^{\prime}(q)\right|\leq\int^{q}_{p}\left|(F\circ\gamma)^{\prime\prime}\right|ds\leq
C(K+1)\,a.$

We conclude that that the length $a$ has a lower bound $C/(K+1)$.

From the Gauss equations and $|\tilde{A}|<C(K+1)$, the sectional curvatures of
$L$ are bounded above by $C^{2}(K+1)^{2}$. We conclude $\mbox{inj}(L)\leq
C(K+1)^{-1}$ [Pet06, p.178].

Step 2. Take a normal neighbourhood $U\subset L$ around a given point $p\in L$
and assume $U$ is contained in a normal neighbourhood $V$ of $M$ at $p$. We
will use $C(g)$ for constants only depending on the ambient geometry of
$(M,g)$. Now, on $V$ we will use
$\delta=\langle\cdot,\cdot\rangle_{\mathbb{R}^{k+l}}$, to measure length of
various geometric quantities already defined in $(V,g)$. First,

$|A|_{\delta}\leq C(g)|A|_{g}\leq C(g)K.$

Identify $T_{p}L$ with $\mathbb{R}^{k}\times\\{0\\}\subset\mathbb{R}^{k+l}$.
Let $e_{1}(x),...,e_{k}(x)$ be the orthonormal frame on $U$ obtained by
parallel transporting an orthonormal frame $e_{1}(0),...,e_{k}(0)$ at $T_{p}L$
along the unique radial geodesic $r_{x}(s)$ in $(U,f^{\ast}g)$ from $0$ to an
arbitrary point $x\in L$, and let $e_{1+k}(0),...,e_{l+k}(0)$ be the
orthonormal frame of $(T_{p}L)^{\perp}$. Integrating along $\gamma_{x}(s)$
leads to

$\displaystyle\left|\langle e_{i}(x),e_{j+l}(0)\rangle\right|$
$\displaystyle=\left|\langle e_{i}(x),e_{j+l}(0)\rangle-\langle
e_{i}(0),e_{j+l}(0)\rangle\right|$
$\displaystyle=\left|\int_{0}^{|x|}\frac{d}{ds}\langle
e_{i}(\gamma_{x}(s)),e_{j+l}(0)\rangle ds\right|$
$\displaystyle=\left|\int_{0}^{|x|}\langle e_{i}^{\prime}(s),e_{j+l}(0)\rangle
ds\right|$
$\displaystyle\leq\int_{0}^{|x|}\left|\langle\nabla_{\partial_{r}}^{g}e_{i},e_{j+l}(0)\rangle\right|ds$
$\displaystyle=\int_{0}^{|x|}\left|\langle
A(\partial_{r},e_{i})+\nabla_{\partial_{r}}^{L}e_{i},e_{j+l}(0)\rangle\right|ds$
$\displaystyle\leq C(g)K\,|x|$

as $\nabla_{\partial_{r}}^{L}e_{i}=0$ on $L$. Therefore, there exists
$r_{0}=C(g)K^{-1}$ (where $C(g)$ may differ from the one above) such that for
any $x\in B_{r_{0}}(0)$ the projection of each $e_{i}(x)$ in each fixed normal
direction $e_{j+l}(0)$ is at most $c_{n}/\sqrt{l}$ and the norm of the
projection is no more than some universal constant $c_{n,l}$ that we get to
choose. It is known that such $T_{x}L$ projects bijectively to $T_{p}L$.
Therefore, locally around any $x\in B_{r_{0}}(0)$, implicit function theorem
asserts that $U$ can be written as a graph over a ball in $T_{x}L$, hence as a
graph over a ball in $T_{p}L$ from the projection. The graphing functions over
the fixed reference plane $T_{P}L$ must coincide on the overlap of any pair of
such balls. This yields a global graphing function $\mathcal{F}$ over
$B_{r_{0}}^{n}(p)\subset T_{p}L$. Moreover, $|D\mathcal{F}|\leq C(g,l)$
because $D\mathcal{F}$ is close to $T_{x}L$ which is close (measured in $l$)
to $T_{p}L$ via the projection. ∎

## References

  * [EI05] Joachim Escher and Kazuo Ito, _Some dynamic properties of volume preserving curvature driven flows_ , Math. Ann. 333 (2005), no. 1, 213–230. MR 2169834
  * [EL78] J. Eells and L. Lemaire, _A report on harmonic maps_ , Bull. London Math. Soc. 10 (1978), no. 1, 1–68. MR 495450
  * [EM02] Y. Eliashberg and N. Mishachev, _Introduction to the $h$-principle_, Graduate Studies in Mathematics, vol. 48, American Mathematical Society, Providence, RI, 2002. MR 1909245
  * [ES64] James Eells, Jr. and J. H. Sampson, _Harmonic mappings of Riemannian manifolds_ , Amer. J. Math. 86 (1964), 109–160. MR 164306
  * [FFRS21] Alberto Fiorenza, Maria Rosaria Formica, Tomáš G. Roskovec, and Filip Soudský, _Detailed proof of classical Gagliardo-Nirenberg interpolation inequality with historical remarks_ , Z. Anal. Anwend. 40 (2021), no. 2, 217–236. MR 4237368
  * [Fif00] Paul C. Fife, _Models for phase separation and their mathematics_ , Electron. J. Differential Equations (2000), No. 48, 26. MR 1772733
  * [HL82] Reese Harvey and H. Blaine Lawson, Jr., _Calibrated geometries_ , Acta Math. 148 (1982), 47–157. MR 666108
  * [JLS11] Dominic Joyce, Yng-Ing Lee, and Richard Schoen, _On the existence of Hamiltonian stationary Lagrangian submanifolds in symplectic manifolds_ , Amer. J. Math. 133 (2011), no. 4, 1067–1092. MR 2823871
  * [Lee13] John M. Lee, _Introduction to smooth manifolds_ , second ed., Graduate Texts in Mathematics, vol. 218, Springer, New York, 2013. MR 2954043
  * [May01] Uwe F. Mayer, _A singular example for the averaged mean curvature flow_ , Experiment. Math. 10 (2001), no. 1, 103–107. MR 1822855
  * [MM12] Carlo Mantegazza and Luca Martinazzi, _A note on quasilinear parabolic equations on manifolds_ , Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 11 (2012), no. 4, 857–874. MR 3060703
  * [MS17] Dusa McDuff and Dietmar Salamon, _Introduction to Symplectic Topology_ , Oxford University Press, 03 2017.
  * [Mul99] W. W. Mullins, _Two-dimensional motion of idealized grain boundaries [MR0078836 (17,1252g)]_ , Fundamental contributions to the continuum theory of evolving phase interfaces in solids, Springer, Berlin, 1999, pp. 70–74. MR 1770893
  * [Pet06] Peter Petersen, _Riemannian geometry_ , second ed., Graduate Texts in Mathematics, vol. 171, Springer, New York, 2006. MR 2243772
  * [PW16] Scott Parkins and Glen Wheeler, _The polyharmonic heat flow of closed plane curves_ , J. Math. Anal. Appl. 439 (2016), no. 2, 608–633. MR 3475940
  * [Wei71] Alan Weinstein, _Symplectic manifolds and their Lagrangian submanifolds_ , Advances in Math. 6 (1971), 329–346. MR 286137
  * [Whe13] Glen Wheeler, _On the curve diffusion flow of closed plane curves_ , Ann. Mat. Pura Appl. (4) 192 (2013), no. 5, 931–950. MR 3105957
  * [Woo20] Albert Wood, _Singularities of lagrangian mean curvature flow_ , Doctoral thesis, UCL (University College London) (2020).