Notation and Definitions In this section we shall introduce a number of the
definitions and key assumptions upon which our results depend. Let
$(\,\tilde{\mathcal{M}},g\,)$ be a $D$-dimensional $C^{3}$-Riemannian manifold
without boundary for some $D\in\mathbb{N}$, where $\mathbb{N}$ denotes the set
of positive integers. Throughout the article any submanifold
$\mathcal{M}\subset\tilde{\mathcal{M}}$ is assumed to be an embedded
submanifold, i.e. such that the map
$\iota:~{}\mathcal{M}\rightarrow\tilde{\mathcal{M}}$, $x\mapsto x$, is an
embedding. We shall denote the dimension of a (sub)-manifold $\mathcal{N}$ by
$\text{dim}(\mathcal{N})\in\\{\,1,\ldots,D\,\,\\}$. Recall that a compact
manifold without boundary is called closed. For any
$\mathcal{A}\subset\tilde{\mathcal{M}}$ we shall denote the toplogical
closure/interior of $\mathcal{A}$ as $\text{cl}(A)/\text{int}(A)$
respectively.

In order for our results to hold we shall require the following notion of
extendability. We say that a submanifold
$\mathcal{N}\subset\tilde{\mathcal{M}}$ is extendable, if there exists a
$\text{dim}(\mathcal{N})$-dimensional, $C^{3}$-submanifold
$\mathcal{A}\subset\tilde{\mathcal{M}}$,
$\text{dim}(\mathcal{N})\in\\{\,1,\ldots,D\,\\}$, without boundary, such that
$\cl\big{(}\mathcal{N}\big{)}$ is a $C^{3}$-submanifold of $\mathcal{A}$. This
extendability condition is widely satisfied. First, $\tilde{\mathcal{M}}$
itself is trivially extendable. Secondly, if $\cl\big{(}\mathcal{N}\big{)}$ is
a $D$-dimensional compact $C^{3}$-submanifold with boundary
$\partial\mathcal{N}$ such that $\partial\mathcal{N}$ is a $(D-1)$-dimensional
orientable $C^{3}$-submanifold of $\tilde{\mathcal{M}}$, then $\mathcal{N}$ is
extendable, since $\cl\big{(}\mathcal{N}\big{)}$ can be embedded into the
union of $\cl\big{(}\mathcal{N}\big{)}$ and a tubular neighbourhood, which
exists by [Lee1997, Exercise 8-5.]. Thirdly, any open $d$-dimensional polygon
$\mathcal{P}$, $d\in\\{\,1,\ldots,D\,\\}$, where
$\cl\big{(}\mathcal{P}\big{)}$ is contained in an open subset $\mathcal{O}$ of
an $d$-dimensional affine plane of $\mathbb{R}^{D}$ is extendable, since
$\mathcal{O}$ can be used for $\mathcal{A}$. In particular this implies that
the interior, all faces and all edges of the unit cube
$[-1,1]^{D}\subset\mathbb{R}^{D}$ are extendable. Moreover the same applies to
many curved submanifolds which are themselves proper $d$-dimensional
submanifolds of a $d$-dimensional submanifold of $\tilde{\mathcal{M}}$, for
example,
$\mathbb{S}^{D-1}=\\{\,x\in\mathbb{R}^{D}:~{}x_{1}^{2}+\ldots+x_{D}^{2}=1\,,~{}x_{D}>c\,\\}$
for some $c\in(-1,1)$. Finally, if $\cl\big{(}\mathcal{N}\big{)}$ is a
$d$-dimensional closed $C^{3}$-submanifold, then $\mathcal{N}$ is extendable.
This implies that spaces like the unit sphere $\mathbb{S}^{D-1}$ and
$\mathbb{S}^{D-1}\backslash\\{\,(1,0,\ldots,0)\,\\}$ are extendable. To define
critical values of a function $h\in C^{1}(\mathcal{N})$ recall that any vector
field $X:~{}\mathcal{N}\rightarrow T\mathcal{N}$, where $T\mathcal{N}$ denotes
the tangent bundle of $\mathcal{N}$, defines a function
$X\\!h:\mathcal{N}\rightarrow\mathbb{R}$. Let $\mathcal{N}$ be a
$d$-dimensional submanifold of $\tilde{\mathcal{M}}$, then the set of critical
points of $h\in C^{1}(\mathcal{N})$ is defined as

$\mathcal{C}_{h}(\mathcal{N})=\left\\{s\in\mathcal{N}:~{}X\\!h(s)=0\text{forallvectorfields}X\text{on}\mathcal{N}\,\right\\}\,.$
(1)

We will denote the number of critical points by

$\mu_{h}\big{(}\mathcal{N}\big{)}=\\#\mathcal{C}_{h}\big{(}\mathcal{N}\big{)}\,.$
(2)

Let $\mathcal{U}\subset\mathcal{N}$ and
$\mathcal{V}\subset\mathbb{R}^{\text{dim}(\mathcal{N})}$ be open and
$\varphi:\mathcal{U}\rightarrow\mathcal{V}$ be a differentiable chart of
$\mathcal{N}$. For $i\in\\{\,1,\ldots,{\text{dim}(\mathcal{N})}\,\\}$ we
define the vector field $E_{i}:\mathcal{U}\rightarrow T\mathcal{U}$ by
$E_{i}h(s)=\frac{d}{dt}\left(\,(h\circ\varphi^{-1})(\varphi(s)+te_{i})\,\right)\big{|}_{t=0}$,
where $e_{i}$ is the $i$-th standard Euclidean basis vector. For all
$s\in\mathcal{U}$, the tangent vectors
$E_{1}(s),\ldots,E_{\text{dim}(\mathcal{N})}(s)$ form a basis of the tangent
space $T_{s}\mathcal{U}$ and so it follows that $E_{i}h(s)=0$ for all
$i\in\\{\,1,\ldots,{\text{dim}(\mathcal{N})}\,\\}$ if and only if $X\\!h(s)=0$
for all vector fields $X$. Moreover, $E_{i}h(s)=0$ if and only if
$\frac{\partial(h\circ\varphi^{-1})}{\partial
x_{i}}\big{(}\varphi(s)\big{)}=0$. Therefore $s\in\mathcal{N}$ is a critical
point in the manifold sense defined by $X\\!h(s)=0$ for all vector fields $X$
if and only if there exists a chart $\varphi$ such that
$\varphi(s)\in\mathbb{R}^{\text{dim}(\mathcal{N})}$ is a critical point of
$h\circ\varphi^{-1}$ in the usual Euclidean sense. The set of critical values
and number of critical values can be naturally extended to functions $h\in
C^{1}(\mathcal{M})$ defined on disjoint unions
$\mathcal{M}=\bigsqcup_{d=0}^{D}\pdM\subset\tilde{\mathcal{M}}$, where each
$\pdM$ is either empty or a $d$-dimensional $C^{3}$-submanifold without
boundary. Letting $h|_{\pdM}$ denote the restriction of $h$ to $\pdM$, this is
done by setting

$\mathcal{C}_{h}(\mathcal{M})=\bigcup_{d=0}^{D}\mathcal{C}_{h|_{\pdM}}(\pdM)\,,~{}~{}~{}\mu_{h}\big{(}\mathcal{M}\big{)}=\sum_{d=0}^{D}\mu_{h|_{\pdM}}\big{(}\pdM\big{)}\,.$
(3)

This broader definition means that our results will apply, among others, to
manifolds with boundary, manifolds with corners and more general Whitney
stratified manifolds (e.g., [Adler:RFT2009, Chapter 8.1]). In particular note
that, if $\mathcal{M}$ is a Whitney stratified manifold, the set of critical
points defined in eq:DefCritsII is the same as the set of critical points
defined in [Adler:RFT2009, p. 194]. To see this recall that if
$\tilde{\mathcal{M}}$ has a Riemannian metric $g$, then $g$ induces a
Riemannian metric on any $C^{3}$-submanifold
$\mathcal{N}\subset\tilde{\mathcal{M}}$ without boundary. We will also denote
this induced Riemannian metric by $g$. The gradient of $h\in
C^{1}(\,\mathcal{N}\,)$, denoted by $\nabla^{\mathcal{N}}\\!h$, is the unique
continuous vector field on $\mathcal{N}$ such that
$g\big{(}\,\nabla^{\mathcal{N}}\\!h,X\,\big{)}=X\\!h$ for every vector field
$X$ on $\mathcal{N}$. Hence given $s\in\mathcal{N}$,
$\nabla^{\mathcal{N}}\\!h(s)=0$ if and only if $X\\!h(s)=0$ for every vector
field $X$ on $\mathcal{N}$, which establishes the connection between the two
definitions. Recall that $(\,\mathcal{U},\varphi\,)$ is a $C^{3}$-chart around
$s\in\tilde{\mathcal{M}}$ if $\mathcal{U}\subset\tilde{\mathcal{M}}$ is open
and contains $s$ and $\varphi\in C^{3}(\,\mathcal{U},\mathcal{V}\,)$ is a
diffeomorphism onto an open set $\mathcal{V}\subset\mathbb{R}^{D}$. For such a
chart and a function $h\in C^{2}(\tilde{\mathcal{M}})$ we define the functions
$h^{\varphi}=h\circ\varphi^{-1}$ and its derivatives as
$h^{\varphi}_{i}=\partial h^{\varphi}/\partial x_{i}$ and
$h^{\varphi}_{ij}=\partial^{2}h^{\varphi}/\partial x_{i}\partial x_{j}$ for
$i,j=1,\ldots,D$. Moreover, we write $\nabla
h^{\varphi}=\big{(}\,h^{\varphi}_{1},\ldots,h^{\varphi}_{D}\,\big{)}$ for the
gradient of $h^{\varphi}$ and $\nabla^{2}{h^{\varphi}}$ for the Hessian of
$h^{\varphi}$, which has as the $(i,j)$-th entry $h^{\varphi}_{ij}$.
Furthermore, given $D,D^{\prime}\in\mathbb{N}$ and a matrix
$A\in\mathbb{R}^{D\times D^{\prime}}$ we define its _vectorization_ as
$\mathbf{vec}(A)=(\,A_{11},\ldots,A_{D^{\prime}1},\ldots,A_{1D},\ldots,A_{DD^{\prime}}\,)^{T}$,
and the _half-vectorization_ of a symmetric matrix $A\in\mathbb{R}^{D\times
D}$ as
$\mathbf{vech}(A)=(\,A_{11},\ldots,A_{D1},A_{22},\ldots,A_{D2},\ldots,A_{D-1D-1},A_{DD-1},A_{DD}\,)^{T}$.
With these definitions we formulate the following assumption on a random field
$f$ on $\tilde{\mathcal{M}}$. $f$ has almost surely
$C^{2}(\tilde{\mathcal{M}})$ sample paths and for all
$s\in\tilde{\mathcal{M}}$ there exists a $C^{3}$-chart
$(\,\mathcal{U},\varphi\,)$ around $s$ mapping
$\mathcal{U}\subset\tilde{\mathcal{M}}$ to an open set
$\mathcal{V}\subset\mathbb{R}^{D}$, such that the following conditions hold.

  * (C1)

There exist positive constants $L\in(0,\infty)$ and $\eta>0$ such that

$\mathbb{E}\big{[}\,f_{ij}^{\varphi}(x)-f_{ij}^{\varphi}(y)\,\big{]}^{2}\leq
L\|\,x-y\,\|^{2\eta}\,,$

for all $x,y\in\mathcal{V}$ and all $i,j\in\\{\,1,\ldots,D\,\\}$.

  * (C2)

For each $(x,y)\in\mathcal{V}\times\mathcal{V}$ with $x\neq y$, the Gaussian
random vector

$\Big{(}\,\nabla
f^{\varphi}(x),\mathbf{vech}\big{(}\nabla^{2}{f^{\varphi}}(x)\big{)}^{T},\nabla
f^{\varphi}(y),\mathbf{vech}\big{(}\nabla^{2}{f^{\varphi}}(y)\big{)}^{T}\,\Big{)}^{T}$

is non-degenerate in the sense that its covariance matrix is positive
definite.

Given a $C^{3}$-chart $(\,\mathcal{U},\varphi\,)$ such that (C1)/(C2) holds
with respect to a random field $f$ on $\tilde{\mathcal{M}}$, we shall say that
$f^{\varphi}$ satisfies (C1)/(C2) on $\mathcal{U}$ and when the existence of
$f$ is clear we will simply say that $(\,\mathcal{U},\varphi\,)$ satisfies
(C1)/(C2) and take $f$ to be implicit. [Cheng:2015extremes] introduced
slightly different conditions (their (C1’) and (C2’)) in order to obtain a
manifold version of their Lemma 4.2. These are not quite what we want, since
if they are assumed for a random field $f$ over $\tilde{\mathcal{M}}$, they do
not necessarily hold for the restriction of $f$ to arbitrary submanifolds
$\mathcal{N}$ of $\tilde{\mathcal{M}}$. This is because for an arbitrary
submanifold $\mathcal{N}\subset\tilde{\mathcal{M}}$ in general there is no
collection of $\text{dim}(\mathcal{N})$ orthonormal fields of the $D$
orthonormal fields chosen on $\tilde{\mathcal{M}}$ such that all of these
fields are orthonormal fields for $\mathcal{N}$. However, in Lemma 1 we show
that, if (C1), (C2) hold in any chart containing $s\in\tilde{\mathcal{M}}$,
then they hold in all charts containing $s$. Thus, in particular they hold for
the the special charts chosen in [Cheng:2015extremes]. Note also that the
inclusion of $f(x)$ and $f(y)$ in the vector in (C2) is unnecessary, despite
its inclusion in the conditions of [Cheng:2015extremes], as it was not used in
the proof of their Lemma 4.2. If $\tilde{\mathcal{M}}\subset\mathbb{R}^{D}$
and $f$ is stationary and satisfies (C1) then it satisfies the generalised
Geman condition, see Lemma LABEL:lem:Geman.

##  1 Proof of the main result

In order to prove our main result we first prove an important technical lemma.
This lemma justifies the use of Conditions (C1) and (C2) since it shows that
they are compatible with changing coordinates. We will use this, in Lemma 1,
to show that if these conditions hold for a random field $f$ on
$\tilde{\mathcal{M}}$, then they also hold for the restriction of $f$ to any
$C^{3}$-submanifold $\mathcal{N}$ of $\tilde{\mathcal{M}}$. Let $f$ be a
Gaussian random field on $\tilde{\mathcal{M}}$ satisfying Assumption 1 for the
$C^{3}$ chart $(\,\mathcal{U},\varphi\,)$. Let
$(\,\mathcal{U}^{\prime},\phi\,)$ be another $C^{3}$-chart such that
$\mathcal{W}=\mathcal{U}\cap\mathcal{U}^{\prime}$ is non-empty. Then for any
$s\in\mathcal{W}$ there exists an open set
$s\in\mathcal{W}_{s}\subset\mathcal{W}$ such that $(\,\mathcal{W}_{s},\phi\,)$
satisfies Assumption 1. Since $f^{\phi}=f^{\varphi}\circ\varphi\circ\phi^{-1}$
on $\phi(\mathcal{W})$, the chain rule implies that for
$\psi=(\,\psi_{1},\ldots,\psi_{D}\,)=\varphi\circ\phi^{-1},$

$f^{\phi}_{i}=\big{(}f^{\varphi}\circ\psi\big{)}_{i}=\sum_{d=1}^{D}\frac{\partial
f^{\varphi}\circ\psi}{\partial\psi_{d}}\frac{\partial\psi_{d}}{\partial
s_{i}}=\sum_{d=1}^{D}f^{\varphi}_{d}\circ\psi\frac{\partial\psi_{d}}{\partial
s_{i}}\,,~{}~{}i=1,\ldots,D\,,$ (4)

which shows that $\nabla f^{\phi}(x)=\nabla f^{\varphi}(\tilde{x})J_{\psi}(x)$
for all $x\in\phi(\mathcal{W})$, where $\tilde{x}=\psi(x)$ and $J_{\psi}(x)$
is the Jacobian matrix of $\psi$ at $x$. For the second order derivatives we
obtain:

$\split f^{\phi}_{ij}&=\sum_{d=1}^{D}\frac{\partial
f^{\varphi}\circ\psi}{\partial\psi_{d}}\frac{\partial^{2}\psi_{d}}{\partial
s_{i}\partial
s_{j}}+\sum_{d,d^{\prime}=1}^{D}\frac{\partial^{2}f^{\varphi}\circ\psi}{\partial\psi_{d}\partial\psi_{d^{\prime}}}\frac{\partial\psi_{d}}{\partial
s_{i}}\frac{\partial\psi_{d^{\prime}}}{\partial s_{j}}\\\
=\sum_{d=1}^{D}f^{\varphi}_{d}\circ\psi\frac{\partial^{2}\psi_{d}}{\partial
s_{i}\partial
s_{j}}+\sum_{d,d^{\prime}=1}^{D}f^{\varphi}_{dd^{\prime}}\circ\psi\frac{\partial\psi_{d}}{\partial
s_{i}}\frac{\partial\psi_{d^{\prime}}}{\partial
s_{j}}\,,~{}~{}i,j=1,\ldots,D\,.$ (5)

This can be written in vector notation as

$\mathbf{vech}\Big{(}\,\nabla^{2}{f^{\phi}}(x)\,\Big{)}=L\,\mathbf{vec}\Bigg{(}\,\sum_{d=1}^{D}f_{d}^{\varphi}\big{(}\tilde{x}\big{)}\nabla^{2}{\psi_{d}}(x)\,\Bigg{)}+L\,\big{(}\,J_{\psi}(x)\otimes
J_{\psi}(x)\,\big{)}\,R\,\mathbf{vech}\Bigg{(}\nabla^{2}{f^{\varphi}}\big{(}\tilde{x}\big{)}\Bigg{)}\,.$
(6)

Here $L\in\mathbb{R}^{D(D+1)/2\times D^{2}}$ is the elimination matrix and
$R\in\mathbb{R}^{D^{2}\times D(D+1)/2}$ is the duplication matrix, the precise
definitions of which can be found in [Magnus1980]. The matrix
$L\big{(}\,J_{\psi}(x)\otimes J_{\psi}(x)\,\big{)}R$ is invertible by Lemma
4.4.iv of [Magnus1980], and the fact that $J_{\psi}(x)$ is invertible because
$\psi$ is a diffeomorphism. Furthermore note that, letting ${\rm
detcov}(U)=\text{det}\big{(}\,\Cov[U]\,\big{)}$, for any random vector
$U\in\mathbb{R}^{D}$,

${\rm detcov}(U)={\rm detcov}(MU)\,/\det(M)^{2D}$ (7)

for any invertible matrix $M\in\mathbb{R}^{D\times D}$, see Lemma
LABEL:lem:MU. Proof of (C2) for the chart $(\,\mathcal{W},\phi\,)$: We shall
show that for all $x,y\in\phi(\mathcal{W})$ the distribution of the random
vector $V$ defined by

$\split
V&=\Big{(}\,\nabla^{T}f^{\phi}(x);\mathbf{vech}\Big{(}\,\nabla^{2}{f^{\phi}}(x)\,\Big{)};\nabla^{T}f^{\phi}(y);\mathbf{vech}\Big{(}\,\nabla^{2}{f^{\phi}}(y)\,\Big{)}\,\Big{)}\\\
=\Bigg{(}J_{\psi}(x)^{T}\nabla^{T}f^{\varphi}(\tilde{x});L\,\mathbf{vec}\Bigg{(}\,\sum_{d=1}^{D}f_{d}^{\varphi}\big{(}\tilde{x}\big{)}\nabla^{2}{\psi_{d}}\,\Bigg{)}+L\,\big{(}\,J_{\psi}(x)\otimes
J_{\psi}(x)\,\big{)}\,R\,\mathbf{vech}\Big{(}\,\nabla^{2}{f^{\varphi}}\big{(}\tilde{x}\big{)}\,\Big{)};\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}J_{\psi}(y)^{T}\nabla^{T}f^{\varphi}(\tilde{y});L\,\mathbf{vec}\Bigg{(}\,\sum_{d=1}^{D}f_{d}^{\varphi}\big{(}\tilde{y}\big{)}\nabla^{2}{\psi_{d}}\,\Bigg{)}+L\,\big{(}\,J_{\psi}(y)\otimes
J_{\psi}(y)\,\big{)}\,R\,\mathbf{vech}\Big{(}\,\nabla^{2}{f^{\varphi}}\big{(}\tilde{y}\,\big{)}\,\Big{)}\Bigg{)},$

where $\tilde{y}=\psi(y)$ (and we have used ; to vertically stack vectors), is
non-degenerate. Let $I_{D\times D}$ denote the identity matrix in
$\mathbb{R}^{D\times D}$. Multiplying $V$ by the the block diagonal matrix $B$
which has diagonal blocks

$\big{(}J_{\psi}^{T}(x)\big{)}^{-1},I_{D(D+1)/2\times
D(D+1)/2},\big{(}J_{\psi}^{T}(y)\big{)}^{-1},I_{D(D+1)/2\times D(D+1)/2}$

only changes $\text{detcov}[V]$ by a positive scalar by applying eq:detcov.
Scaling the gradient appropriately and adding it to the vectors in the third
and sixth components it follows that

$\split&\Bigg{(}\nabla^{T}f^{\varphi}(\tilde{x});L\,\big{(}\,J_{\psi}(x)\otimes
J_{\psi}(x)\,\big{)}\,R\,\mathbf{vech}\Big{(}\,\nabla^{2}{f^{\varphi}}\big{(}\tilde{x}\big{)}\,\Big{)};\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\nabla^{T}f^{\varphi}(\tilde{y});L\big{(}\,J_{\psi}(y)\otimes
J_{\psi}(y)\,\big{)}\,R\,\mathbf{vech}\Big{(}\,\nabla^{2}{f^{\varphi}}\big{(}\tilde{y}\big{)}\Big{)}\,\Bigg{)}$

has, up to a positive scalar, the same ${\rm detcov}$ as the random vector
$V$. For all $x\in\phi(\mathcal{W})$, $L\big{(}\,J_{\psi}(x)\otimes
J_{\psi}(x)\,\big{)}R$ is invertible, and so applying eq:detcov once more with
an appropriate block diagonal matrix shows that, up to a positive scalar, the
random vector

$\split&\Big{(}\,\nabla^{T}f^{\varphi}(\tilde{x});\mathbf{vech}\big{(}\,\nabla^{2}{f^{\varphi}}(\tilde{x})\,\big{)};\nabla^{T}f^{\varphi}(\tilde{y});\mathbf{vech}\big{(}\,\nabla^{2}{f^{\varphi}}(\tilde{y})\,\big{)}\,\Big{)}$

has the same ${\rm detcov}$ as $V$. Since (C2) holds for the chart $\varphi$
this last vector is non-degenerate for all
$\tilde{x},\tilde{y}\in\varphi(\mathcal{W})$ and hence $V$ is non-degenerate.
Proof of (C1) for a chart $(\,\mathcal{W}_{s},\phi\,)$ for $s\in\mathcal{W}$:
Since $\mathcal{W}$ is non-empty and open there exists an open set
$\mathcal{W}_{s}$ around $s$ such that
$\cl\left(\mathcal{W}_{s}\right)\subset\mathcal{W}$. Using equation
eq:2ndDeriv and the fact that $\big{(}\sum_{n=1}^{N}a_{n}\big{)}^{2}\leq
2^{N-1}\sum_{n=1}^{N}a_{n}^{2}$ for all $a_{1},\ldots,a_{N}\in\mathbb{R}$,
proving (C1) reduces to showing that for all
$d,d^{\prime},i,j\in\\{\,1,\ldots,D\,\\}$ E[ ( f^φ_d( ψ(x) ) ∂2ψd∂si∂sj(x) -
f^φ_d( ψ(y) ) ∂2ψd∂si∂sj(y) )^2 ] ≤M ∥ x - y ∥^2η’ and E[ ( f^φ_dd’( ψ(x) )
∂ψd∂si(x)∂ψd∂sj(x) - f^φ_dd’( ψ(y) ) ∂ψd∂si(y)∂ψd’∂sj(y) )^2 ] ≤M ∥ x - y
∥^2η’ for some $\eta^{\prime}>0$, some $M\in(0,\infty)$ and all
$x,y\in\phi(\mathcal{W}_{s})$. In order to bound the left hand side of these
expressions, notice that given a function
$g:\tilde{\mathcal{M}}\rightarrow\mathbb{R}$ and a random field
$Y:\tilde{\mathcal{M}}\rightarrow\mathbb{R}$, &E[ ( g(x)Y(ψ(x)) \- g(y)Y(ψ(y))
)^2 ]
≤2g(x)^2 E[ ( Y(ψ(x)) \- Y(ψ(y)) )^2 ] +2E[ Y^2(ψ(y)) ]( g(x) - g(y) )^2
≤2 max_x ∈ϕ(W_s){ g^2(x) } E[ ( Y(ψ(x)) \- Y(ψ(y)) )^2 ] +2max_x∈φ(W_s){ E[
Y^2(x) ] } ( g(x) - g(y) )^2. Taking $g=\frac{\partial\psi_{d}}{\partial
s_{i}}\frac{\partial\psi_{d^{\prime}}}{\partial s_{j}}$ and
$Y=f^{\varphi}_{dd^{\prime}}$ or $g=\frac{\partial^{2}\psi_{d}}{\partial
s_{i}\partial s_{j}}$ and $Y=f^{\varphi}_{d}$ allows us to provide the desired
bounds. Both possible choices of $g$ are continuous on
$\phi\left(\cl\left(\mathcal{W}_{s}\right)\right)$ and hence
$\max_{x\in\phi(\mathcal{W}_{s})}\\{\,g^{2}(x)\,\\}<\infty$. A similar
argument establishes finiteness for the maximum of the expected value of $Y$
squared. Furthermore both choices of $g$ are differentiable, as $\psi$ is
$C^{3}$, and thus Lipschitz on
$\phi\left(\cl\left(\mathcal{W}_{s}\right)\right)$. Also,

$\mathbb{E}\Bigg{[}\,\Bigg{(}f^{\varphi}_{dd^{\prime}}\big{(}\psi(x)\big{)}-f^{\varphi}_{dd^{\prime}}\big{(}\psi(y)\big{)}\Bigg{)}^{2}\,\Bigg{]}\leq
L\|\,\psi(x)-\psi(y)\,\|^{2\eta}\leq\tilde{L}\|\,x-y\,\|^{2\eta}$ (8)

for some $L,\tilde{L}\in(0,\infty)$ by (C1) and Lipschitz continuity of
$\psi$. A similar argument works for $Y=f^{\varphi}_{d}$ since $f$ is $C^{2}$
and Gaussian. The key observation, that is needed to prove our main result, is
the following extension of Lemma 4.2 from [Cheng:2015extremes] to certain
submanifolds of $\tilde{\mathcal{M}}$. Let
$\mathcal{N}\subset\tilde{\mathcal{M}}$ be a relatively compact and extendable
$d$-dimensional submanifold for some $d\in\mathbb{N}.$ Let $\tilde{f}$ be a
Gaussian random field on $\tilde{\mathcal{M}}$ satisfying Assumption 1 and let
$f$ denote the restriction of $\tilde{f}$ to $\mathcal{N}$. Then
$\mE\big{[}\,\mu_{f}\big{(}\mathcal{N}\big{)}^{2}\,\big{]}<\infty$. Since
$\mathcal{N}$ is extendable there exists a $d$-dimensional submanifold
$\mathcal{A}$ without boundary such that $\cl(\mathcal{N})\subset\mathcal{A}$.
For $s\in\cl(\mathcal{N})$, let $(\,\mathcal{U}_{s},\varphi\,)$ be a chart
around $s$ satisfying Assumption 1. Let
$\left(\,\mathcal{U}^{\mathcal{A}}_{s},\varphi_{\mathcal{A}}\,\right)$ be a
submanifold chart of $\mathcal{A}$ around $s$, i.e. such that,

$\varphi_{\mathcal{A}}\big{(}\mathcal{U}^{\mathcal{A}}_{s}\cap\mathcal{A}\big{)}=\left\\{\,(\,x_{1},\ldots,x_{D}\,)\in\mathbb{R}^{D}:~{}x_{d+1}=\ldots=x_{D}=0\,\right\\}\cap\varphi_{\mathcal{A}}\big{(}\mathcal{U}^{\mathcal{A}}_{s}\big{)}\,.$
(9)

By Lemma 1, there exists some non-empty open set
$\mathcal{W}_{s}\subset\mathcal{U}_{s}^{\mathcal{A}}$ such that
$f^{\varphi_{\mathcal{A}}}$ satisfies assumptions (C1) and (C2) on
$\mathcal{W}_{s}$. By eq:submfChart it follows that

$\varphi_{\mathcal{A}}\big{(}\mathcal{W}_{s}\cap\mathcal{A}\big{)}=\left\\{\,(x_{1},\ldots,x_{D})\in\mathbb{R}^{D}:~{}x_{d+1}=\ldots=x_{D}=0\,\right\\}\cap\varphi_{\mathcal{A}}\big{(}\mathcal{W}_{s}\big{)}\,.$
(10)

Define the projection
$\pi:\varphi_{\mathcal{A}}\big{(}\mathcal{W}_{s}\cap\mathcal{A}\big{)}\rightarrow\mathcal{V}_{s},(\,x_{1},\dots,x_{D}\,)\mapsto(\,x_{1},\dots,x_{d}\,)$,
where
$\mathcal{V}_{s}=\\{\,(\,e_{1}^{T}\varphi_{\mathcal{A}}(y),\ldots,e_{d}^{T}\varphi_{\mathcal{A}}(y)\,):~{}y\in\mathcal{U}^{\mathcal{A}}_{s}\cap\mathcal{A}\,\\}\subset\mathbb{R}^{d}$
and $\\{e_{i}\\}_{i=1,\dots,d}$ is the standard basis on $\mathbb{R}^{d}$.
Then
$\pi\circ\varphi_{\mathcal{A}}:\mathcal{W}_{s}\cap\mathcal{A}\rightarrow\mathcal{V}_{s}$
is a diffeomorphism since
$\pi^{-1}:~{}\mathcal{V}_{s}\rightarrow\varphi_{\mathcal{A}}(\mathcal{W}_{s})$
can be defined as the differentiable map
$(x_{1},\ldots,x_{d})\mapsto(x_{1},\ldots,x_{d},0,\ldots,0)\in\mathbb{R}^{D}$.
Now,
$f^{\pi\circ\varphi_{\mathcal{A}}}=f^{\varphi_{\mathcal{A}}}\circ\pi^{-1}:\mathcal{V}_{s}\rightarrow\mathbb{R}$
satisfies conditions (C1) and (C2) from [Cheng:2015extremes] since the same
calculation as in equations eq:1stDeriv and eq:2ndDeriv shows that, for
$x\in\mathcal{V}_{s}$, the random vectors obtained from
$f^{\varphi_{\mathcal{A}}}\circ\pi^{-1}(x)$ in these conditions are identical
to the random vectors obtained from $f^{\varphi_{\mathcal{A}}}$, evaluated at
$\pi^{-1}(x)$, if the coordinates with indices $i,j\in\\{\,d+1,\ldots,D\,\\}$
are removed. Therefore Lemma 4.2 from [Cheng:2015extremes] can be applied and
as such there exists $\epsilon(s)>0$ such that
$\mu_{f^{\pi\circ\varphi_{\mathcal{A}}}}\big{(}\,B_{\epsilon(s)}\big{(}\pi\circ\varphi_{\mathcal{A}}(s)\big{)}\,\big{)}<\infty$.
For what follows, let
$\mathcal{O}_{s}=\varphi_{\mathcal{A}}^{-1}\circ\pi^{-1}\big{(}\,B_{\epsilon(s)}\big{(}\pi\circ\varphi_{\mathcal{A}}(s)\big{)}\,\big{)}$.
The family
$\left\\{\,\mathcal{O}_{s}\cap\mathcal{A}\,\right\\}_{s\in\cl(\mathcal{N})}$
forms an open cover of the compact space $\cl(\mathcal{N})$. Hence there exist
$s_{1},\ldots,s_{K}$, $K\in\mathbb{N}$, such that
$\bigcup_{k=1}^{K}\mathcal{O}_{s_{k}}\cap\mathcal{A}\supset\cl(\mathcal{N})$.
Now,

$\split\mu_{f}\big{(}\mathcal{N}\big{)}\leq\mu_{f}\big{(}\cl(\mathcal{N})\big{)}\leq\sum_{k=1}^{K}\mu_{f}\Big{(}\mathcal{O}_{s_{k}}\cap\mathcal{A}\Big{)}=\sum_{k=1}^{K}\mu_{f^{\pi\circ\varphi_{\mathcal{A}}}}\Big{(}B_{\epsilon(s_{k})}\big{(}\pi\circ\varphi_{\mathcal{A}}(s)\big{)}\Big{)},$
(11)

since $\pi\circ\varphi_{\mathcal{A}}$ is a diffeomorphism, and so it follows
that

$\split\mE\Big{[}\,\mu_{f}\big{(}\mathcal{N}\big{)}^{2}\,\Big{]}\leq&\sum_{k=1}^{K}\mE\Bigg{[}\,\mu_{f^{\pi\circ\varphi_{\mathcal{A}}}}\Big{(}B_{\epsilon(s_{k})}\big{(}\pi\circ\varphi_{\mathcal{A}}(s)\big{)}\Big{)}^{2}\,\Bigg{]}\\\
~{}~{}~{}~{}+\sum_{k=1}^{K}\sum_{\substack{k^{\prime}=1\\\ k^{\prime}\neq
k}}^{K}\mE\Bigg{[}\,\mu_{f^{\pi\circ\varphi_{\mathcal{A}}}}\Big{(}B_{\epsilon(s_{k})}\big{(}\pi\circ\varphi_{\mathcal{A}}(s)\big{)}\Big{)}\mu_{f^{\pi\circ\varphi_{\mathcal{A}}}}\Big{(}B_{\epsilon(s_{k^{\prime}})}\big{(}\pi\circ\varphi_{\mathcal{A}}(s)\big{)}\Big{)}\,\Bigg{]}\\\
\leq\sum_{k=1}^{K}\mE\Bigg{[}\,\mu_{f^{\pi\circ\varphi_{\mathcal{A}}}}\Big{(}B_{\epsilon(s_{k})}\big{(}\pi\circ\varphi_{\mathcal{A}}(s)\big{)}\Big{)}^{2}\,\Bigg{]}\\\
~{}~{}~{}~{}+\sum_{k=1}^{K}\sum_{\substack{k^{\prime}=1\\\ k^{\prime}\neq
k}}^{K}\sqrt{\mE\Bigg{[}\,\mu_{f^{\pi\circ\varphi_{\mathcal{A}}}}\Big{(}B_{\epsilon(s_{k})}\big{(}\pi\circ\varphi_{\mathcal{A}}(s)\big{)}\Big{)}^{2}\,\Bigg{]}}\sqrt{\mE\Bigg{[}\,\mu_{f^{\pi\circ\varphi_{\mathcal{A}}}}\Big{(}B_{\epsilon(s_{k^{\prime}}t)}\big{(}\pi\circ\varphi_{\mathcal{A}}(s)\big{)}\Big{)}^{2}\,\Bigg{]}}\,.$

The right hand side of this inequality is finite since all expectations in the
sums are finite by Lemma 4.2 from [Cheng:2015extremes]. Let
$\mathcal{M}=\bigsqcup_{d=0}^{D}\pdM\subset\tilde{\mathcal{M}}$, where $\pdM$,
for $d=\\{\,1,\ldots,D\,\\}$, is either empty or a disjoint union of finitely
many $d$-dimensional relatively compact and extendable $C^{3}$-submanifolds of
$\tilde{\mathcal{M}}$ without boundary and ${\partial_{0}\cM}$ is either empty
or is the union of finitely many points of $\tilde{\mathcal{M}}$. Let
$\tilde{f}$ be a Gaussian random field on $\tilde{\mathcal{M}}$ satisfying
Assumption 1 and let $f$ denote its restriction to $\mathcal{M}$. Then
$\mE\Big{[}\mu_{f}^{2}\big{(}\mathcal{M}\big{)}\Big{]}<\infty$. The number of
critical points of $\mathcal{M}$ is given by
$\mu_{f}\big{(}\mathcal{M}\big{)}=\sum_{d=1}^{D}\mu_{f|_{\partial_{d}\mathcal{M}}}\big{(}\partial_{d}\mathcal{M}\big{)}$.
Thus, applying Lemma 1 to $\partial_{d}\mathcal{M}$ and the Cauchy-Schwartz
inequality it follows that

$\split\mE\Big{[}\,\mu_{f}^{2}\big{(}\mathcal{M}\big{)}\,\Big{]}=\sum_{d=1}^{D}\sum_{d^{\prime}=1}^{D}\mE\Big{[}\,\mu_{f|_{\partial_{d}\mathcal{M}}}\big{(}\partial_{d}\mathcal{M}\big{)}\mu_{f|_{\partial_{d}\mathcal{M}}}\big{(}\partial_{d^{\prime}}\mathcal{M}\big{)}\,\Big{]}\leq\left(\,\sum_{d=1}^{D}\sqrt{\mE\Big{[}\,\mu_{f|_{\partial_{d}\mathcal{M}}}^{2}\big{(}\partial_{d}\mathcal{M}\big{)}\,\Big{]}}\,\right)^{2}<\infty\,.$


Convergence of Type IIA manifolds and application to the Type IIA flow.

Nikita Klemyatin111Supported in part by the National Science Foundation under
grant DMS-1855947.

###### Abstract

The Type IIA flow is a flow of 3-forms on a 6-dimensional symplectic manifold.
It was introduced by Fei, Phong, Picard, and Zhang in [20]. There is little
known about the singularities of this flow in general. We formulate and prove
convergence theorems for this flow. We also describe the singularity models
for this flow.

##  1 Introduction

The problem of finding canonical geometric structures has a long history with
roots both in geometry and physics. There are many examples of such geometric
structures: Einstein and Kähler-Einstein metrics on Riemannian and Kähler
manifolds, Hermitian-Einstein metrics on stable holomorphic vector bundles
(see [35]), and Ricci-flat metrics on Calabi-Yau manifolds ([36]). However,
many more examples of canonical geometric structures arise from the string
theory. Starting with the celebrated work of Candelas, Horowitz, Strominger,
and Witten [6], various geometric structures arise from the string theory were
studied extensively (see [10, 8, 9, 12, 17, 11, _FP21, 19, 24, 28, 27, 29, 4,
5] for some examples). The Type IIA geometry is one such structure. It was
introduced by Fei, Phong, Picard, and Zhang in [20] as a geometric structure
emerging from the Type IIA string theory (see [32], [33] and [34]). This is a
specific geometrical structure on a 6-dimensional symplectic manifold
$(M,\omega)$ together with a primitive 3-form $\varphi$ on $M$, which
satisfies an additional nondegeneracy condition. Such form $\varphi$ defines
an almost complex structure $J_{\varphi}$ on $M$, which is compatible with
$\omega$ and satisfies the following system of equations:

$\displaystyle d\Lambda d(|\varphi|^{2}\star\varphi)=\rho,$ (1.1)
$\displaystyle d\varphi=0.$ (1.2)

Here $\Lambda$ is the usual contraction with the symplectic form $\omega$,
$\star$ is the usual Hodge-star operator with respect to the metric
$g_{\varphi}$ defined by $\omega$ and $J_{\varphi}$. Also, $\rho$ is a source
form, which is Poincare dual to a linear combination of special Lagrangian
cycles, calibrated by $\varphi$.

In [20] the authors suggest a geometric flow approach to solve this system.
Namely, they considered the following flow of closed, primitive 3-forms on a
6-dimensional symplectic manifold $(M,\omega)$:

$\displaystyle\partial_{t}\varphi=d\Lambda d(|\varphi|^{2}\star\varphi)-\rho.$
(1.3)

One can check that this flow preserves the primitiveness and the closedness of
$\varphi$ if the initial data is primitive.

In [20] the authors established some important properties of this flow,
including the short-time existence, the uniqueness, and the Shi-type estimates
for the source-free flow (i.e. $\rho=0$). In other papers [21], [22] and [18]
authors established additional properties of the flow, such as dynamical
stability on compact symplectic 6-manifolds, as well as applications to
symplectic geometry. However, it is little known about the formation of
singularities of the Type IIA flow. Once we have the Shi-type estimates, we
can study the formation of singularities, as it was done in the case of the
Ricci flow by Hamilton [14], for the Laplacian flow in [25], and in the case
of the Type IIB flow in [23].

In order to do it, we need to establish the compactness properties of the
solutions to the Type IIA flow. First, we need to establish convergence of
manifolds, which carry the Type IIA manifolds:

###### Theorem 1

Let $\\{(M_{j},g_{j},\omega_{j},\varphi_{j},J_{j},p_{j})\\}^{\infty}_{j=1}$ be
a sequence of complete, pointed Type IIA manifolds. Denote $u_{j}:=\,{\rm
log}\,|\phi_{j}|_{g_{j}}$. Assume there are positive constants
$C_{k},\iota_{0}>0$, independent of $j$, such that the following holds:

(1)

$\displaystyle|({\nabla}^{g_{j}})^{k}Rm_{j}|_{g_{j}}\leq C_{k},$ (1.4)
$\displaystyle|({\nabla}^{g_{j}})^{k}N_{j}|_{g_{j}}\leq C_{k},$ (1.5)
$\displaystyle|({\nabla}^{g_{j}})^{k}\varphi_{j}|_{g_{j}}\leq C_{k},$ (1.6)
$\displaystyle|({\nabla}^{g_{j}})^{k}u_{j}|_{g_{j}}\leq C_{k}$ (1.7)

(2) The injectivity radius satisfies the following inequality:

$\displaystyle\mathrm{inj}(M_{j},g_{j})\geq\iota_{0}$ (1.8)

Then there is a complete, pointed manifold $(M,g,\omega,\varphi,J,p)$, a
subsequence
$\\{(M_{j_{k}},g_{j_{k}},\omega_{j_{k}},\varphi_{j_{k}},J_{j_{k}},p_{j_{k}})\\}$
which converge to $(M,g,\omega,\phi,J,p)$ in the sense of Definition 2 below.

Once we have Theorem 1, we can prove the theorem about the convergence of
solutions to the Type IIA flow:

###### Theorem 2

Let $(M_{j},g_{j}(t),\omega_{j},\varphi_{j}(t),J_{j}(t),p_{j})$ be the family
of complete solutions of the Type IIA flow for $t\in(\alpha;\beta)\subset{\bf
R}$. Assume that the following holds:

(1) There is a constant $\iota_{0}>0$ such that
$\mathrm{inj}(M_{j},g_{j}(t))\geq\iota_{0}$ along the flow;

(2) There is a constant $C>0$ indepentent of $j$, such that
$|u_{j}(t)|+|Rm_{j}(t)|\leq C$.

Then there is a complete solution $(M,g(t),\omega,\varphi(t),J(t),p)$ and a
subsequence
$\\{(M_{j_{k}},g_{j_{k}}(t),\omega_{j_{k}},\varphi_{j_{k}}(t),J_{j_{k}}(t),p_{j_{k}})\\}$
which converge to $(M,g(t),\omega,\varphi(t),J(t),p)$.

Finally, we apply Theorem 2 to the study of singularities of the Type IIA
flow. On this path, we have a new phenomena, as the formation of singularities
is controlled by $\,{\rm log}\,|\varphi|_{g_{\varphi}}+|Rm|_{g_{\varphi}}$.
Since this quantity is not homogeneous under scaling of metric, we need to
distinguish two possibilities: in one case the norm $|\varphi|_{g_{\varphi}}$
may be bounded, while in the other it is not.

###### Theorem 3

Let $(M,g(t),\omega,\varphi(t),J(t))$ be a solution of the Type IIA flow on a
compact manifold $M$, and assume $0<T\leq+\infty$ be its maximum time of
existence. Assume that the solution admits an injectivity radius estimate,
that is $\mathrm{inj}(M,g(t))\geq{c\over|u|+|Rm|}$ for some $c>0$. Then for
suitable times $t_{j}\to T$ and the rescalings
$g_{j}(t)=C_{j}g(t_{j}+C_{j}^{-1}t)$, the sequence $g_{j}(t)$ admits a
convergent subsequence to a solution of the Type IIB flow with a corresponding
function $f(t)=\sup_{M}(|u|+|Rm|)$ satisfying the following properties:

[Type I] For the Type I the solution exists on the interval $(-\infty,c)$ and
$f(t)\leq{C\over c-t}$;

[Type IIa] For the Type IIa the solution exists for all time and $f(t)\leq C$;

[Type IIb] For the Type IIb the solution exists for all time and $f(t)\leq C$
(the same as the previous case);

[Type III] For the Type III singular solution the singular model solution
exists on the interval $(-C,\infty)$ and $f(t)\leq{C\over c+t}$.

[Type IVa] $\sup_{[0;T]}(T-t)|Rm|=C<\infty$, $|\varphi|$ is bounded and
$T<\infty$

[Type IVb] $\sup_{[0;T]}(T-t)|Rm|=\infty$, $|\varphi|$ is bounded and
$T<\infty$

[Type IVc] $\sup_{[0;T]}t|Rm|=\infty$,$|\varphi|$ is bounded and $T=\infty$

[Type IVd] $\sup_{[0;T]}t|Rm|=C<\infty$, $|\varphi|$ is bounded, and
$T=\infty$

The paper is organized as follows: in Section 2, we describe the basic
properties of the Type IIA geometry. In Section 3. we formulate important
facts about the Type IIA flow. Sections 4 and 5 contain the proofs of Theorem
1 and Theorem 2 respectively. Finally, Section 6 contains proof of Theorem 3.

Acknowledgments: I am would like to thank my advisor Professor Duong H. Phong
for suggesting this problem and his constant attention, help and patience
while the paper was in preparation. I also want to thank Shuang Liang for
helpful discussions, and Vladislav Cherepanov and Yulia Gorginyan for reading
the text and pointing out typos.

##  2 Preliminaries.

###  2.1 The Type IIA geometry.

In this section, we recall all necessary definitions and results about the
Type IIA geometry. We are following the exposition in [20].

Let $(M,\omega,\varphi)$ be a 6-dimensional symplectic manifold with a
symplectic form $\omega$ and a closed 3-form $\varphi$. Assume that
$\omega\wedge\varphi=0$. Then, as shown in [20, 16], one can define the map
$K_{\varphi}:TM\rightarrow\Lambda^{5}(M)\cong\Lambda^{6}(M)\otimes TM$ by the
formula:

$\displaystyle K_{\varphi}(v)=-\iota_{v}\varphi\wedge\varphi.$ (2.9)

We define the following ”trace” of the square of $K_{\varphi}$:

$\displaystyle\lambda_{\varphi}={1\over 6}{\rm
tr}(K_{\varphi}^{2})\in(\Lambda^{6}(T^{*}M))^{\otimes 2}.$ (2.10)

When $\lambda_{\varphi}<0$ we will say that $\varphi$ is nondegenerate. In
this case, we can extract a positive square root
$\sqrt{-\lambda_{\varphi}}\in\Lambda^{6}(M)$. Now we define the following
operator:

$\displaystyle J_{\varphi}:={K_{\varphi}\over\sqrt{-\lambda_{\varphi}}}.$
(2.11)

In [16], Hitchin shows that $J_{\varphi}$ is an almost complex structure. It
is showed in [20, Lemma 3] that $J_{\varphi}$ is compatible with respect to
$\omega$ if and only if $\omega\wedge\varphi=0$. This means that in this case
the symmetric bilinear form $g_{\varphi}(X,Y):=\omega(X,J_{\varphi}Y)$ is
positive definite and $\omega$ is invariant under $J_{\varphi}$.

We can summarize it all in the following definition.

###### Definition 1

Let $(M,\omega,\varphi,J_{\varphi},g_{\varphi})$, where $(M,\omega)$ is a
6-dimensional symplectic manifold, $\varphi$ is a closed 3-form, which
satisfies $\omega\wedge\varphi=0$ and $(J_{\varphi},g_{\varphi})$ are defined
above. We will call such structure the Type IIA structure (or the Type IIA
geometry).

It is shown in [20, Lemma 4] that the following identity holds for
$\lambda_{\varphi}$:

$\displaystyle\sqrt{-\lambda_{\varphi}}={1\over
2}|\varphi|^{2}{\omega^{3}\over 3!}.$ (2.12)

It is easy to see that $d\mu_{g_{\varphi}}=\frac{\omega^{3}}{3!}$, where
$d\mu_{g_{\varphi}}$ is the volume form of $g_{\varphi}$. Furthermore, we can
give the explicit formula for $g_{\varphi}$ in coordinates [20, Lemma 5]:

$\displaystyle(g_{\varphi})_{jk}=12|\varphi|^{2}\frac{\iota_{\partial_{j}}\varphi\wedge\iota_{\partial_{k}}\varphi\wedge\omega}{\omega^{3}}=-|\varphi|^{2}\varphi_{jab}\varphi_{kcd}\omega^{ac}\omega^{bd}.$
(2.13)

Here and henceforth we will write $g_{jk}$ for $(g_{\varphi})_{jk}$.

###  2.2 The Gauduchon line of connections and the holonomy of conformally
rescaled metric.

The formula 2.13 implies that
$\tilde{g}_{jk}=-\varphi_{jab}\varphi_{kcd}\omega^{ac}\omega^{bd}$ is also a
Riemannian metric compatible with $J_{\varphi}$. The metric
$\tilde{g}_{\varphi}=|\varphi|^{2}g_{\varphi}$ has rather special properties,
which we are going to explain here. In order to do that, we need an additional
ingredient: the Gauduchon line of connections in almost Hermitian geometry.
This family of connections was introduced in [13], but our exposition and
notations are close to [20].

Let $\tilde{{\nabla}}$ be the Levi-Civita connection for the metric
$\tilde{g}_{\varphi}$. Consider a $TM$-valued 2-form
$(d^{c}\tilde{\omega})^{k}_{jp}=\tilde{g}^{ks}(d^{c}\tilde{\omega})_{sjp}$,
where $\tilde{\omega}(X,Y)=\tilde{g}(J_{\varphi}X,Y)$. We can decompose
${1\over 2}(d^{c}\tilde{\omega})^{k}_{jp}$ with respect to the action of
$J_{\varphi}$ as follows:

$\displaystyle U^{k}_{jp}={1\over
4}((d^{c}\tilde{\omega})^{k}_{jp}+(d^{c}\tilde{\omega})^{k}_{ab}J^{a}_{j}J^{b}_{p})$
(2.14) $\displaystyle V^{k}_{jp}={1\over
4}((d^{c}\tilde{\omega})^{k}_{jp}-(d^{c}\tilde{\omega})^{k}_{ab}J^{a}_{j}J^{b}_{p})$
(2.15) $\displaystyle U^{k}_{jp}+V^{k}_{jp}={1\over
2}(d^{c}\tilde{\omega})^{k}_{jp}.$ (2.16)

This is the standard decomposition of 2-forms on an almost complex manifold.
The form $U$ is the $(1,1)$-part of ${1\over
2}(d^{c}\tilde{\omega})^{k}_{jp}$, and the form $V$ is $(2,0)+(0,2)$-part of
${1\over 2}(d^{c}\tilde{\omega})^{k}_{jp}$. By using this decomposition, we
can define the family of unitary connections $D^{t}$ (which is called the
Gauduchon line of connections – see [13] and [20]) by the following formula:

$\displaystyle
D^{t}_{j}X^{k}=\tilde{{\nabla}}_{j}X^{k}-N^{k}_{jp}X^{p}-U^{k}_{jp}X^{p}-tV^{k}_{jp}X^{p},$
(2.17)

where $N$ is the Nijenhuis tensor of $J_{\varphi}$. The connection
$D^{0}=D^{t}\vline_{t=0}$ is called the projected Levi-Civita connection and
we denote it simply by $D$. The projected Levi-Civita connection has a number
of special properties in the Type IIA geometry, as it is shown in the
following theorem (see [20, Theorem 3]):

###### Theorem 4

Let $(M,\omega,\varphi)$ be a Type IIA geometry, and
$g_{\varphi},\tilde{g}_{\varphi}$ are the metrics we defined above. Let $D$
and ̵̃$\tilde{D}$ be the projected Levi-Civita connections of $g_{\varphi}$
and $\tilde{g}_{\varphi}$ respectively, $\Omega=\varphi+iJ_{\varphi}\varphi$,
and $|\Omega|_{\tilde{g}_{\varphi}}$ the norm of $\Omega$ with respect to
$\tilde{g}_{\varphi}$. Then

(1) ̵̃$\tilde{D}({\Omega\over|\Omega|_{\tilde{g}_{\varphi}}})=0$. Hence,
$\tilde{D}$ has holonomy in SU(3);

(2) $D^{0,1}\Omega=0$, so $\Omega$ is formally holomorphic;

(3) The Nijenhuis tensor of $J_{\varphi}$ has only 6 independent components.

The theorem above shows that the Type IIA geometry is an apt analog of the
Calabi-Yau geometry in the almost complex setting. Moreover, it shows the
importance of the metric $\tilde{g}_{\varphi}$. Also, it is shown in [20] that
the metric $\tilde{g}_{\varphi}$ defines the norm
$|\varphi|^{2}_{g_{\varphi}}$, and hence, metric $g_{\varphi}$ itself.
However, for our purposes it is irrelevant which metric we choose, so
throughout this paper, we mainly use $g_{\varphi}$.

###  2.3 Convergence in the sense of Gromov-Cheeger-Hamilton.

In this section, we give a definition of convergence of Type IIA manifolds.
Our definition closely follows the definitions from [15] and [23].

###### Definition 2

Let $\\{(M_{j},g_{j},p_{j})\\}$ be a sequence of complete Riemannian
manifolds, where $p_{j}\in M_{j}$ is a point. We say that this sequence
converges to a complete pointed manifold $(M,g,p)$ in $C^{k}$-sense if there
is a sequence of compact sets $\\{U_{j}\\}$ in $M_{\infty}$ with the following
properties:

(1) The sequence $\\{U_{j}\\}$ is an exhaustion of $M_{\infty}$;

(2) There are diffeomorphisms $\Phi_{j}:U_{j}\rightarrow V_{j}\subset M_{j}$
such that $\Phi_{j}(p)=p_{j}$ and the sequence of metrics
$\\{\Phi_{j}^{*}g_{j}\\}$ converges to $g_{\infty}$ uniformly in
$C^{k}$-topology on compact subsets in $U_{j}$.

We also recall the theorem of Hamilton from [15]:

###### Theorem 5

Let $\\{M_{j},g_{j},p_{j}\\}$ be a sequence of complete pointed Riemannian
manifolds and $\nabla_{j}$ be the Levi-Civita connection on $(M_{j},g_{j})$.
Assume that the following holds uniformly in $j$:

(1) There are constants $C_{k}>0$, such that $|\nabla^{k}_{j}Rm_{j}|\leq
C_{k}$;

(2) There is a positive uniform lower bound $\iota_{0}$ for the injectivity
radii of $M_{j}$ at $p_{j}$, $\mathrm{inj}(M_{j},p_{j})\geq\iota_{0}$.

Then there is a subsequence $\\{M_{j_{k}},g_{j_{k}},p_{j_{k}}\\}$, which
converges to a complete pointed Riemannian manifold
$(M_{\infty},g_{\infty},p_{\infty})$.

This theorem is an essential ingredient in the proof of Theorem 1.

##  3 The Type IIA flow.

The Type IIA flow was introduced by Fei, Phong, Picard, and Zhang in [20]. It
is the flow of 3-forms on a 6-dimensional symplectic manifold with a Type IIA
structure, which can be written as follows:

$\displaystyle\partial_{t}\varphi=d\Lambda d(|\varphi|^{2}\star\varphi).$
(3.18)

Here $\star$ is the Hodge star operator with respect to the metric
$g_{\varphi}$. By [20, Theorem 1] we can rewrite this flow as a Laplacian flow

$\displaystyle\partial_{t}\varphi=-dd^{\dagger}(|\varphi|^{2}\varphi)+2d(N^{\dagger}\cdot\varphi),$
(3.19)

where $N$ is the Nijenhuis tensor of $J_{\varphi}$ and
$(N^{\dagger}\cdot\varphi)_{kj}=N^{a~{}b}_{~{}j}\varphi_{akb}-N^{a~{}b}_{~{}k}\varphi_{ajb}$.
Here $N^{a~{}b}_{~{}j}=N^{a}_{~{}js}g^{sb}$. This formulation is similar to
the Bryant’s Laplacian $G_{2}$-flow [4, 5].

We also can rewrite the Type IIA flow in terms of metric $g_{\varphi}$ and
$u=\,{\rm log}\,|\varphi|$ (see either [20, Theorem 4], or [21] for the
derivation):

$\displaystyle\partial_{t}g_{ij}=E_{ij}:=e^{u}(-2R_{ij}+2{\nabla}_{i}{\nabla}_{j}u-4(N^{2}_{-})_{ij}+u_{i}u_{j}+u_{Ji}u_{Jj}+4u_{s}(N^{~{}s}_{i~{}j}+N^{~{}s}_{j~{}i}))$
(3.20) $\displaystyle(\partial_{t}-e^{u}\Delta)u=(2|{\nabla}u|^{2}+|N|^{2})$
(3.21)

In [20, Proposition 7 and Theorem 7] the authors proved the Shi-Type estimates
for the Type IIA flow.

###### Theorem 6

Let $(g(t),u(t))$ evolve along the Type IIA flow. Denote by $Rm$ the curvature
tensor of the Levi-Civita connection with respect to $g_{\varphi}$, and
$u=\,{\rm log}\,|\varphi|$ as usual. Assume that $|u|+|Rm|\leq C$ for some
uniform constant $C>0$ on the interval $[0;T]$. Then the following holds

(1) There is a constant $C_{0}$ such that

$\displaystyle\sup_{M\times[0;T]}(|Rm|+|N|^{2}+|{\nabla}N|+|{\nabla}^{2}u|+|{\nabla}u|+|u|)\leq
C_{0}$ (3.22)

(2) There are constants $C_{k}$ for $k\in\bf N$, such that

$\displaystyle\sup_{M\times[0;T]}(|{\nabla}^{k}Rm|+|{\nabla}^{k+1}N|+|{\nabla}^{k+2}u|)\leq
C_{k}$ (3.23)

(3) Assume that the bound $|u|+|Rm|\leq C_{0}$ holds on $[0;T)$. Then
$|{\nabla}^{k}\varphi|\leq C(C_{0},k,T,\varphi(0))$ and the flow can be
continued to $[0;T+\varepsilon)$ for some $\varepsilon>0$.

The third statement of Theorem 6 justifies the following definition.

###### Definition 3

Consider the solution $(M,g(t),\omega,\varphi(t),J(t))$ for the Type IIA flow
on a finite time interval $[0;T)$ for $T<+\infty$. We say that the solution of
the Type IIA flow becomes singular at time $T$ if
$\sup_{M\times[0;T)}(|u|+|Rm|)=+\infty$.

###### Remark 1

It is worthy to mention that in [18] the authors showed that the Type IIA flow
is very close to the Hitchin flow (see [16, 8] for the details). However, the
Hitchin flow is more degenerate than the Type IIA flow, and it is hard to
prove the short-time existence of the solution for the Hitchin flow. Also, the
evolution equation for $\,{\rm log}\,|\varphi|^{2}$ implies that the Hitchin
functional $H(\varphi)=\int\varphi\wedge\hat{\varphi}$ is monotone increasing
along the Type IIA flow.

##  4 Proof of Theorem 1.

Here we give the proof of Theorem 1.

Step 1: Compactness for the underlying Riemannian manifolds.

This is the content of Theorem 5. Hence, there exists a complete, pointed
Riemannian manifold $(M,g,p)$, such that $(M_{j},g_{j},p_{j})$ converge to it
in the sense of Definition 2.

Step 2: Convergence of complex structures.

By the formula ${\nabla}^{g_{j}}\omega_{j}=-2N_{j}$ (this follows from the
formula for the Gauduchon connection for $g_{j}$) we see that for any $k\geq
1$ the norms of covariant derivatives
$|({\nabla}^{g_{j}})^{k}\omega_{j}|_{g_{j}}$ are bounded. We will argue like
in [25] and [23]. Let $\Phi_{j,k}$ be restriction of $\Phi_{j+k}$ on $U_{j}$.
Denote $g_{j,k}:=\Phi_{j,k}^{*}g_{j+k}$. The metrics $g_{j,k}$ converge to
$g_{j,\infty}$ as $k\rightarrow\infty$. It is easy to see that $g_{j,\infty}$
is the restriction of $g_{\infty}$ on $U_{j}$. Similarly, the notation
$\omega_{i,j}$ means $\Phi^{*}_{i,j}\omega_{j}$ and $\varphi_{i,j}$ stands for
$\Phi^{*}_{i,j}\varphi_{j}$.

Consider the quantity $A_{i,j}={\nabla}^{g}-{\nabla}^{g_{i,j}}$. It is bounded
together with all covariant derivatives with respect to ${\nabla}^{g}$,
because $g_{i,j}$ converges to $g$ in $C^{\infty}$. Using this, together with
the fact that $g_{i,j}$ is uniformly equivalent to $g$ for big $j$, we obtain
that there are constants $C^{\prime}_{k}$ such that
$|({\nabla}^{g})^{k}\omega_{i,j}|_{g}\leq C^{\prime}_{k}$. By the Arzela-
Ascoli theorem, after passing to a subsequence, there is a form $\omega_{i}$,
such that the subsequence converges in $C^{\infty}$ to $\omega_{i}$. The form
$\omega_{i}$ satisfies the identity $\omega_{i}^{3}=6d\mu_{g}$, hence it is
nondegenerate. Moreover, there is a global form $\omega$ on $M$, such that
$\omega|_{U_{i}}=\omega_{i}$.

Finally, since $J_{j}=g^{-1}_{j}\omega_{j}$, we see that
$J_{i,j}=g^{-1}_{i,j}\omega_{i,j}$ converge to the operator
$J_{i,\infty}=g^{-1}\omega_{i}$, which satisfies
$J^{2}_{i,\infty}=-\mathrm{Id}$. Moreover, all $J_{i,\infty}$ are gluing
together to a globally-defined almost complex structure $J=g^{-1}\omega$.

Step 3: Convergence of 3-forms.

The argument that proves the convergence of $\varphi_{i,j}$ to
$\varphi_{i,\infty}$ and the existence of a globally defined limit form
$\varphi$ carries over verbatim. We only need to prove the nondegeneracy of
the form $\varphi$.

We use the following identities in Type IIA geometry (see Section 2.1):

$\displaystyle g(\xi_{1},\xi_{2}){\omega^{3}\over
3!}=\iota_{\xi_{1}}\varphi\wedge\iota_{\xi_{2}}\varphi\wedge\omega$ (4.24)
$\displaystyle K_{\varphi}\xi=-\iota_{\xi}\varphi\wedge\varphi$ (4.25)
$\displaystyle J\xi={K_{\varphi}\xi\over\sqrt{-{\rm tr}(K^{2})}}$ (4.26)
$\displaystyle\sqrt{-{\rm tr}(K^{2})}={1\over 2}|\varphi|^{2}{\omega^{3}\over
3!}$ (4.27) $\displaystyle\Lambda_{\omega}\varphi=\omega\wedge\varphi=0$
(4.28)

These identities hold for any tuple
$(g_{i,j},\omega_{i,j},J_{i,j},\varphi_{i,j})$ and for any vector fields
$\xi,\xi_{1}$ and $\xi_{2}$.

Again, by the Arzela-Ascoli theorem, there are limiting 3-forms $\varphi_{i}$
on $U_{i}$, and these forms define a global form $\varphi$ on $M$. Next, the
positive lower bound for $|\varphi_{i,j}|$ implies that $|\varphi|>0$
pointwise, and ${1\over 2}|\varphi|^{2}{\omega^{3}\over 3!}>0$. This implies
that all the identities 4.24 holds for $(g,\omega,J,\varphi)$. Hence, the form
$\varphi$ is a nondegenerate 3-form, and the tuple $(g,\omega,J,\varphi)$
defines the Type IIA geometry on $M$. Q.E.D.

##  5 Compactness of solutions of the Type IIA flow.

###  5.1 Estimates for mixed derivatives.

In this section we are going to prove the estimates for
$\partial_{t}^{m}{\nabla}^{k}g_{j}$ and $\partial_{t}{\nabla}^{k}\varphi_{j}$.
These estimates are the same as in the case of the Ricci flow, the Laplacian
$G_{2}$ flow, and the Type IIB flow.

Next, we prove that metrics are equivalent under the assumptions of Theorem 2:

$\displaystyle
C_{1}:=C|t_{1}-t_{0}|\geq\int_{t_{0}}^{t_{1}}|\partial_{t}\,{\rm
log}\,g(t)(V,V)|dt\geq|\,{\rm log}\,{g(t_{1})(V,V)\over g(t_{0})(V,V)}|,$
(5.29)

where $C$ depends on the bounds for $Rm,N$ and $u$. This implies that norms
defined by $g(t_{0})$ and $g_{(}t_{1})$ are equivalent.

Consider $\Omega\times[a;b]\subset M\times(\alpha,\beta)$ such that
$\Omega\subset U_{j}$ for j large enough. By abusing the notation, we will
denote $\Phi_{j}^{*}\varphi_{j}(t)$ and $\Phi_{j}^{*}g_{j}(t)$ on $\Omega$ by
$\varphi_{j}$ and $g_{j}$ respectively. To stress the difference between
$g_{j}$ and $g$ on $M$ we will denote $g$ by $\hat{g}$ and $\varphi,\omega$ by
$\hat{\varphi}$ and $\hat{\omega}$ respectively.

Now we prove the estimates for
$|\partial^{l}_{t}\hat{{\nabla}}^{k}g_{j}|_{\hat{g}}$ and
$|\partial^{l}_{t}\hat{{\nabla}}^{k}\varphi_{j}|_{\hat{g}}$. To begin with,
consider $|\partial_{t}\hat{{\nabla}}g_{j}|_{\hat{g}}$. Let
$A_{j}:=\hat{{\nabla}}-{\nabla}^{g_{j}}$ and $E_{j}=\partial_{t}g_{j}(t)$. We
have:

$\displaystyle|\partial_{t}\hat{{\nabla}}g_{j}|_{\hat{g}}\leq|\hat{{\nabla}}\partial_{t}g_{j}|_{\hat{g}}\leq|A_{j}|_{\hat{g}}|E_{j}|_{\hat{g}}+|{\nabla}^{g_{j}}E_{j}|_{\hat{g}}\leq
C(1+|\hat{{\nabla}}g_{j}|_{\hat{g}})$ (5.30)

Here we can estimate the norm $|E_{j}|_{\hat{g}}$ because the metrics $g_{j}$
and $\hat{g}$ are equivalent. Using the following formula, we can proceed by
induction:

$\displaystyle\partial_{t}\hat{{\nabla}}^{k}g_{j}=\sum_{j_{1}+\dots+j_{p}=k}{\nabla}^{j_{1}}E_{j}*\hat{{\nabla}}^{j_{2}}g_{j}*\dots*\hat{{\nabla}}^{j_{p}}g_{j}$
(5.31)

which can be proved in a similar manner as in the case of the Ricci flow (see
[2, Lemma 8.6]). Now the induction by $k$ shows that for each $k$ there is a
constant $C$, independent of $j$, such that

$\displaystyle|\partial_{t}\hat{{\nabla}}^{k}g_{j}|_{\hat{g}}\leq
C(1+|\hat{{\nabla}}^{k}g_{j}|_{\hat{g}}),$ (5.32)

so this implies that $|\partial_{t}\hat{{\nabla}}^{k}g_{j}|_{\hat{g}}$ is
bounded. Finally, note that $\partial_{t}^{l}g_{j}=\partial_{t}^{l-1}E_{j}$,
and formula 3.20 together with evolution formulas in [20, Proposition 7 and
Theorem 7] imply that the covariant derivatives of $\partial_{t}^{l-1}E_{j}$
are uniformly bounded.

The similar arguments holds for
$|\partial^{l}_{t}\hat{{\nabla}}^{k}\varphi_{j}|_{\hat{g}}$. Consider the case
$l=1$. We have:

$\displaystyle|\partial_{t}\hat{{\nabla}}^{k}\varphi_{j}|_{\hat{g}}=|\hat{{\nabla}}^{k}\partial_{t}\varphi_{j}|_{\hat{g}}=|\hat{{\nabla}}^{k}(dd^{\dagger}(|\varphi_{j}|_{j}^{2}\varphi_{j})+2d(N_{j}^{\dagger}\cdot\varphi_{j}))|_{\hat{g}}\leq
C(|\hat{{\nabla}}^{k+2}\varphi_{j}|_{\hat{g}}+|\hat{{\nabla}}^{k+1}\varphi_{j}|_{\hat{g}}).$
(5.33)

By the equivalence of metrics and Theorem 6, the RHS is bounded. For $l>1$ we
have

$\displaystyle|\partial^{l}_{t}\hat{{\nabla}}^{k}\varphi_{j}|_{\hat{g}}\leq|\partial^{l-1}_{t}\hat{{\nabla}}^{k}dd^{\dagger}(|\varphi_{j}|_{j}^{2}\varphi_{j})|_{\hat{g}}+2|\partial^{l-1}_{t}\hat{{\nabla}}^{k}(d(N_{\varphi_{j}}^{\dagger}\cdot\varphi_{j}))|_{\hat{g}}\leq$
(5.34) $\displaystyle
C(|\partial^{l-1}_{t}\hat{{\nabla}}^{k+1}(|\varphi_{j}|_{j}^{2}\varphi_{j})|_{\hat{g}}+|\partial^{l-1}_{t}\Sigma_{j=0}^{k+1}\hat{{\nabla}}^{j}N^{\dagger}\hat{{\nabla}}^{k-j+1}\varphi_{j}|_{\hat{g}})$
(5.35)

The second summand can be estimated in terms of
$|\partial^{m}_{t}\hat{{\nabla}}^{p}\varphi_{j}|_{\hat{g}}$ and
$|\partial^{m}_{t}\hat{{\nabla}}^{p}N_{j}|_{\hat{g}}$ for $m\leq l-1$.
However, the Nijenhuis tensor $N_{j}$ can be expressed via $J_{j}$ and
$\hat{{\nabla}}J_{j}$, while the time derivatives of $J_{j}$ can be expressed
in terms of time derivatives of $g_{j}$. More precisely,
$\partial^{l}_{t}J^{a}_{b}=\omega^{as}\partial^{l-1}_{t}(\partial_{t}g_{sb})$.
Hence, the second summand in the formula above can be bounded by a finite
expression, involving
$|\partial^{m}_{t}\hat{{\nabla}}^{p}\varphi_{j}|_{\hat{g}}$ and
$|\partial^{m}_{t}\hat{{\nabla}}^{p}g_{j}|_{\hat{g}}$. The induction by $l$
shows that the norms
$|\partial^{l}_{t}\hat{{\nabla}}^{k}\varphi_{j}|_{\hat{g}}$ are also bounded
for each $l>1$.

Thus we proved the following proposition

###### Proposition 1

Under the assumptions of Theorem 2 there are constants $C_{l,k}$, such that
$|\partial^{l}_{t}\hat{{\nabla}}^{k}g_{j}|_{\hat{g}}\leq C_{l,k}$ and
$|\partial^{l}_{t}\hat{{\nabla}}^{k}\varphi_{j}|_{\hat{g}}\leq C_{l,k}$ on
$\Omega\times[a;b]\subset M\times(\alpha,\beta)$.

###  5.2 Proof of Theorem 2.

First of all, Theorem 1 guarantee that, after taking a subsequence, there is a
limit $(M,g,\omega,\varphi,J,p)$ of
$(M_{j},g_{j}(0),\omega_{j},\varphi_{j}(0),J_{j},p_{j})$.

Next, by Proposition 1 and the Arzela-Ascoli theorem (see [2, Corollary
9.14]), the sequence $\\{(g_{j}(t),\varphi_{j}(t),J_{j}(t))\\}$ converge
(again after taking a subsequence) to the triple $(g(t),\varphi(t),J(t))$ on
$\Omega\times[a;b]\subset M\times(\alpha,\beta)$ in $C^{\infty}$. The
$C^{\infty}$ convergence implies that the couple $(g(t),\varphi(t))$ satisfies
the Type IIA flow equation. Now the standard diagonalization argument implies
that there exists $(M,g(t),\varphi(t),J(t),\omega)$ for all times
$t\in(\alpha,\beta)$, where $\varphi(t),\gamma(t)$ are the solutions to the
Type IIA flow. Q.E.D.

##  6 The singularities of the Type IIA flow.

In this section, we describe the singularities for the Type IIA flow. Recall
that from Definition 3 singularities are characterized by the condition
$\sup_{M\times[0;T)}(|u|+|Rm|)=+\infty$. From this, we can see a significant
difference between the Type IIA flow and many other geometric flows such as
the Ricci flow, the Type IIB flow, and the Laplacian $G_{2}$-flow. The
singularities of all these flows are characterized by quantities, which are
homogeneous under the rescaling of metric. However, in the case of the Type
IIA flow, we have a different situation. If we multiply metric $g$ by a
positive constant, then the norm of the Riemann curvature tensor and the
function $|u|=|\,{\rm log}\,|\varphi|^{2}|$ behave differently under this
operation. Hence, the usual approach to the singularities doesn’t work here
and we need to consider another quantity, which behaves nicely under the
rescaling.

Consider the function $f(t):=\sup_{M}(|\varphi|^{1\over 3}+|Rm|)$. This
quantity has the desired homogeneous properties but it is now clear whenever
$f(t)$ has the same rate of growth as $|u|+|Rm|$. There are two possible
cases:

1\. The norm $|\varphi|$ is bounded along the flow;

2\. The norm $|\varphi|$ tends to $+\infty$ as $t\rightarrow T$.

The latter case is simpler: if $|\varphi|\nearrow+\infty$, we can find a
sequence of times $t_{j}\rightarrow T$, such that $(|u|+|Rm|)(t_{j})\leq
f(t_{j})$. The former case is a bit more delicate: if $|\varphi|$ remains
small enough along the flow, we cannot say that $|\varphi|^{1\over
3}\geq\,{\rm log}\,|\varphi|^{2}$. However, if $|\varphi|$ is bounded, the
condition $\sup_{M\times[0;T)}(|u|+|Rm|)=+\infty$ is equivalent to the
condition $\sup_{M\times[0;T)}|Rm|=+\infty$.

With all these considerations we can give the following definition.

###### Definition 4

Suppose $(M,g(t),\omega,\varphi(t),J)$ be a solution of the Type IIA flow. Let
$f(t)$ be as above, and $T$ is the maximal time for which flow exists. We
define the following type of singularities

[Type I] $\sup_{[0;T]}(T-t)f(t)=C<\infty$, and $T<\infty$

[Type IIa] $\sup_{[0;T]}(T-t)f(t)=\infty$, and $T<\infty$

[Type IIb] $\sup_{[0;T]}tf(t)=\infty$, $T=\infty$

[Type III] $\sup_{[0;T]}tf(t)=C<\infty$, $T=\infty$

[Type IVa] $\sup_{[0;T]}(T-t)|Rm|=C<\infty$, $|\varphi|$ is bounded and
$T<\infty$

[Type IVb] $\sup_{[0;T]}(T-t)|Rm|=\infty$, $|\varphi|$ is bounded and
$T<\infty$

[Type IVc] $\sup_{[0;T]}t|Rm|=\infty$,$|\varphi|$ is bounded and $T=\infty$

[Type IVd] $\sup_{[0;T]}t|Rm|=C<\infty$, $|\varphi|$ is bounded, and
$T=\infty$

###### Remark 2

Here we have a problem with terminology: we study the Type IIA flow, but the
singularity types are named like Type I, Type II, etc. Our terminology follows
closely the terminology from [14], and we hope it is not misleading.

With all these preparations, we can prove Theorem 3.

Proof of Theorem 3

The proof is essentially the same as the proof of the similar theorem for the
Ricci flow and [23, Theorem 3].

Consider the family of metrics $g_{j}(t):=C_{j}g(t_{j}+{t\over C_{j}})$, where
we determine the constants $C_{j}$ and $t_{j}$ later. In addition, we rescale
the symplectic form $\omega$ by $\omega_{j}=C_{j}\omega$. Also, we need to
rescale the form $\varphi$ as following: $\varphi_{j}(t)=\varphi(t_{j}+{t\over
C_{j}})$.

The case of Type I singularity

Let $C=\sup_{[0;T]}(T-t)f(t)$ and pick a sequence $(x_{j},t_{j})\in
M\times[0;T)$, and assume $C_{j}:=F(x_{j},t_{j})$. Note that
$(T-t_{j})C_{j}\geq c>0$ for some $c$. Now we consider rescaling of the metric
and forms as described above. We have

$\displaystyle|\varphi_{j}|^{1\over 3}_{g_{j}}+|Rm_{j}|_{g_{j}}\leq
C_{j}^{-1}{C\over T-(t_{j}+C_{j}^{-1}t)}\leq{C\over c-t}$ (6.36)

Now we applying the compactness theorem and extract a limit of the sequence
$(M,g_{j}(t),\omega_{j},\varphi_{j}(t),J_{j})$.

The case of Type IIa singularity

The proof is again an adaptation of the proof in the case of the Ricci flow.
Pick $T_{j}\nearrow T$ and points $(x_{j},t_{j})$ such that
$(T_{j}-t_{j})F(x_{j},t_{j})=\max_{M\times[0;T_{j}]}F(x,t)$. Taking $C_{j}$ as
before, we have

$\displaystyle|\varphi_{j}|^{1\over 3}_{g_{j}}+|Rm_{j}|_{g_{j}}={1\over
C_{j}}(|\varphi|+|Rm|)\leq{C_{j}(T_{j}-t_{j})\over C_{j}(T_{j}-t_{j}-{t\over
C_{j}})}={1\over 1-{t\over C_{j}(T_{j}-t_{j})}}\leq 1$ (6.37)

The case of Type IIb singularities.

We again pick up the sequence of points $(x_{j},t_{j})$, such that
$t_{j}(T_{j}-t_{j})F(x_{j},t_{j})=\max_{M\times[0;T_{j}]}t(T_{j}-t)F(x,t)$. By
rescaling, we obtain

$\displaystyle|\varphi_{j}|^{1\over 3}_{g_{j}}+|Rm_{j}|_{g_{j}}={1\over
C_{j}}(|\varphi|^{1\over 3}+|Rm_{j}|)(t_{j}+{t\over
C_{j}})\leq{C_{j}t_{j}(T_{j}-t_{j})\over C_{j}(t_{j}+{t\over
C_{j}})(T_{j}-t_{j}-{t\over C_{j}})}=$ (6.38) $\displaystyle={1\over(1+{t\over
C_{j}t_{j}})(1-{t\over C_{j}(T_{j}-t_{j})})}={C_{j}t_{j}\over
C_{j}t_{j}-t}{C_{j}(T_{j}-t_{j})\over C_{j}(T_{j}-t_{j})-t}\rightarrow 1$
(6.39)

Applying the compactness theorem, we obtain the desired statement.

The case of Type III singularity

As in the case of Type I singularities, let
$C=\sup_{M\times[0;+\infty)}tf(t)$, and $(x_{j},t_{j})\in M\times[0;\infty)$
as before. Then

$\displaystyle|\varphi_{j}|^{1\over 3}_{g_{j}}+|Rm_{j}|_{g_{j}}\leq{C\over
C_{j}(t_{j}+{t\over C_{j}})}\leq{C\over c+t},$ (6.40)

since $C_{j}t_{j}\geq c>0$. Hence, after passing to a subsequence and applying
Theorem 2, the limit satisfies the inequality $|\varphi|^{1\over
3}+|Rm|\leq{C\over c+t}$.

The case of the Type IV singularities.

Let and assume that $g_{j}(t)$ and $\omega_{j}$ are the same as before. We
also rescale $\varphi_{j}=C^{3\over 2}_{j}\varphi(t_{j}-{t\over C_{j}})$. Then
$|\varphi_{j}|_{g_{j}}(t)$ remains bounded, so we can apply the compactness
theorem and extract the limit. The classification part carries out verbatim as
in [15]. Q.E.D.

Finally, we consider a few examples from [20] and

######  Example 1 (The Type IIa singularity)

Consider the 6-dimensional solvmanifold $M$ with the complex structure,
constructed by Tomassini and Vezzoni in [31]. The invariant forms satisfy the
following identities:

$\displaystyle de^{1}=-\lambda e^{15},~{}de^{2}=\lambda
e^{25},~{}de^{3}=-\lambda e^{36}$ (6.41) $\displaystyle de^{4}=\
e^{46},~{}de^{5}=0,~{}de^{6}=0.$ (6.42)

and the symplectic form is given by $\omega=e^{12}+e^{34}+e^{56}$. Here
$e^{ij}=e^{i}\wedge e^{j}$, and $\lambda=\,{\rm log}\,{3+\sqrt{5}\over 2}$.

The nondegenerate 3-form can be defined as follows:

$\displaystyle\varphi=\alpha(e^{135}+e^{136})+\beta(e^{145}-e^{146})+\gamma(e^{235}-e^{236})-\delta(e^{245}-e^{246}),$
(6.43)

and $\alpha,\beta,\gamma,\delta$ depends on $t$. The nongedeneracy condition
is $|\varphi|^{4}=64\alpha\beta\gamma\delta>0$.

In [20] the authors showed that the Type IIA flow preserves this ansatz and
exists for a finite time $T={1\over 32\lambda^{2}}{\,{\rm
log}\,(\beta_{0}\gamma_{o})-\,{\rm
log}\,(\alpha_{0}\delta_{0})\over\beta_{0}\gamma_{o}-\alpha_{0}\delta_{0}}$,
which is defined by the initial data. Moreover, the norm of $\varphi$ tends to
infinity. However, the metric $g_{\varphi}$, the complex structure
$J_{\varphi}$ and the Nijenhuis tensor $N$ extends smoothly to $t=T$, hence
the curvature is bounded as $t\rightarrow T$. Using the calculations from
[20], we can directly compute $|u|$:

$\displaystyle|\,{\rm log}\,|\varphi|^{2}|=|\,{\rm
log}\,(8\sqrt{\alpha\beta\gamma\delta})|=\left|-16\lambda^{2}t+\,{\rm
log}\,{8|\beta_{0}\gamma_{0}-\alpha_{0}\delta_{0}|\sqrt{\alpha_{0}\beta_{0}\gamma_{0}\delta_{0}}\over|\beta_{0}\gamma_{0}e^{\alpha_{0}\delta_{0}}-\alpha_{0}\delta_{0}e^{\gamma_{0}\beta_{0}}|}\right|.$
(6.44)

Hence, ${|u|\over T-t}$ tends to infinity unless $T=16\lambda^{2}\,{\rm
log}\,{8|\beta_{0}\gamma_{0}-\alpha_{0}\delta_{0}|\sqrt{\alpha_{0}\beta_{0}\gamma_{0}\delta_{0}}\over|\beta_{0}\gamma_{0}e^{\alpha_{0}\delta_{0}}-\alpha_{0}\delta_{0}e^{\gamma_{0}\beta_{0}}|}$.
In the latter case, ${|u|+|Rm|\over T-t}$ is bounded, so this is the Type I
singularity.

######  Example 2 (The Type IIb singularity)

Consider the nilmanifold $M$ from [3, Example 5.2] (see also [20, Section
9.3.2]). The Lie algebra of the nilpotent Lie group can be characterized by
invariant 1-forms:

$\displaystyle de^{1}=de^{2}=de^{3}=de^{5}=0$ (6.45) $\displaystyle
de^{4}=e^{15},de^{6}=e^{13}.$ (6.46)

The symplectic form is standard: $\omega=e^{12}+e^{34}+e^{56}$. The family of
nondegenerate primitive 3-forms can be defined as follows:

$\displaystyle\varphi_{a,b}=(1+a)e^{135}-e^{146}-e^{245}-e^{236}+b(e^{134}-e^{156}),$
(6.47)

where $a$ and $b$ are functions of $t$. From [20] we know, that the positivity
condition means $|\varphi_{a,b}|^{4}=16(1+a-b^{2})>0$, and in this case the
flow exists for all time $t>0$. Moreover, it was shown that $a(t)=a_{0}+8t$
and $b=b_{0}$, where $a_{0},b_{0}$ are initial data.

Since $|u|=|\,{\rm log}\,4(1+8t+a_{0}-b^{2}_{0})|$, we see that
$t(|u|+|Rm|)\rightarrow+\infty$ as $t\rightarrow+\infty$. Hence, the solution
forms Type IIb singularity at $t=+\infty$.

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Department of Mathematics, Columbia University, New York, NY 10027 USA

<EMAIL_ADDRESS>