Convergence of Hermitian manifolds and the Type IIB flow

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

###### Abstract

The Type IIB flow is a flow of conformally balanced complex manifolds
introduced by Phong, Picard, and Zhang, about whose singularities little is as
yet known. We formulate convergence criteria for the Gromov-Cheeger-Hamilton
convergence of sequences of Hermitian manifolds, and apply them to
precompactness theorems and the existence of singularity models for the Type
IIB flow, in analogy with Hamilton’s classic compactness theorems and
classification of singularities for the Ricci flow.

##  1 Introduction

The earliest, and still most influential, geometric structure which emerged as
a candidate for vacuum configuration of unified string theories was that of a
Calabi-Yau manifold, equipped with a holomorphic vector bundle admitting a
Hermitian-Einstein metric. This was proposed by Candelas, Horowitz,
Strominger, and Witten [6] for the heterotic string, and it could rely on the
fundamental existence theorems for such structures proved earlier by Yau [36]
and Donaldson-Uhlenbeck-Yau [9, 34]. But other geometric structures have
emerged since, motivated by the other string theories, for which analogous
existence theorems are not yet available. Such theorems can be expected to
require the full theory of non-linear partial differential equations and, as
advocated in [25, 24], notably the theory of geometric flows. In particular,
flow approaches have been proposed in [25] for the Hull-Strominger system in
heterotic string theory, in [24, 19] for the Type IIB string, in [20] for the
Type IIA string, and in [12] for $11$-dimensional supergravity.

These new flows share main features with the Ricci flow, with typically some
additional structure, but they remain more complicated than, say, the Kähler-
Ricci flow. A few known cases of convergence are in [27, 28, 29], but
otherwise very little is known as yet about their possible singularities. It
is a natural first step to establish the existence of singularity models, in
analogy with the case of the Ricci flow. In this paper, we carry this out for
the Type IIB flow.

We state now our main results, The definitions of all relevant notions are
given in §2 below. The first two results are analogues for non-Kähler Calabi-
Yau manifolds and for the Type IIB flow of Hamilton’s compactness theorems for
the Ricci flow on Riemannian manifolds [16]:

###### Theorem 1

Let $\\{(M_{j},g_{j},J_{j},\Omega_{j},x_{j})\\}_{j=1}^{\infty}$ be the
sequence of compact marked Hermitian Calabi-Yau manifolds. Assume that we have
the following properties:

(1) The sequence has bounded Hermitian geometry, in the sense of Definition 1
below;

(2) There are constants $C_{1}\geq C_{0}>-\infty$, such that $C_{1}\geq\,{\rm
log}\,\|\Omega_{j}\|^{2}_{g_{j}}\geq C_{0}$.

(3) All manifolds are conformally balanced:
$d(\|\Omega_{j}\|_{g_{j}}\omega^{n-1}_{j})=0$.

Then, after passing possibly to a subsequence, there is a complete marked
Hermitian Calabi-Yau manifold
$(M_{\infty},g_{\infty},J_{\infty},\Omega_{\infty},x_{\infty})$, an exhaustion
$\\{U_{j}\\}_{j\in\bf N}$ of $M_{\infty}$ and a sequence of diffeomorphisms
$\Phi_{j}:U_{j}\rightarrow V_{j}\subset M_{j}$, $\Phi_{j}(x_{\infty})=x_{j}$,
such that the triple
$(\Phi_{j}^{*}g_{j},\Phi_{j}^{*}J_{j},\Phi_{j}\Omega_{j})$ converge to the
limit triple $(g_{\infty},J_{\infty},\Omega_{\infty})$. The limit complex
structure is integrable and $\Omega_{\infty}$ is a holomorphic $(n,0)$-form
with respect to it. Moreover, the limit metric is conformally balanced.

###### Theorem 2

Let $\\{(M_{j},g_{j}(t),J_{j},\Omega_{j},x_{j})\\}_{j=1}^{\infty}$ be a
sequence of solutions of the Type IIB flow on the interval
$[\tau_{0};\tau_{1}]$, such that:

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

$|Rm_{j}|_{g_{j}}+|{\nabla}T|_{g_{j}}+|T|^{2}_{g_{j}}\leq C_{0}$

(2) There is a constant $\iota_{0}>0$, such that
$\mathrm{inj}(M_{j})\geq\iota_{0}$;

(3)There is a constant $C_{1}$, independent of $j$, such that $C_{1}\leq\,{\rm
log}\,||\Omega_{j}||^{2}_{g_{j}}(\tau_{0})$;

Then, after passing possibly to a subsequence, there is a complete conformally
balanced manifold
$(M_{\infty},g_{\infty}(t),J_{\infty},\Omega_{\infty},x_{\infty})$, an
exhaustion $\\{U_{j}\\}_{j\in\bf N}$ of $M_{\infty}$, and a sequence of
diffeomorphisms $\Phi_{j}:U_{j}\rightarrow V_{j}\subset M_{j}$, such that
$(M_{j},g_{j}(t),J_{j},\Omega_{j},x_{j})$ converge to a complete solution
$(M_{\infty},g_{\infty}(t),J_{\infty},\Omega_{\infty},x_{\infty})$ of the Type
IIB flow.

Our third result is the existence of singularity models for the Type IIB flow,
in analogy with Hamilton’s theorem for the Ricci flow [17]. A singularity
model is a solution of the Type IIB flow on a (possibly non-compact)
conformally balanced complex manifold, which arises as a blow-up limit of a
Type IIB flow, as the time parameter $t$ approaches its maximum time of
existence.

###### Theorem 3

Let $\omega(t)$ be a solution of the Type IIB flow on a compact conformally
balanced manifold $M$, and let $[0,T)$ be its maximum time of existence.
Assume that the solution admits an injectivity radius estimate, in the sense
of Definition 4 below. Then for suitable times $t_{j}\to T$ and suitable
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 $f(t)$, as defined in equation (4.1) below, satisfying
the following properties:

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

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

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

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

Finally, we point out a new way of expressing the Type IIB flow, which reveals
its similarities with Bryant’s $G_{2}$ Laplacian flow [4, 5] as well as the
Laplacian formulation of the Type IIA flow [20]:

###### Theorem 4

Let $\langle\alpha,\beta\rangle_{\omega}$ the standard pointwise Hermitian
product of forms and consider the following $L^{2}$ scalar product on forms:

$\displaystyle(\alpha,\beta)=\int_{M}\langle\alpha,\beta\rangle_{\omega}i^{n^{2}}\Omega\wedge\overline{\Omega}=\int_{M}\alpha\wedge\|\Omega\|^{2}_{\omega}\overline{*\beta}.$
(1.1)

Let $\partial_{\omega,\Omega}^{\dagger}$ the operator conjugated to $\partial$
with respect to $(\alpha,\beta)$. Then the conformally balanced condition for
$\omega$ is equivalent to the condition
$\partial_{\omega,\Omega}^{\dagger}\omega=0$, and the Type IIB flow, as
defined originally in (2.27) below, can be rewritten as

$\displaystyle\partial_{t}\omega=-\partial_{\omega,\Omega}^{\dagger}\partial\omega.$
(1.2)

Acknowledgments: I am indebted to Professor Duong H. Phong for suggesting this
problem and his constant attention and help while the paper was in
preparation. I am also grateful to Misha Verbitsky for the fruitful
discussions about the Gromov-Hausdorff convergence.

I was partially supported by the HSE University Basic Research Program,
Russian Academic Excellence Project ’5-100’, by the Möbius Contest Foundation
for Young Scientists, and by the ”Young Russian Mathematics” award.

##  2 Preliminaries

We begin by describing our notations and conventions. Let $(M,g,J)$ be a
Hermitian manifold, and $\omega$ be the Kähler form:, $\omega(X,Y)=g(X,JY)$.
In local holomorphic coordinates, we write

$\omega=ig_{j\bar{k}}dz^{j}\wedge d\bar{z}^{k}$

Let $V=V^{j}\partial_{j}$ be a section of $T^{1,0}M$. The Chern connection
${\nabla}$ is given by

$\displaystyle\nabla_{j}V^{k}=\partial_{j}V^{k}+\Gamma^{k}_{js}V^{s},\quad\nabla_{\bar{j}}V^{k}=\partial_{\bar{j}}V^{k},$
$\displaystyle\Gamma^{k}_{js}=g^{k\bar{p}}\partial_{j}g_{s\bar{p}}.$ (2.1)

The curvature tensor $Rm=R_{\bar{k}j}{}^{p}{}_{q}$ is then given by

$\displaystyle R_{\bar{k}j}{}^{p}{}_{q}=-\partial_{\bar{k}}\Gamma^{p}_{jq}$
(2.2)

and the first and second Chern-Ricci curvatures are defined by

$\displaystyle
R_{\bar{k}j}=R_{\bar{k}j}{}^{p}{}_{p}=-\partial_{\bar{k}}\partial_{j}\,{\rm
log}\,{\rm det}(g)$
$\displaystyle\tilde{R}_{\bar{p}q}=g_{\bar{p}\ell}g^{j\bar{k}}R_{\bar{k}j}{}^{\ell}{}_{q}.$
(2.3)

The torsion is defined by

$\displaystyle T=i\partial\omega,~{}\bar{T}=-i\bar{\partial}\omega.$ (2.4)

If we express $T$ in components as $T={1\over 2}T_{\bar{l}kj}dz^{j}\wedge
dz^{k}\wedge d\bar{z}^{l}$, then
$T_{\bar{l}kj}=\partial_{k}g_{\bar{l}j}-\partial_{j}g_{\bar{l}j}$. We also
define the contracted torsion $\tau=\Lambda i\partial\omega$ which is a
$(1,0)$-form. In coordinates, $\tau=\tau_{j}dz^{j}$, with
$\tau_{j}=g^{k\bar{l}}T_{\bar{l}kj}$.

###  2.1 Connections on Hermitian manifolds

For our purposes, the Chern connection may not be the most convenient
Hermitian connection to use. The reason is that we would like to make use of
convergence theorems of Riemannian manifolds. These require lower bounds on
the injectivity radius with respect to the Levi-Civita connection, which is
different from the injectivity radius defined with respect to the Chern
connection.

Recall that, given a complex structure $J$ and a Hermitian metric on the
manifold, the Chen connection is only one connection in a whole line of
Hermitian connections $\nabla^{t}$, known as the Gauduchon line, and given
explicitly by [15, 2],

$\displaystyle
g(\nabla^{s}_{X}Y,Z)=g(\nabla^{g}_{X}Y,Z)+\frac{1-t}{4}d^{c}\omega(X,Y,Z)+\frac{1+t}{4}d\omega(JX,Y,Z).$
(2.5)

Here $\nabla^{g}$ denotes the Levi-Civita connection. We shall however be
interested in the Bismut connection ${\nabla}^{+}$, which corresponds to
$t=-1$. Explicitly, it is given by

$\displaystyle\nabla^{+}_{j}V^{k}=\partial_{j}V^{k}+\Gamma^{k}_{mj}V^{m}=\nabla_{j}V^{k}+T^{k}_{mj}V^{m}$
$\displaystyle\nabla^{+}_{\bar{j}}V^{k}=\partial_{\bar{j}}V^{k}+g^{k\bar{m}}\bar{T}_{p\bar{j}\bar{m}}V^{p}.$
(2.6)

Using the Chern and the Bismut connections as reference points, one can write
the Gauduchon line as follows:

$\displaystyle{\nabla}^{t}=(1-\kappa){\nabla}+\kappa{\nabla}^{+},\qquad\kappa={1-t\over
2}.$ (2.7)

We are interested in the Bismut connection because of its following property:

###### Lemma 1

The notions of geodesic with respect to the Levi-Civita connection and with
respect to the Bismut connection coincide. In particular, both connections
define the same notion of injectivity radius.

Proof. By definition, the geodesics for the Bismut connection are those curves
$x(t)$, which satisfy the equation $\nabla^{+}_{\dot{x}}\dot{x}=0$. For any
vector field $Z$ we have

$\displaystyle
g(\nabla^{+}_{\dot{x}}\dot{x},Z)=g(\nabla^{g}_{\dot{x}}\dot{x},Z)+\frac{1}{2}d^{c}\omega(\dot{x},\dot{x},Z).$
(2.8)

But $d^{c}\omega(\dot{x},\dot{x},Z)=(-Jd\omega)(\dot{x},\dot{x},Z)=0.$ Hence
$\nabla^{+}_{\dot{x}}\dot{x}=0$ iff $\nabla^{g}_{\dot{x}}\dot{x}=0.$ The lemma
is proved. Q.E.D.

In view of this lemma, we can now introduce a notion of bounded Hermitian
geometry formulated entirely in terms of the Bismut connection:

###### Definition 1

Let $\\{(M_{j},g_{j},J_{j})\\}_{j\in\bf N}$ be a sequence of complete
Hermitian manifolds. Denote by $Rm^{+}_{j}$, and $T^{+}_{j}$ the curvature and
torsion of the Bismut connection on $(M_{j},g_{j},J_{j})$. We say that this
sequence has bounded Hermitian geometry, if the following holds:

(1) There is a constant $\iota_{0}>0$, such that the injectivity radii of
these manifolds are bounded below by $\iota_{0}$;

(2) There are constants $C_{m}>0$, independent on $j$, such that

$\displaystyle|({\nabla}^{+})^{m}Rm^{+}|_{g_{j}}\leq
C_{m},\qquad|({\nabla}^{+})^{m}T^{+}|_{g_{j}}\leq C_{m}$ (2.9)

for any $m\in\bf N$.

###  2.2 Basic facts about Hermitian Calabi-Yau manifolds

In this section, we collect some facts about the non-Kähler Calabi-Yau
manifolds. Recall that a Hermitian Calabi-Yau manifold is a 4-tuple
$(M,g,J,\Omega)$, where $(M,g,J)$ is a Hermitian manifold, and $\Omega$ is a
nonvanishing $(n,0)$-form.

A Hermitian Calabi-Yau manifold does not always possess a Kähler metric. There
are many examples of such manifolds: Calabi-Yau manifolds obtained from the
conifold transition (see [8]), hyperkähler fibrations over Riemannian surfaces
([11], [10]), factors of Lie groups, etc. A less stringent notion than Kähler
is that of balanced metric, introduced by Michelson [23]. These are metrics
satisfying $d\omega^{n-1}=0$. Closely related is the notion of conformally
balanced metric, defined by the condition
$d(\|\Omega\|_{\omega}\omega^{n-1})=0$. The conformally balanced condition was
recognized by Li and Yau [21] as one of the equations in the Hull-Strominger
system in string theory. In terms of the metric $\eta=\|\Omega\|\omega$, it
can also be expressed as $d(\|\Omega\|^{2}\eta^{n-1})=0$.

###  2.3 Pointed Gromov-Cheeger-Hamilton convergence

We recall the notion of convergence due to Hamilton [16], which is an
adaptation of a notion of $C^{\infty}$ convergence of manifolds due to Gromov
and Cheeger:

###### Definition 2

Let $\\{(M_{j},g_{j},O_{j})\\}$ be a sequence of complete Riemannian
manifolds, where $O_{j}\in M_{j}$ is a point. We say that this sequence
converges to the complete pointed manifold
$(M_{\infty},g_{\infty},O_{\infty},q_{\infty})$ in $C^{k}$-sense, if there is
a sequence of open 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 $\Phi_{j}(O_{\infty})=O_{j}$ and the sequence of metrics
$\\{\Phi_{j}^{*}g_{j}\\}$ converges to $g_{\infty}$ uniformly in
$C^{k}$-topology on compact subsets in $U_{j}$.

The following theorem of Hamilton [16, Theorem 1.2] gives an important set of
conditions implying the existence of a convergent subsequence:

###### Theorem 5

Let $\\{M_{j},g_{j},O_{j}\\}$ 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 $O_{j}$, $inj(M_{j},O_{j})\geq\iota_{0}$.

Then there is a subsequence of the initial sequence, which converges to a
complete pointed Riemannian manifold $(M_{\infty},g_{\infty},O_{\infty})$.

###  2.4 Curvature estimates

In this section, we fix a smooth compact manifold $M$ with metric $g_{ij}$.
Let $E$ be a smooth vector bundle over $M$ equipped with a metric
$H_{\alpha\beta}$. Let $A_{1},A_{2}$ be two connections on $E$, and
$F_{1},F_{2}$ be their curvatures. We are interested in obtaining bounds for
the curvature of one of them from bounds for the curvature of the other.

Let $C=A_{1}-A_{2}$, which is a $End(E)$-valued 1-form. We have

$\displaystyle F_{1}$ $\displaystyle=$ $\displaystyle dA_{1}+F_{1}\wedge
A_{1}=dA_{2}+A_{2}\wedge A_{2}+dC+[A_{2},C]+C\wedge C$ (2.10) $\displaystyle=$
$\displaystyle F_{2}+d_{A_{2}}C+C\wedge C.$

Here $d_{A_{2}}C$ is the covariant derivative with respect to $A_{2}$. Let
$x^{1},\dots,x^{n}$ be a system of local coordinates on $M$. Denote by
$\nabla^{j}$ the covariant derivative with respect to $A_{j}$. Then for any
section $\psi$ of $E$ we have

$\displaystyle\nabla^{j}_{a}\psi=\partial_{a}\psi+(A_{j})_{a}\psi.$ (2.11)

For our endomorphism-valued 1-form $C=C_{a}dx^{a}$ we obtain the following
formula:

$\displaystyle\nabla^{j}_{a}C_{b}=\partial_{j}C_{b}+[(A_{j})_{a},C_{b}].$
(2.12)

In this notation,
$d_{A_{j}}C=(\nabla^{j}_{a}C_{b}-\nabla^{j}_{b}C_{a})dx^{a}\wedge dx^{b}$. Let
$F_{j}=(F_{j})_{ab}dx^{a}\wedge dx^{b}$ for $j=1,2$. Combining all this, we
obtain the following formula:

$\displaystyle(F_{1})_{ab}=(F_{2})_{ab}+(\nabla^{2}_{a}C_{b}-\nabla^{2}_{b}C_{a})+[C_{a},C_{b}].$
(2.13)

Assume now that we have bounds for all covariant derivatives of $F_{2}$ and
$C$ with respect to $\nabla^{2}$. More presicely, for any natural $p$, there
are constants $C_{p}>0$, such that $|(\nabla^{2})^{p}F_{2}|<C_{p}$ and
$|(\nabla^{2})^{p}C|<C_{p}$. We want to establish the estimates of
$|(\nabla^{1})^{p}F_{1}|$ in terms of $|(\nabla^{2})^{q}C|$ and
$|(\nabla^{2})^{p}F_{2}|$.

In order to do this, we recall some basic facts about the norms of tensors.
First of all, for any two $E$-valued tensors $T_{1},T_{2}$ on $M$, we have a
pointwise inequality:

$\displaystyle|T_{1}*T_{2}|\leq C_{1}|T_{1}||T_{2}|$ (2.14)

where $\star$ denotes any bilinear operation involving the metrics $g_{ij}$
and $H_{\alpha\beta}$, and the constant $K_{1}=K_{1}(dim(M),rk(E))$ is a
universal constant depends only on the dimension $n$ of $M$, the rank of $E$,
and the type of $T_{1}$ and $T_{2}$, but independent of $g_{ij}$ and
$H_{\alpha\beta}$. This can be proven by Linear Algebra.

Next, for any tensor $T$ we have the following

$\displaystyle|(\nabla^{2}+C)T|\leq|\nabla^{2}T|+|CT|\leq|\nabla^{2}T|+C_{1}|C||T|.$
(2.15)

Iterating this process, we obtain the following inequality:

$\displaystyle|(\nabla^{2}+C)^{p}T|\leq|(\nabla^{2})^{p}T|+\Sigma_{k=0}^{p-1}|C*\dots*(\nabla^{2})^{k}(C*\dots*C*T)|$
$\displaystyle\leq$
$\displaystyle|(\nabla^{2})^{p}T|+K_{p}\Sigma_{k_{1}+\dots+k_{p}=p-1}|(\nabla^{2})^{k_{1}}C|\dots|(\nabla^{2})^{k_{p-1}}C||(\nabla^{2})^{k_{p}}T|,$
(2.16)

where $K_{p}$ are universal, and independent of the metrics $g_{ij}$ and
$H_{\alpha\beta}$ Now we can state the main lemma of this section.

###### Lemma 2

Let $M$ be a smooth manifold, E is a vector bundle over $M$, $A_{1},A_{2}$ are
arbitrary connections and $F_{1},F_{2}$ are they curvatures. Let
$C=A_{1}-A_{2}$ (as above). Suppose that there are constants $C_{m}>0$, such
that $|({\nabla}^{2})^{m}F_{2}|<C_{m}$ and $|({\nabla}^{2})^{m}C|<C_{m}$ for
any natural $m$. Then there are constants $K_{m}$ depending on $C_{p}$ and
independent on the metrics $g_{ij}$ and $H_{\alpha\beta}$ such that
$|({\nabla}^{1})^{m}F_{1}|<K_{m}$

The proof is just applying the inequality above to $T=F_{2}$, $T=d_{A_{2}}C$
and $T=[C,C]$. Note that there is a constant $C_{1,1}$, such that
$|[C,C]|<C_{1,1}|C|^{2}$. This lemma has very important consequences:

###### Lemma 3

Let $(M,g,J)$ be a Hermitian manifold. Let ${\nabla}$ and ${\nabla}^{g}$ be
the Chern and the Levi-Civita connections on $M$ with curvatures $Rm,Rm^{g}$.
Suppose there are constants $C_{m}$, $m\in\bf N$, such that
$|{\nabla}^{m}Rm|_{g}\leq C_{m}$ and $|{\nabla}^{m}T|_{g}\leq C_{m}$. Then
there are constants $K_{m}$, such that $|({\nabla}^{g})^{m}Rm^{g}|_{g}\leq
K_{m}$.

Proof. Let $C(X,Y)={\nabla}_{X}Y-{\nabla}^{g}_{X}Y$ for any vector fields
$X,Y$ on $M$. This is a bilinear form with values in the tangent bundle. By
the formula 2.5 , we have the following:

$\displaystyle g(C(X,Y),Z)$ $\displaystyle=$ $\displaystyle
g({\nabla}_{X}Y,Z)-g({\nabla}^{g}_{X}Y,Z)=d\omega(JX,Y,Z)$ (2.17)
$\displaystyle=$ $\displaystyle(T^{{\bf R}}*J*J)(X,Y,Z)$

where $T^{\bf R}=T+\bar{T}=i(\partial-\bar{\partial})\omega$. Hence, there is
a constant $P$, such that $|C|_{g}\leq P|T|_{g}|J|_{g}$. Similarly,
$|{\nabla}^{m}C|_{g}=|J*{\nabla}^{m}T|_{g}\leq
P_{m}|{\nabla}^{m}T|_{g}|J|_{g}$ for some universal constant $P_{m}$. Note
that $|J|_{g}=\sqrt{n}$, where $n=\dim_{\bf C}M$. Hence, we obtain bounds for
$|{\nabla}^{m}C|_{g}$. Now we apply Lemma 2 with ${\nabla}^{2}={\nabla}$,
${\nabla}^{1}={\nabla}^{g}$. Q.E.D.

###### Lemma 4

Let $(M,g(t),J)$ be a family Hermitian manifold, $t\in{\bf R}$. Assume that
$|{\nabla}^{m}T|_{g(t)}\leq C_{m}$, where $C_{m}$ are independent on $g(t)$.
Then there are constants $K_{m}$, such that $|({\nabla}^{g})^{m}J|\leq K_{m}$,
and these constants are independent on $g(t)$.

Proof. For notational simplicity, we omit the dependence on $t$ of $g(t)$. By
definition of the Chern connection, ${\nabla}J=0$. Hence,

$\displaystyle{\nabla}^{g}J={\nabla}^{g}J-{\nabla}J=T^{\bf R}*J*J*J$ (2.18)

and

$\displaystyle|{\nabla}^{g}J|_{g}\leq P_{1}|T|_{g}|J|^{3}_{g}.$ (2.19)

The constant $P_{1}$ depends only on the dimension $n=\dim_{\bf C}M$. The norm
$|J|_{g}$ is equal to $\sqrt{n}$, hence

$\displaystyle|{\nabla}^{g}J|_{g}\leq P_{1}C_{1}n^{3\over 2}.$ (2.20)

In particular, this bound does not depend on the metric. For the case of
higher derivatives we can use the formula (2.4). We have

$\displaystyle|({\nabla}^{g})^{m}J|_{g}\leq
P_{m}\sum_{k_{1}+\dots+k_{m-1}=m-1}|{\nabla}^{k_{1}}T|_{g}\dots|{\nabla}^{k_{m-1}}T|_{g}|J|^{q(m)}_{g},$
(2.21)

and $P_{m},q(m)$ does not depend on metric. Since the all derivatives of $T$
are bounded, we conclude, what there are constants $K_{m}=P_{m}n^{q(m)\over
2}\sum_{k_{1}+\dots+k_{m-1}=m-1}C_{k_{1}}\dots C_{k_{m-1}}$ Q.E.D.

###### Lemma 5

Let $\\{(M_{j},g_{j}(t),J_{j})\\}_{j\in\bf N}$ be the sequence of Hermitian
manifolds with bounded geometry in the sense of the Definition 1. Denote by
$RM_{g_{j}}$ the curvature of the Levi-Civita connection on $(M_{j},g_{j})$.
Then there are constants $K_{m}>0$, independent of $j$, such that
$|({\nabla}^{g_{j}})^{m}Rm_{g_{j}}|_{g_{j}}\leq K_{m}$.

Proof. In general, the Bismut connection and the Levi-Civita connection are
related by the following formula:

$\displaystyle{\nabla}^{+}={\nabla}^{g}+{1\over 2}g^{-1}H$ (2.22)

and $H=d^{c}\omega$. On the other hand, $H=g(T^{+}(X,Y),Z)$. Applying to the
case at hand, and omitting the dependence on $t$ from the notation, we can
write ${\nabla}^{+,j}-{\nabla}^{g}={1\over 2}g^{-1}_{j}H_{j}$. The desired
statement follows now from Lemma 2 with ${\nabla}^{1}={\nabla}^{g_{j}}$ and
${\nabla}^{2}={\nabla}^{+,j}$. Q.E.D.

###### Lemma 6

Let $\\{(M_{j},g_{j}(t),J_{j})\\}_{j\in\bf N}$ be a sequence of Hermitian
manifolds, and ${\nabla}^{j},Rm_{j},T_{j}$ be the Chern connection, its
curvature and torsion respectively. Denote by
${\nabla}^{s,j},Rm_{s,j},T_{s,j}$ be the Gauduchon connection, its curvature
and torsion with parameter $s$ on the Gauduchon line. Assume that there are
constants $C_{m}>0$, independent on $j$, such that
$|({\nabla}^{j})^{m}Rm_{j}|_{g_{j}}\leq C_{m}$ and
$|({\nabla}^{j})^{m}T_{j}|_{g_{j}}\leq C_{m}$. Then for any $s\in{\bf R}$
there are constants $K_{s,m}>0$, independent of $j$, such that

$\displaystyle|({\nabla}^{s,j})^{m}Rm_{t,j}|_{g_{j}}\leq
K_{s,m},\qquad|({\nabla}^{s,j})^{m}T_{t,j}|_{g_{j}}\leq K_{s,m}.$ (2.23)

Proof. We know that ${\nabla}^{s}={\nabla}^{g}+{1+s\over 4}g^{-1}H+{1-s\over
4}g^{-1}S$, where $H(X,Y,Z)=d^{c}\omega(X,Y,Z)$ and
$S(X,Y,Z)=d\omega(JX,Y,Z)$. Next, ${\nabla}^{g}={\nabla}-{1\over 2}g^{-1}S$.
Hence

$\displaystyle{\nabla}^{s}={\nabla}+{1+s\over 4}g^{-1}H-{1+s\over 4}g^{-1}S.$
(2.24)

This implies that the difference ${\nabla}^{t}-{\nabla}$ has the form of
$T^{\bf R}*J*J$ and $T^{\bf R}$, where $T^{R}=d^{c}\omega$. The bounds on
$T_{j}$ implies the bounds on $T^{\bf R}_{j}$ with all derivatives. Thus, the
difference ${\nabla}^{t}-{\nabla}$ has bounded covariant derivatives w.r.t.
${\nabla}^{j}$. Lemma 2 implies bounds for
$|({\nabla}^{t,j})^{m}Rm_{t,j}|_{g_{j}}$. The bounds for
$|({\nabla}^{t,j})^{m}T_{t,j}|_{g_{j}}$ follow from the inequality (2.15) with
${\nabla}^{2}={\nabla}^{j}$ and ${\nabla}^{1}={\nabla}^{t,j}$. Q.E.D.

Finally, we can prove an analogue of Lemma 4 for any Gauduchon connection.

###### Lemma 7

Let $(M,g(t),J)$ be a family of Hermitian manifolds, $s\in{\bf R}$,
${\nabla}^{s}$ as above. Assume that $|({\nabla}^{s})^{m}T_{s}|_{g(t)}\leq
C_{m}$, where $C_{m}$ are independent of $g(t)$. Then there are constants
$K_{m}$, such that $|({\nabla}^{g})^{m}J|\leq K_{m}$, and these constants are
independent of $g(t)$.

Proof. The proof is similar to the proof of Lemma 4. We have the formula
${\nabla}^{s}={\nabla}^{g}+{1+s\over 4}g^{-1}H+{1-s\over 4}g^{-1}S$, and
${\nabla}^{s}J=0$. The bound on torsion implies the bound on $d\omega$ by the
following formula

$\displaystyle(d\omega)_{kjl}={\nabla}^{s}_{j}\omega_{kl}+{\nabla}^{s}_{k}\omega_{lj}+{\nabla}^{s}_{l}\omega_{jk}+3(T_{s}*\omega)_{kjl}=3(T_{s}*\omega)_{kjl},$
(2.25)

where our indices are real indices. The same is true for $d^{c}\omega$. Hence
we obtain $d^{c}\omega$, and both $H=d^{c}\omega$ and $S=d\omega*J$. This
implies a bound on the difference $C_{s}={\nabla}^{g}-{\nabla}^{s}$ in terms
of norms of $H$ and $S$, which are bounded by the norm of the torsion
multiplied by a universal constant. Now we apply the inequality (2.4) for
$({\nabla}^{g})^{m}J$ with ${\nabla}^{2}={\nabla}^{t}$ and $C=C_{s}$.

###  2.5 The Type IIB flow

Let $M$ be a complex manifold of complex dimension $n$ with trivial canonical
bundle $K_{M}={\rm det}(T^{(1,0)}M)^{*}$, and let $\Omega$ be a nowhere
vanishing holomorphic $(n,0)$-form. The Type IIB flow is the flow of
$(n-1,n-1)$-forms defined as follows [24]

$\displaystyle\partial_{t}(\|\Omega\|_{\eta}\eta^{n-1})=i\partial\overline{\partial}\eta^{n-2}.$
(2.26)

with conformally balanced intiial data $\omega_{0}$. It can also be viewed as
a special case with vanishing slope parameter $\alpha^{\prime}$ of the Anomaly
flow introduced in [25]. If we define the metric $\omega$ by
$\omega=\|\Omega\|_{\eta}\eta=ig_{j\bar{k}}dz^{j}\wedge d\bar{z}^{k}$, then
the above flow for $\eta$ is equivalent to the following flow for $\omega$,

$\displaystyle\partial_{t}(\|\Omega\|^{2}_{\omega}\omega^{n-1})=i\partial\overline{\partial}(\|\Omega\|^{2}_{\omega}\omega^{n-2})$
(2.27)

A motivating property for the Type IIB flow is the fact that it preserves the
conformally balanced property of the initial data. For estimates, it is useful
to express it as follows as a flow of $(1,1)$-forms (see [18] for the detailed
derivation)

$\displaystyle\partial_{t}g_{\bar{k}j}=-\tilde{R}_{\bar{k}j}-{1\over
2}T_{\bar{k}pq}\bar{T}_{j}^{~{}pq}$ (2.28)

where $\tilde{R}_{j\bar{k}}=g_{s\bar{k}}g^{p\bar{q}}R^{s}_{~{}j\bar{q}p}$ is
the second Chern-Ricci curvature of the metric. The Type IIB flow reduces to
the Kähler-Ricci flow on Kähler manifolds and one can think about it as a
particular generalization of the Kähler-Ricci flow to non-Kähler manifolds
111Other generalizations have been considered by Streets and Tian [33] and
Ustinovsky [35].. In particular, the Type IIB flow shares many properties with
the Kähler-Ricci flow. As described in Theorem 4, we can provide yet another
formulation of the Type IIB flow in the spirit of Laplacian flows:

Proof of Theorem 4. Recall that for any (2,1)-form $\psi$ the usual operator
$\partial^{\dagger}$ acts as follows (see [18, Appendix D]):

$\displaystyle(\partial^{\dagger}\psi)_{\bar{\alpha}\beta}=-g^{\gamma\bar{j}}{\nabla}_{\bar{j}}\psi_{\bar{\alpha}\beta\gamma}+g^{\gamma\bar{j}}\bar{\tau}_{\bar{j}}\psi_{\bar{\alpha}\beta\gamma}-{1\over
2}\bar{T}_{\beta\bar{j}\bar{m}}\psi_{\bar{\alpha}\gamma\delta}g^{\gamma\bar{j}}g^{\delta\bar{m}}$
(2.29)

The second summand appears from the integration by parts formula for the
divergence of the Chern connection:

$\displaystyle\int_{M}\nabla_{j}V^{j}\omega^{n}=\int_{M}\tau_{j}V^{j}\omega^{n}$
(2.30)

for any vector field $V^{j}$. On the other hand, by the conformally balanced
condition, we have
$\nabla_{\|}\Omega\|^{2}_{\omega}=\|\Omega\|^{2}_{\omega}\tau_{j}$ (see [26,
Lemma 4]). This implies

$\displaystyle\int_{M}\nabla_{j}V^{j}i^{n^{2}}\Omega\wedge\overline{\Omega}=\int_{M}(\nabla_{j}V^{j})||\Omega||^{2}_{\omega}{\omega^{n}\over
n!}=0.$ (2.31)

Hence, by the same argument as in [18, Appendix D], we can show that for any
(2,1)-form $\psi$ we have

$\displaystyle(\partial^{\dagger}_{\omega,\Omega}\psi)_{\bar{\alpha}\beta}=-g^{\gamma\bar{j}}{\nabla}_{\bar{j}}\psi_{\bar{\alpha}\beta\gamma}-{1\over
2}\bar{T}_{\beta\bar{j}\bar{m}}\psi_{\bar{\alpha}\gamma\delta}g^{\gamma\bar{j}}g^{\delta\bar{m}}.$
(2.32)

In fact, $\partial^{\dagger}_{\omega,\Omega}=\partial^{\dagger}-\iota_{\tau}$,
where $\iota_{\tau}$ is the contraction with the vector field
$g^{j\bar{k}}\bar{\tau}_{k}$. The equivalence between the conformally balanced
condition and the condition $\partial_{\omega,\Omega}^{\dagger}\omega=0$
follows. Next, let $\psi=T=i\partial\omega$. Then

$\displaystyle(\partial^{\dagger}_{\omega,\Omega}T)_{\bar{\alpha}\beta}=-{\nabla}^{\gamma}T_{\bar{\alpha}\beta\gamma}-{1\over
2}\bar{T}_{\beta\bar{j}\bar{m}}T_{\bar{\alpha}\gamma\delta}g^{\gamma\bar{j}}g^{\delta\bar{m}}=-\tilde{R}_{\bar{\alpha}\beta}-{1\over
2}T_{\bar{\alpha}}^{\bar{j}\bar{m}}\bar{T}_{\beta\bar{j}\bar{m}}$ (2.33)

where in the second equality we used the fact
$\tilde{R}_{\bar{\alpha}\beta}=-{\nabla}^{\gamma}T_{\bar{\alpha}\beta\gamma}$
(see [26, Lemma 5,iii]). Combining all together, and using the fact that
$\omega=ig_{\bar{k}j}dz^{j}\wedge d\bar{z}^{k}$, we obtain that

$\displaystyle\partial_{t}(ig_{\bar{k}j})=i(\partial^{\dagger}_{\omega,\Omega}i\partial\omega)_{\bar{k}j}=-(\partial^{\dagger}_{\omega,\Omega}\partial\omega)_{\bar{k}j}.$
(2.34)

Q.E.D.

We observe that the evolution equation for the torsion was derived in [19,
Equation 4.6]. Using our Laplacian formulation, we can readily rederive this
result:

$\displaystyle\partial_{t}T=\partial_{t}i\partial\omega=-i\partial\partial^{\dagger}_{\omega,\Omega}\partial\omega=-\partial\partial^{\dagger}_{\omega,\Omega}T.$
(2.35)

###  2.6 Shi-type estimates for connections on the Gauduchon line

Shi-type estimates for the Type IIB flow were established in [19, 26]:

###### Theorem 6

Let $(M,\omega)$ be a Calabi-Yau manifold and $\omega$ evolve by the Type IIB
flow. Suppose there is a constant $A$, such that $|Rm|+|\nabla T|+|T|^{2}\leq
A$ for $t\in[0;{1\over A}]$. Then for any $k>0$ there is $C_{k}>0$ (depending
on the lower bound for $||\Omega||$), such that
$|\nabla^{k}Rm|\leq\frac{C_{k}A}{t^{k/2}}$ and
$|\nabla^{k+1}T|\leq\frac{C_{k}A}{t^{k/2}}$.

In the above theorem, the curvature and torsion were those of the Chern
connection. For our purposes, in order to apply later compactness theorems for
sequences of manifolds with bounded Hermitian geometry, we need rather bounds
for the curvature and torsion of the Bismut connection. But the passage from
the Chern connection to any connection on the Gauduchon line can now be
obtained from Lemma 6, applied to the case of fixed $t$. Thus we obtain the
following Shi-estimates for the Type IIB flow for any connection
${\nabla}^{s}$ on the Gauduchon line:

###### Lemma 8

Let $(M,g(t),J)$ be a solution of the Type IIB flow on the interval
$[0;\frac{1}{A}]$. As before, we denote by $Rm_{s}$ and $T_{s}$ the curvature
and the torsion tensors for $\nabla^{s}$. We denote $Rm_{1}$ by $Rm$, and
$T_{1}$ by $T$. Assume that $|Rm|+|\nabla T|+|T|^{2}\leq A$ for any point in
$M\times[0;\frac{1}{A}]$. Then, for any $s\in[-1;1]$ there are constants
$Q_{s,k},P_{s,k}$, such that
$|(\nabla^{s})^{k}Rm_{s}|\leq\frac{Q_{s,k}}{t^{k/2}}$, and
$|(\nabla^{s})^{k+1}T|\leq\frac{P_{s,k}}{t^{k/2}}$.

##  3 Compactness theorem for the Type IIB flow

We give now the proof of Theorem 1. We break it into several steps.

Convergence of the metrics

The $C^{\infty}$ convergence of the Riemannian manifolds $(M_{j},g_{j})$ to
$M_{\infty},g_{\infty}$, the existence of an exhaustion $U_{j}$ for
$M_{\infty}$ and the existence of diffeomorphisms $\Phi_{j}:U_{j}\rightarrow
V_{j}\subset M_{j}$ all follow from Lemma 2 and Hamilton’s compactness
theorem. Indeed, by Lemma 2 and Lemma 3, we have uniform bounds for the
covariant derivatives of Riemannian curvature w.r.t. to the Levi-Civita
connection, so we can apply Hamilton’s compactness theorem.

Convergence of the complex structures

Consider the forms $\omega_{j}(X,Y)=g_{j}(J_{j}X,Y)$. It is well known that
the pair $g_{j},\omega_{j}$ defines $J_{j}$. Let $\Phi_{j,k}$ will be
restriction $\Phi_{j+k}$ 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
$U_{j}$. Due to this, we omit the index $j$ in $g_{j,\infty}$ and write
$g_{\infty}$ everywhere. We need to show that the forms
$\omega_{j,k}:=\Phi_{j,k}^{*}\omega_{j+k}$ converge uniformly with all
derivatives on $U_{j}$ as $k\rightarrow\infty$ to some nondegenerate forms
$\omega_{j,\infty}$. In order to do this, we need to obtain uniform bounds for
$|({\nabla}^{\infty})^{m}\omega_{j,k}|_{g_{\infty}}$.

Let $v_{j,k}:=g_{\infty}-g_{j,k}$ and
$A_{j,k}:={\nabla}^{g^{\infty}}-{\nabla}^{g_{j,k}}$. In a local coordinate
chart $(x^{1},\dots,x^{2n})$ on $U_{j}$, the tensor $A_{j,k}$ can be written
as follows:

$\displaystyle(A_{j,k})^{\alpha}_{\beta\gamma}={1\over
2}(g^{j,k})^{\alpha\delta}({\nabla}^{\infty}_{\beta}(v_{j,k})_{\delta\gamma}+{\nabla}^{\infty}_{\gamma}(v_{j,k})_{\beta\delta}-{\nabla}^{\infty}_{\delta}(v_{j,k})_{\beta\gamma}).$
(3.1)

One can check this formula in the geodesic coordinates for $g_{\infty}$.

Since $g_{j,k}$ converge to $g_{\infty}$ in $C^{\infty}$, the norms
$|({\nabla}^{g_{\infty}})^{m}v_{j,k}|_{g_{\infty}}$ tends to zero for any
$m\in\bf N$. The argument from [22, Theorem 7.1] shows, that for any $m\geq 0$
there is a constant $K_{m}$, such that
$|({\nabla}^{g_{\infty}})^{m}A_{j,k}|_{g_{\infty}}\leq K_{m}$.

Now we can obtain bounds for $\omega_{jk}$. These bounds follow from the
boundedness of the covariant derivatives of $J_{j}$ (which follows from the
Lemma 4), and the bounds for $A_{j,k}$. We have

$\displaystyle{\nabla}^{g_{\infty}}\omega_{j,k}={\nabla}^{g_{j,k}}\omega_{j,k}+({\nabla}^{g_{\infty}}-{\nabla}^{g_{j,k}})\omega_{j,k}.$
(3.2)

Since $g_{j,k}$ converge to $g_{\infty}$ with all derivatives, there is a
constant $K_{\infty}$ independent of metrics, such that ${1\over
K_{\infty}}g_{j,k}\leq g_{\infty}\leq K_{\infty}g_{j,k}$ for all $k$ on
$U_{j}$, and

$\displaystyle|{\nabla}^{g_{\infty}}\omega_{j,k}|_{g_{\infty}}\leq(K_{\infty})^{3\over
2}|{\nabla}^{g_{j}}\omega_{j,k}|_{g_{j,k}}+K_{0}|\omega_{j,k}|_{g_{j,k}}.$
(3.3)

The quantity $|{\nabla}^{g_{j}}\omega_{j,k}|_{g_{j,k}}$ is bounded because the
covariant derivatives of $J_{j}$ are bounded. The quantity
$|\omega_{j,k}|_{g_{j,k}}$ is a constant.

Next, we obtain bounds for the higher derivatives of ${\nabla}_{j,k}$. As in
[22], we can write:

$\displaystyle|({\nabla}^{g_{\infty}})^{l}\omega_{j,k}|_{g_{\infty}}$
$\displaystyle\leq$ $\displaystyle
C\Sigma_{m=0}^{l}|A_{j,k}|^{m}_{g_{\infty}}|({\nabla}^{g_{j,k}})^{m-l}\omega_{j,k}|_{g_{j,k}}$
(3.4)
$\displaystyle\quad+C\Sigma_{m=1}^{l-1}|({\nabla}^{g_{\infty}})^{m}A_{j,k}|_{g_{\infty}}|({\nabla}^{g_{j,k}})^{m-l-1}\omega_{j,k}|_{g_{j,k}}$

The right hand side is bounded by a constant, because each summand is bounded
by a constant. By the Arzela-Ascoli theorem (see [3, Corollary 9.14]), we can
extract a subsequence, which converge to the form $\omega_{j,\infty}$ in
$C^{\infty}$ on compacts in $U_{j}$. The form $\omega_{j,\infty}$ is
compatible with $g_{\infty}$ on $U_{j}$. This follows from the compatibility
of $g_{j,k}$ and $\omega_{j,k}$. In particular,

$\displaystyle\omega_{j,\infty}^{n}=\lim_{k\rightarrow\infty}\omega_{j,k}^{n}=n!\lim_{k\rightarrow\infty}d\mu_{g_{j,k}}=n!d\mu_{g_{\infty}},$
(3.5)

hence $\omega_{j,\infty}$ is nondegenerate. The pair
$(g_{j,\infty},\omega_{j,\infty})$ defines an almost complex structure
$J_{j,\infty}:=g_{j,\infty}^{-1}\omega_{j,\infty}$, which is the
$C^{\infty}$-limit of $J_{j,k}:=g_{j,k}^{-1}\omega_{j,k}$. In fact,
$J_{j,\infty}$ is integrable, because all $J_{j,k}$ are integrable, and
$\lim_{k\rightarrow\infty}J_{j,k}=J_{j,\infty}$ in $C^{\infty}$.

Let $I_{jk}:U_{j}\rightarrow U_{k}$ be the inclusion map for $k\geq j$. We see
that $I^{*}_{jk}\omega_{k,l}=\omega_{j,l}$, and
$I^{*}_{jk}\omega_{k,\infty}=\omega_{j,\infty}$. Since the set of smooth
sections of any vector bundle on $M_{\infty}$ is a sheaf, there is a
nondegenerate form $\omega_{\infty}$ on the whole $M_{\infty}$, such that the
restriction of this form on each $U_{j}$ is $\omega_{j,\infty}$. Furthermore,
the pair $(g_{\infty},\omega_{\infty})$ defines an almost complex structure
$J_{\infty}:=g_{\infty}^{-1}\omega_{\infty}$ on $M_{\infty}$.

In is not hard to see that the restriction of $J_{\infty}$ on $U_{j}$ is equal
to $J_{j,\infty}$, defined by the pairs $(g_{\infty},\omega_{j,})$. Hence, the
Nijenhuis tensor of $J_{\infty}$ is zero, and
$(M_{\infty},g_{\infty},J_{\infty})$ is a complete Hermitian manifold.

Convergence of the holomorphic forms

By [31, Lemma 2.5], we know that forms
$\psi_{j}:={\Omega_{j}\over||\Omega_{j}||_{g_{j}}}$ are parallel w.r.t. Bismut
connection ${\nabla}^{+,g_{j}}$. Recall that the Bismut connection on $(n,0)$
forms is written as follows:

$\displaystyle{\nabla}^{+}_{j}\psi=\partial_{j}\psi-\Gamma^{s}_{js}\psi-T^{s}_{js}\psi$
$\displaystyle{\nabla}^{+}_{\bar{j}}\psi=\partial_{\bar{j}}\psi-g^{s\bar{p}}\bar{T}_{s\bar{j}\bar{p}}\psi=\partial_{\bar{j}}\psi+\bar{\tau}_{\bar{j}}\psi$
(3.6)

Hence, all derivatives of these forms are bounded, and we again apply the
Arzela-Ascoli theorem, and show that the pullbacks
$\psi_{j,k}:=\Phi^{*}_{j+k}\psi_{j+k}$ preconverge to the limit form
$\psi_{j,\infty}$ on $U_{j}$ in the $C^{\infty}$ sense. Moreover, the form
$\psi_{j,\infty}$ will be parallel $(n,0)$-form with respect to the Bismut
connection of the pair $(g_{\infty},J_{\infty})$ and
$|\psi_{j,\infty}|_{g_{\infty}}=1$. Indeed, we have the following estimate:

$\displaystyle|\nabla^{+,g_{\infty}}\psi_{j,\infty}-\nabla^{+,g_{j,k}}\psi_{j,k}|_{g_{\infty}}\leq|\nabla^{+,\infty}(\psi_{j,\infty}-\psi_{j,k})|_{g_{\infty}}+|(\nabla^{+,\infty}-\nabla^{+,g_{j,k}})\psi_{j,k}|_{g_{\infty}}.$
(3.7)

The first summand is bounded by $\varepsilon$, because because metrics,
complex structures and $(n,0)$ forms $\psi_{j,k}$ converge to
$g_{\infty},J_{\infty}$ and $\psi_{j,\infty}$ in $C^{\infty}$ sense. The
second summand is bounded by $\varepsilon$ by the same reason. As in previous
section, there is a globally defined, nondegenerate $(n,0)$-form
$\psi_{\infty}$ on the whole $M_{\infty}$, which is parallel w.r.t. the Bismut
connection on $(M_{\infty},g_{\infty},J_{\infty})$.

Let $f_{j,\infty}=\lim_{k\rightarrow\infty}\|\Omega_{j,k}\|_{g_{j,k}}$. This
limit exists and will be the $C^{\infty}$ function, because the differentials
of $\|\Omega_{j,k}\|$ is the 1-form
$\|\Omega_{j,k}\|\tau(g_{j,k})=-\|\Omega_{j}\|_{g_{j,k}}J_{j,k}d^{\dagger}_{j,k}\omega_{j,k}$
(see [26, Lemma 4]; the operator $d^{\dagger}_{j,k}$ is defined via
$\omega_{j,k}$). Hence all derivatives of $\|\Omega_{j}\|$ are bounded, and we
again apply the Arzela-Ascoli theorem. By our assumption on $\,{\rm
log}\,\|\Omega_{j}\|_{g_{j}}$, this is a $C^{\infty}$ positive function and
$\nabla^{+,g_{j,k}}\|\Omega_{j}\|_{g_{j,k}}\rightarrow{\nabla}^{+,g_{j,\infty}}f_{j,\infty}=f_{j,\infty}\tau(g_{j,\infty})$,
and there is a $C^{\infty}$ positive function $f_{\infty}$, defined on the
whole $M_{\infty}$, such that its restriction on $U_{j}$ is $f_{j,\infty}$.
Consider the form $\Omega_{\infty}:=f_{\infty}\psi_{\infty}$. This form will
be a nonvanishing holomorphic form on $(M,J_{\infty})$. Indeed, in a local
coordinate system on $(M_{\infty},J_{\infty})$ we have

$\displaystyle\partial_{\bar{a}}\Omega_{\infty}={\nabla}^{+}_{\bar{a}}\Omega_{\infty}-\tau_{\bar{a}}\Omega_{\infty}={\nabla}^{+}_{\bar{a}}(f_{\infty}\phi_{\infty})-\tau_{\bar{a}}f_{\infty}\phi_{\infty}=\tau_{\bar{a}}f_{\infty}\phi_{\infty}-\tau_{\bar{a}}f_{\infty}\phi_{\infty}=0,$
(3.8)

where bar over indices takes w.r.t. $J_{\infty}$, and all $\tau$ is the
torsion of the Chern connection on $(M_{\infty},g_{i}.J_{i})$. Note that the
function $f_{\infty}$ satisfies the following identity:
$f_{\infty}=||\Omega_{\infty}||_{g_{\infty}}$.

The conformally balanced property of the limit metric

Finally, the limit metric satisfies the conformally balanced condition. This
follows from the convergence of derivatives of metrics and complex structures.
Q.E.D.

Clearly, condition (4) in the theorem can be replaced by the condition
$d(\|\Omega_{j}\|^{2}_{g_{j}}\omega^{n-1}_{j})=0$, and the conclusion would
remain the same, with the limiting triple
$(g_{\infty},J_{\infty},\Omega_{\infty})$ satisfying the same condition.

Finally, we note that a compactness theorem for sequences of Kähler manifolds
can be found in [30].

###  3.1 Estimates for sequences of metrics and for $\|\Omega\|$

In order to prove Theorem 2 we need to estimate the mixed time and spatial
derivatives for the metric $g(t)$, which evolves as
$\partial_{t}g_{\bar{k}j}=-\tilde{R}_{\bar{k}j}-{1\over
2}T_{\bar{k}pq}\bar{T}_{j}^{pq}$. We need a fixed Hermitian metric $\hat{g}$,
which is independent on time $t$. All geometric quantities (e.g. the Chern
connection, the curvature, etc) related to $\hat{g}$ we will denote with the
hat symbol.

Now we are going to prove the following lemma, which is analogous to the Lemma
3.11 from Chow et al.[7].

###### Lemma 9

Let $(M,J,\hat{g})$ be a Hermitian manifold and $K$ be a compact subset of
$M$. Assume that $\\{g_{m}(t)\\}$ be a sequence of solutions of the flow
$\partial_{t}g_{\bar{k}j}=-\tilde{R}_{\bar{k}j}-{1\over
2}T_{\bar{k}pq}\bar{T}_{j}^{pq}$ on a neighborhood of
$K\times[\tau_{0};\tau_{1}]$. Fix some $t_{0}\in[\tau_{0};\tau_{1}]$. Assume
also that the following conditions holds:

(1) For our $t_{0}\in[\tau_{0};\tau_{1}]$, and for any $m$ we have the
estimate

$C^{-1}\hat{g}\leq g_{m}(t_{0})\leq C\hat{g}$

for the constant $C$ is independent of $k$.

(2) There are constants $C^{{}^{\prime}}_{p}>0$, independent of $k$, such that

$\displaystyle|\hat{\nabla}^{p}g_{m}(t_{0})|_{\hat{g}}\leq
C^{{}^{\prime}}_{p}$ (3.9)

for all $p\geq 1$.

(3) There are constants
$C^{{}^{\prime\prime}}_{p},C^{{}^{\prime\prime\prime}}_{p}>0$, independent of
$k$, such that the covariant derivatives of Chern curvature tensor $Rm$ are
bounded

$\displaystyle|({\nabla}^{m})^{p}Rm_{m}|_{g_{m}}\leq
C^{{}^{\prime\prime}}_{p}$ (3.10)

Moreover, all covariant derivatives of the torsion $T_{m}$ are bounded as well

$\displaystyle|{\nabla}^{p}T_{m}|_{g_{m}}\leq
C^{{}^{\prime\prime\prime}}_{p}.$ (3.11)

Here ${\nabla}^{m}$ denotes the Chern connection of $g_{m}(t)$.

Then the following holds

(1) There are constant $N$ independent of $m$, such that

$\displaystyle N^{-1}\hat{g}\leq g_{m}(t)\leq N\hat{g}$ (3.12)

(2) There are constants $N^{{}^{\prime}}_{p}>0$, independent of $k$, such that

$\displaystyle|\hat{\nabla}^{p}g_{m}(t)|_{\hat{g}}\leq N^{{}^{\prime}}_{p}$
(3.13)

for all $p\geq 1$.

(3) There are constants $N^{{}^{\prime\prime}}_{p,q}>0$, such that

$\displaystyle|\frac{\partial^{q}}{\partial
t^{q}}\hat{\nabla}^{p}g_{m}(t)|_{\hat{g}}\leq N^{{}^{\prime\prime}}_{p,q}$
(3.14)

Proof. (1) For any vector $V$ and for any solution $g$ of the Type IIB flow we
have the following identity

$\displaystyle\partial_{t}g(V,V)=-\tilde{R}_{j\bar{k}}V^{j}\overline{V}^{k}-{1\over
2}(T\circ\bar{T})_{j\bar{k}}V^{j}\overline{V}^{k}.$ (3.15)

From the definition we see that
$|\tilde{R}_{j\bar{k}}V^{j}\overline{V}^{k}|\leq\sqrt{n}|Rm|g(V,V)$, and the
norm of $T\circ\bar{T}$ is bounded.

Hence, by the standard trick, we can write the following:

$\displaystyle C|t_{1}-t_{0}|\geq\int_{t_{0}}^{t_{1}}|\partial_{t}\,{\rm
log}\,(g(V,V))|dt\geq|\,{\rm log}\,\frac{g(t_{1})(V,V)}{g(t_{0})(V,V)}|$
(3.16)

for some constant $C$, which depends on the dimension, and the bounds for $T$
and $Rm$. Now we can take the exponent and obtain the desired bound with
$N=e^{C(t_{1}-t_{0})}$

(2) Throughout this section ${\nabla}^{m}$ and $\Gamma_{m}$ denote the Chern
connection with respect to $g_{m}$.

We want to obtain bounds for $|\hat{{\nabla}}^{p}g_{m}(t)|_{\hat{g}}$ for any
$p$. First of all (see [19] for details),
$h^{-1}_{m}\hat{{\nabla}}h_{m}={\nabla}^{m}h_{m}h^{-1}_{m}={\nabla}^{m}-\hat{{\nabla}}=g^{-1}\hat{{\nabla}}g$,
where $h_{m}$ is a relative endomorphism
$(h_{m})^{a}_{b}=\hat{g}^{a\bar{s}}(g_{m})_{b\bar{s}}$. Next, since we proved
the equivalence for $\hat{g}$ and $g_{m}$, the norms on tensors, induced by
these metrics are equivalent (see [Lemma 3.13][7] for the proof).

We have the following identity:

$\displaystyle\partial_{t}\hat{{\nabla}}g_{m}=\hat{{\nabla}}\dot{g}=-\hat{{\nabla}}(\tilde{Ric}_{m}+(T\circ\bar{T})_{m})=-{\nabla}(\tilde{Ric}_{m}+(T\circ\bar{T})_{m})+$
$\displaystyle({\nabla}-\hat{{\nabla}})(\tilde{Ric}_{m}+(T\circ\bar{T})_{m})$
(3.17)

The first summand is bounded by our assumptions. The second summand is bounded
by constant times $|\hat{{\nabla}}^{p}g_{m}(t)|_{\hat{g}}$. Altogether, we
obtain the inequality

$\displaystyle|\hat{{\nabla}}^{p}g_{m}(t)|_{\hat{g}}\leq
C(1+|\hat{{\nabla}}^{p}g_{m}(t)|_{\hat{g}}),$ (3.18)

where $C$ is the maximum of bounds for $\tilde{Ric}_{m}+(T\circ\bar{T})_{m}$
and for ${\nabla}^{m}(\tilde{Ric}_{m}+(T\circ\bar{T})_{m})$.

For $p>1$ we proceed by induction. First of all, for any $p\in\bf N$, by the
same way as in [3, Lemma 8.6], one can show that

$\displaystyle\partial_{t}\hat{\nabla}^{p}g_{m}=\sum_{k_{1}+\dots+k_{q}=p}({\nabla}^{m})^{k_{1}}(Rm_{m}+(T*\bar{T})_{m})*\hat{{\nabla}}^{k_{2}}g_{m}*\dots*\hat{{\nabla}}^{k_{q}}g_{m}.$
(3.19)

Suppose we have the estimate for $p-1$. Then, taking the time derivative of
$|{\nabla}^{p}g_{m}|_{\hat{g}}$ and incorporate bounds for the curvature, the
torsion and for the lower derivatives, we obtain:

$\displaystyle\partial_{t}|{\nabla}^{p}g_{m}|_{\hat{g}}\leq
C(1+|{\nabla}^{p}g_{m}|_{\hat{g}}),$ (3.20)

where $C>0$ depends only on bounds for the curvature, the torsion and for the
lower derivatives. Hence,

$\displaystyle\,{\rm log}\,(|{\nabla}^{p}g_{m}|_{\hat{g}}+1)\leq Ct$ (3.21)

for some constant $D$. Now we take exponent and obtain the desired bound.

(3) We have

$\displaystyle|\partial^{q}_{t}\hat{{\nabla}}^{p}g_{m}(t)|_{\hat{g}}=|\hat{{\nabla}}^{p}\partial^{q}_{t}g_{m}(t)|_{\hat{g}}=|\hat{{\nabla}}^{p}\partial^{q-1}_{t}\dot{g}_{m}(t)|_{\hat{g}}.$
(3.22)

The Laplacian formulation of the Type IIB flow (see 1.2) allows us write time
derivatives of the metric $g_{m}$ via the spatial derivatives. Iterating this
process, we are able to estimate all mixed derivatives via spatial
derivatives. However, we proved before the estimates for spatial derivatives.
Q.E.D.

We can derive a useful consequence of this lemma.

###### Lemma 10

Let $(M,\hat{g},J,\Omega)$ be a Hermitian Calabi-Yau manifold and $K\subset M$
is compact. Assume there are complex structures $\\{J_{j}\\}$ and holomorphic
forms $\\{\Omega_{j}\\}$ on $M$, which converge in $C^{\infty}$ to $J$ and
$\Omega$ respectfully. Suppose $g_{j}(t)$ are the solutions of the Type IIB
flow w.r.t. $J_{j}$ and $\Omega_{j}$ on $K\times[\tau_{0};\tau_{1}]$. Suppose
all $g_{j}(t)$ satisfy the assumptions of Lemma 9. Then the results of Lemma 9
also holds for $g_{j}(t)$, i.e. there are constants $N,N_{l,m}$, such that
$|\partial^{l}_{t}\hat{{\nabla}}^{m}g_{j}(t)|_{\hat{g}}\leq N_{l,m}$ and
$N^{-1}\hat{g}\leq g_{j}\leq N\hat{g}$.

The difference between this Lemma and Lemma 9 is that here, the metrics
$g_{j}(t)$ are Hermitian with respect to the different complex structures on
$K$.

Proof. Define the sequence of the metrics $\hat{g}_{j}$ by the formula
$\hat{g}_{j}:={1\over 2}(\hat{g}(X,Y)+\hat{g}(J_{j}X,J_{j}Y))$. The metrics
$\hat{g}_{j}$ converge to $\hat{g}$ in $C^{\infty}$ because $J_{j}$ converge
to $J$ in $C^{\infty}$. Without loss of generality, we can assume that for any
$j$ there is a constant $P$ independent on $j$, such that
$P^{-1}\hat{g}\leq\hat{g}_{j}\leq P\hat{g}$. Applying the estimates from Part
(1) of the previous lemma, we obtain the uniform equivalence between
$g_{j}(t)$ and $\hat{g}$.

Denote by $\hat{\Gamma}_{j}$ and $\hat{\Gamma}$ the Chern connections of
$(\hat{g}_{j},J_{j})$ and $(\hat{g},J)$ respectfully. The difference
$|\hat{\Gamma}_{j}-\hat{\Gamma}|_{\hat{g}}$ between Chern connection
$\hat{\Gamma}_{j}$ of $(\hat{g}_{j},J_{j})$ and $\hat{\Gamma}$ of the pair
$(\hat{g},J)$ tends to zero as $j\rightarrow\infty$. Let
$\hat{S}_{j}(X,Y,Z):=d\hat{\omega}_{j}(J_{j}X,Y,Z)$ and $\hat{S}$ has the same
meaning for $(\hat{g},J)$. The difference of the Chern connections can be
written as follows

$\displaystyle\hat{\Gamma}_{j}-\hat{\Gamma}=({\nabla}^{\hat{g}_{j}}-{\nabla}^{\hat{g}})+(\hat{g}^{-1}_{j}S_{j}-\hat{g}^{-1}\hat{S}).$
(3.23)

The quantity $|{\nabla}^{\hat{g}_{j}}-{\nabla}^{\hat{g}}|_{\hat{g}}$ tends to
zero as $j$ tends to infinity due to $C^{\infty}$ convergence of $\hat{g}_{j}$
and $J_{j}$. The norm of the second summand is again tends to zero by the same
reason.

The same is true for
$|\hat{{\nabla}}^{l}(\hat{\Gamma}_{j}-\hat{\Gamma})|_{\hat{g}}$ for any
$l\in\bf N$. Hence
$|\hat{{\nabla}}^{l}(\hat{\Gamma}_{j}-\hat{\Gamma})|_{\hat{g}}$ is bounded by
a constant, independent on $j$. Now we apply the previous lemma to $g_{j}(t)$
and $\hat{g}_{j}$, and use the fact that
$|\hat{{\nabla}}^{l}(\Gamma_{j}-\hat{\Gamma})|_{\hat{g}}\leq|\hat{{\nabla}}^{l}(\Gamma_{j}-\hat{\Gamma}_{j})|_{\hat{g}}+|\hat{{\nabla}}^{l}(\hat{\Gamma}_{j}-\hat{\Gamma})|_{\hat{g}}$.
Hence, we obtain bounds for $|\hat{{\nabla}}^{l}g_{j}(t)|_{\hat{g}}$ and
$|\partial^{l}\hat{{\nabla}}^{m}g_{j}(t)|_{\hat{g}}$ with respect to
$\hat{g}$. Q.E.D.

Finally, we need to study the evolution of $\,{\rm log}\,\|\Omega\|^{2}$ along
the flow.

###### Lemma 11

Let $(M,g(t),J,\Omega)$ be a compact conformally balanced Calabi-Yau manifold,
and $g(t)$ evolves according to the Type IIB flow:
$\partial_{t}g_{\bar{k}j}=-\tilde{R}_{\bar{k}j}-{1\over
2}T_{\bar{k}pq}\bar{T}_{j}^{pq}$. Assume there is a constant $C>0$, such that
$|Rm|_{g}+|{\nabla}T|_{g}+|T|^{2}_{g}\leq C$. Then there are constants
$C_{1},C_{2}$ and $C_{3}$ such that the following holds:

$\displaystyle C_{1}\leq\,{\rm log}\,\|\Omega\|^{2}_{g}(t)\leq C_{2}t+C_{3}.$
(3.24)

Proof. By a straightforward computation, we can show that
$\tilde{R}=g^{j\bar{k}}\tilde{R}_{\bar{k}j}=g^{j\bar{k}}\partial_{\bar{k}}\partial_{j}\,{\rm
log}\,\|\Omega\|^{2}=\Delta\,{\rm log}\,\|\Omega\|^{2}$ and,

$\displaystyle(\partial_{t}-\Delta)\,{\rm log}\,\|\Omega\|^{2}={1\over
2}|T|^{2}_{g}\geq 0$ (3.25)

(see [19, Section 3.1]). By the maximum principle, $\,{\rm
log}\,\|\Omega\|^{2}$ achieves minimum at $t=0$, hence it bounded below. On
the other hand, by the assumption on $Rm$ and $T$, there is constant $C_{2}$,
such that the following inequality holds:

$\displaystyle\partial_{t}\,{\rm log}\,\|\Omega|\|^{2}=\tilde{R}+{1\over
2}|T|^{2}_{g}\leq C_{2}.$ (3.26)

Now we can integrate it along $t$, and obtain the bound from above. Q.E.D.

###  3.2 Proof of Theorem 2

We proceed as in the proof of Hamilton’s compactness theorem (see [16, 22]).

By Theorem 1, Shi-type estimates from [26], [18], and by Lemma 11, we can
guarantee that the family
$\\{(M_{j},g_{j}(\tau_{0}),J_{j},\Omega_{j},x_{j})\\}_{j=1}^{\infty}$ converge
to the manifold $(M_{\infty},\hat{g},J_{\infty},\Omega_{\infty},x_{\infty})$.
Note that Shi-type estimates, together with Lemma 4 imply the bounds on the
covariant derivatives of complex structures. Moreover, since the bounds on
torsion does not depend on $g_{j}$, the bounds on the covariant derivatives of
$J_{j}$ do not depend on $g_{j}$ either.

Now we apply Lemma 9 and Lemma 10 for $\hat{g}$ and the sequence of metrics
$\tilde{g}_{j}:=\Phi^{*}_{j}g_{j}(t)$ on a compact set $K\subset U_{j}\subset
M_{\infty}$, and obtain uniform $C^{\infty}$ control on these metrics. The
Shi-type estimates from [26] ensure that the conditions of Lemma 9 are
satisfied. Hence, we can consider the sequence $\tilde{g}_{j}(t)+dt^{2}$,
which will converge on $K\times[\tau_{0};\tau_{1}]$ to some metric
$g_{\infty}(t)+dt^{2}$ (by the Arzela-Ascoli theorem). The limit
$g_{\infty}(t)$ will be the solution to the Type IIB flow with the initial
metric $\hat{g}$, i.e. $g_{\infty}(\tau_{0})=\hat{g}$ with respect to the
complex structure $J_{\infty}$ and the holomorphic volume form
$\Omega_{\infty}$. This follows from the $C^{\infty}$ convergence of metrics
to the limit metric. By the diagonalization argument, there is a subsequence,
which converges on the whole $M_{\infty}\times[\tau_{0};\tau_{1}]$ Q.E.D.

##  4 Singularity models for the Type IIB flow

Let $\omega(t)$ be a solution for the Type IIB flow on a complex manifold on
some time interval $[0,T)$. For each $t\in[0,T)$, define the function $f(t)$
by

$\displaystyle f(t):=\sup_{M}(|Rm|^{2}+|\nabla T|^{2}+|T|^{4}).$ (4.1)

######  Definition 3 (Singularity types)

Let $(M,g(t),J,\Omega)$ be a solution of the Type IIB flow on a compact
Hermitian Calabi-Yau manifold $M$. Let $f(t)=\sup_{M}(|Rm|^{2}+|\nabla
T|^{2}+|T|^{4})$ 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] $\sup_{[0;T]}tf(t)=C<\infty$, $T=\infty$

We observe that we could also try to define singularities in terms of
$\|\Omega\|^{2}$, as suggested by examples (see [31, Section 2.3.2] for
instance). However, as Lemma 11 shows, the bounds on $f(t)$ imply bounds on
$\|\Omega\|$.

###### Definition 4

We say that $(M,g(t),J,\Omega)$ satisfies the injectivity radius estimate if
the injectivity radius satisfy the following inequality

$\displaystyle\mathrm{inj}(M,g(t))\geq{c\over\sqrt{f(t)}}$ (4.2)

for some positive constant $c$.

We give now the proof of Theorem 3. Denote $F(x,t):=\sqrt{|Rm|^{2}+|\nabla
T|^{2}+|T|^{4}}$. Recall that $f(t)=\sup_{M}\sqrt{|Rm|^{2}+|\nabla
T|^{2}+|T|^{4}}=\sup_{x\in M}F(x,t)$. Throughout the proof of the theorem we
will use the sequence of rescaled Hermitian metrics $g_{j}(t)$, defined as
follows:

$\displaystyle g_{j}(t):=C_{j}g(t_{j}+C_{j}^{-1}t),$ (4.3)

where the constants $C_{j}$ will be defined later. We also rescale the
holomorphic volume form

$\displaystyle\Omega_{j}:=C_{j}^{-n\over 2}\Omega.$ (4.4)

The case of Type I singularity

Let $C:=\sup_{t}f(t)$. . Pick a sequence $(x_{j},t_{j})\in M\times[0;T]$, such
that $F(x_{j},t_{j})\rightarrow C$. Let $C_{j}:=F(x_{j},t_{j})$, and
$(T-t_{j})F(x_{j},t_{j})\geq c>0$. Consider the rescaled Hermitian metrics
$g_{j}(t)$ and forms $\Omega_{j}$. These metrics are again the solutions of
the Type IIB flow on the interval $(-C_{j}t_{j};C_{j}(T-t_{j}))$. The
curvature and torsion tensors of the metrics $g_{j}$ satisfy the following
bound:

$\displaystyle\sqrt{|Rm|_{g_{j}}^{2}+|\nabla
T|_{g_{j}}^{2}+|T|_{g_{j}}^{4}}\leq C_{j}^{-1}{C\over
T-(t_{j}+C_{j}^{-1}t)}={C\over C_{j}(T-t_{j})-t}\leq{C\over c-t}.$ (4.5)

By the compactness theorem for Type IIB flow, the sequence
$(M,g_{j}(t),J,\Omega_{j},x_{j})$ converge to the limit manifold
$(M_{\infty},g_{\infty}(t),J,\Omega,\infty)$ with
$|Rm|_{g_{\infty}}^{2}+|\nabla
T|_{g_{\infty}}^{2}+|T|_{g_{\infty}}^{4}\leq{C\over c-t}$. The case of Type I
singularity is proved.

The case of Type IIa singularity

Pick a monotone increasing sequence ${T_{j}}$, such that $T_{j}\rightarrow T$
and let $C_{j}=\sup_{t<T_{j}}f(t)$. Next, we pick a sequence of points
$(x_{j},t_{j})$, such that $C^{2}_{j}=(T-t_{j})F(x_{j},t_{j})$. Now we take a
sequence of rescaled metrics $g_{j}(t):=C_{j}g(t_{j}+C_{j}^{-1}t)$ and the
sequence of pointed manifolds $(M,g_{j}(t),J,\Omega_{j},x_{j})$ again converge
to the limit manifold $(M,g_{\infty}(t),J,\Omega,x_{\infty})$. The limit
metric $g_{\infty}(t)$ is the solution of the Type IIB flow on the whole line
$(-\infty,\infty)$, as $C_{j}(T-t_{j})\rightarrow\infty$. The bound for the
curvature is the following:

$\displaystyle\sqrt{|Rm|_{g_{j}}^{2}+|\nabla
T|_{g_{j}}^{2}+|T|_{g_{j}}^{4}}\leq{C_{j}\over C_{j}(T-t_{j})}\rightarrow 1.$
(4.6)

Hence, $f(t)\leq 1$.

The case of Type IIb singularity Pick a monotone sequence
$T_{j}\rightarrow\infty$ and a sequence of points $(x_{j},t_{j}),t_{j}\leq
T_{j}$, such that: $t_{j}(T-t_{j})F(x_{j},t_{j})=\sup t(T_{j}-t)F(x,t)$. Let
$C_{j}$ be as above, and $A_{j}=t_{j}C_{j}$

$\displaystyle\sqrt{|Rm|_{g_{j}}^{2}+|\nabla
T|_{g_{j}}^{2}+|T|_{g_{j}}^{4}}\leq C_{j}^{-1}{A_{j}\over
t_{j}+C_{j}^{-1}t}={A_{j}\over A_{j}+t}\rightarrow 1$ (4.7)

We again apply the compactness theorem and have a limit with the desirable
property.

The case of Type III singularity

Let $A=\lim\sup tf(t)<+\infty$. This number is positive. Indeed, if
$tf(t)<\varepsilon$, then the diameter $L$ satisfies the following inequality:

$\displaystyle{\partial L\over\partial t}\leq{C\varepsilon L\over t},$ (4.8)

because the $f(t)$ is bounded, hence $\tilde{R}_{\bar{k}j}+{1\over
2}T_{\bar{k}pq}\bar{T}_{j}^{pq}$ is bounded. Hence, $\,{\rm log}\,L\leq
C\varepsilon\,{\rm log}\,t$. Hence, $L\leq t^{C\varepsilon}$. Take
$\varepsilon$ smaller than ${1\over 2C}$. This choice implies
$L^{2}f(t)\rightarrow 0$. On the other hand,
$L\geq\mathrm{inj}(M,g(t))\geq{C\over\sqrt{f(t)}}$ by the assumption of the
theorem. This inequality is equivalent to the inequality $L^{2}f(t)\geq C$.
This is a contradiction, hence $A>0$.

Let $C=\sup f(t)$, and $C_{j},(x_{j},t_{j})$ as above. The metrics $g_{j}(t)$
are solutions to the Type IIB flow on the interval $[-C_{j}t_{j};+\infty)$
and, by our assumptions $C_{j}t_{j}\rightarrow A$. We have the following
inequality:

$\displaystyle\sqrt{|Rm|_{g_{j}}^{2}+|\nabla
T|_{g_{j}}^{2}+|T|_{g_{j}}^{4}}\leq C^{-1}_{j}{A\over
t_{j}+C_{j}^{-1}t}={A\over C_{j}t_{j}+t}\rightarrow{A\over A+t}.$ (4.9)

Q.E.D.

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

also

Laboratory of Algebraic Geometry, Faculty of Mathematics, HSE University, 6
Usacheva Str. Moscow, Russia

<EMAIL_ADDRESS>


# Uniqueness theorems for almost periodic objects

S.Yu. Favorov, O.I.Udodova  Sergii Favorov,
iii Karazin’s Kharkiv National University
iii Svobody sq., 4,
iii 61022, Kharkiv, Ukraine<EMAIL_ADDRESS>Olga Udodova,
iii ???
iii ?????,
iii<EMAIL_ADDRESS>
> Abstract. Uniqueness theorems are considered for various types of almost
> periodic objects: functions, measures, distributions, multisets, holomorphic
> and meromorphic functions.
>
> AMS Mathematics Subject Classification: 42A75, 32A60, 32A22
>
> Keywords: almost periodic function, almost periodic measure, almost periodic
> meromorphic function

It easily follows from the definition of almost periodic functions that if the
values of two such functions converge at infinity, then these almost periodic
functions coincide. This effect also manifested itself in [7] for zeros of
holomorphic almost periodic functions, and then in [1] and [2] for Fourier
quasicrystals and some classes of transformable measures on LCA-groups.

In this note, we discuss this effect in detail, show how it can be
strengthened, what form it takes for other almost periodic objects - almost
periodic distributions, almost periodic measures, almost periodic multisets,
$a$ -points of holomorphic and meromorphic almost periodic functions.

##  1\. Almost periodic functions

We start with the simplest almost periodic object - uniformly almost periodic
functions on a finite-dimensional space and on tube sets. The definitions
introduced in this section will also be used in subsequent sections.

Let $B_{\mathbb{C}}(z^{0},R)$ be the open ball
$\\{z\in{\mathbb{C}}:\,|z-z^{0}|<R\\}$ in the space ${\mathbb{C}}^{d}$, and
$B_{\mathbb{R}}(x^{0},R)$ be the open ball
$\\{x\in{\mathbb{R}}:\,|x-x^{0}|<R\\}$ in the space ${\mathbb{R}}^{d}$. The
tube set $T_{K}\subset{\mathbb{C}}^{d}$ means the set of the form

$T_{K}=\left\\{{z=x+iy\in{\mathbb{C}}^{d}:x\in{\mathbb{R}}^{d},\;y\in
K}\right\\},$

where $K$ is a compact subset of ${\mathbb{R}}^{d}$. Clearly,
${\mathbb{R}}^{d}=T_{\\{0\\}}$. Then $T_{\Omega}$ means the domain

$T_{\Omega}=\left\\{{z=x+iy:x\in{\mathbb{R}}^{d},\;y\in\Omega}\right\\}$

where $\Omega$ is a domain in ${\mathbb{R}}^{d}$, maybe
$\Omega={\mathbb{R}}^{d}$. The set $E$ is relatively dense in
${\mathbb{R}}^{d}$, if there exists $R<\infty$ such that each ball
$B_{\mathbb{R}}(x,R)$ intersects with $E$. By $\\#A$ we denote the number of
elements of the finite set $A$.

######  Definition 1.

A continuous complex-valued function $f(z)$ on a tube set $T_{K}$ is called
almost periodic if for every $\varepsilon>0$ the set of its
$\varepsilon$-almost periods

$E_{\varepsilon,K}=E_{\varepsilon,K}(f)=\\{\tau\in{\mathbb{R}}^{d}:\sup_{z\in
T_{K}}|f(z+\tau)-f(z)|<\varepsilon\\}$

is relatively dense in ${\mathbb{R}}^{d}$.

It easily follows from this definition that almost periodic functions on
$T_{K}$ are bounded. Less obvious is the following statement:

######  Theorem 1.

( [9]) A continuous function $f(z)$ on $T_{K}$ is almost periodic iff for any
sequence $\\{{x_{n}}\\}\subset{\mathbb{R}}^{d}$ there is a subsequence
$\\{{x_{n^{\prime}}}\\}$ such that the functions
$f_{n^{\prime}}(z)=f(z+x_{n^{\prime}})$ form the fundamental sequence with
respect to the uniform convergence on $T_{K}$.

######  Definition 2.

A function $f(z)$ on a tube domain $T_{\Omega}$ is called almost periodic if
for every compact set $K\subset\Omega$ its restriction to $T_{K}$ is almost
periodic.

######  Theorem 2.

([9]) A continuous function $f(z)$ is almost periodic on a tube domain
$T_{\Omega}$ iff for any sequence $\\{{x_{n}}\\}\subset{\mathbb{R}}^{d}$ there
is a subsequence $\\{{x_{n^{\prime}}}\\}$ such that the functions
$f_{n^{\prime}}(z)=f(z+x_{n^{\prime}})$ form the fundamental sequence with
respect to the uniform convergence on $T_{K}$ for every $K\subset\Omega$.

Remark. All these definitions and theorems carry over practically unchanged to
the case of mappings $F:T_{K}\to{\mathbb{C}}^{N}$ or
$F:T_{\Omega}\to{\mathbb{C}}^{N}$. Since component-wise convergence is
equivalent to the convergence of mappings, we get that the vector function
$F(z)=(f_{1}(z),\dots,f_{N}(z))$ is almost periodic if and only if its
components are almost periodic. Therefore for any $\varepsilon>0$ the set
$E_{\varepsilon,K}$ of common almost periods of functions $f_{1},\dots,f_{N}$
is also relatively dense. In particular, this implies that a sum or a product
of any finite number of almost periodic functions is also an almost periodic
function.

In the rest of the article, only the cases of functions and sets on
${\mathbb{R}}^{d}$ or on $T_{\Omega}\subset C^{d}$ will be considered.

Now we give the basic definition of our article.

######  Definition 3.

We shall say that functions $f,g$ on ${\mathbb{R}}^{d}$ converge weakly at
infinity, if

$\lim_{x\to\infty,x\in G}|f(x)-g(x)|=0,$

where $G\subset{\mathbb{R}}^{d}$ is a set with the property

(1) $G\supset\bigcup_{k=1}^{\infty}B_{\mathbb{R}}(x_{k},R_{k})\quad\mbox{for
some sequence of balls}\quad B_{\mathbb{R}}(x_{k},R_{k}),\quad
R_{k}\to\infty.$

######  Definition 4.

Functions $f,g$ on $T_{\Omega}$ converge weakly at infinity if for each fixed
$y^{0}\in\Omega$ the functions $f(x+iy^{0}),\ g(x+iy^{0})$ of the variable
$x\in{\mathbb{R}}^{d}$ converge weakly at infinity.

######  Theorem 3.

If almost periodic functions $f,g$ on ${\mathbb{R}}^{d}$ or $T_{\Omega}$
converge weakly at infinity, then they coincide identically.

Proof. Let $f,g$ be almost periodic functions on ${\mathbb{R}}^{d}$. Fix
$x^{0}\in{\mathbb{R}}^{d}$ and $\varepsilon>0$. Let $E_{\varepsilon}$ be the
set of $\varepsilon$-almost periods of the almost periodic function $h=f-g$.
Taking into account (1) and relative density of $E_{\varepsilon}$, we get that
for large $n$ there is a point $\tau_{n}\in E_{\varepsilon}\cap
B_{\mathbb{R}}(x_{n}-x^{0},R_{n})$. Hence $x_{0}+\tau_{n}\in
B_{\mathbb{R}}(x_{n},R_{n})$ and $|h(x_{0}+\tau_{n})|<\varepsilon$. Also,
$|h(x^{0}+\tau_{n})-h(x^{0})|<\varepsilon$, therefore,
$|h(x^{0})|<2\varepsilon$. The choice of $\varepsilon$ and $x^{0}$ was
arbitrary, therefore $h(x)\equiv 0$. In the case of functions on $T_{\Omega}$
we take $z^{0}=x^{0}+iy^{0}\in T_{\Omega}$ and a compact set $K\subset\Omega$
such that $y^{0}\in K$, then replace $E_{\varepsilon}$ by $E_{\varepsilon,K}$
and $x^{0}$ by $z^{0}$. Theorem is proved.

##  2\. Almost periodic distributions, measures, multisets

Let $D({\mathbb{R}}^{d})$ be the space of test functions on
${\mathbb{R}}^{d}$, i.e., $C^{\infty}$-functions with compact supports,
equipped with the topology of uniform convergence of derivatives of all orders
of functions from $D({\mathbb{R}}^{d})$, provided that all their supports are
subsets of some fixed compact from $R^{d}$, let $D^{*}({\mathbb{R}}^{d})$ be
the space of distributions on ${\mathbb{R}}^{d}$, that is, the set of
continuous linear functionals on $D({\mathbb{R}}^{d})$. The distribution space
$D^{*}(T_{\Omega})$ is similarly defined as continuous linear functionals on
the space $D(T_{\Omega})$, consisting of $C^{\infty}$-functions with compact
support in $T_{\Omega}$.

######  Definition 5.

A distribution $f\in D^{*}({\mathbb{R}}^{d})$ is called almost periodic, if
for any test-function $\varphi\in D({\mathbb{R}}^{d})$ the function
$(f,\varphi(\cdot-t))$ is almost periodic in the variable
$t\in{\mathbb{R}}^{d}$.

######  Definition 6.

A distribution $f\in D^{*}(T_{\Omega})$ is called almost periodic, if for any
test-function $\varphi\in D(T_{\Omega})$ the function $(f,\varphi(\cdot-z))$
is almost periodic in the variable $z\in T_{\omega}$. Here $\omega$ is the
open subset of $\Omega$ such that for all $z\in T_{\omega}$ the condition
$\zeta-z\in\operatorname{supp}\varphi$ implies $\zeta\in T_{\Omega}$.

A particular case of distributions are complex-valued measures. Such measures
will be denoted by $\mu$, and the measure, which is the variation of $\mu$, by
$|\mu|$. A measure $\mu$ on ${\mathbb{R}}^{d}$ is called translation bounded
if

$\sup_{x\in{\mathbb{R}}^{d}}|\mu|(B_{\mathbb{R}}(x,1))<\infty.$

Similarly, a measure on $T_{\Omega}$ is called translation bounded if for any
compact $K\subset\Omega$

$\sup_{x\in{\mathbb{R}}^{d}}|\mu|(B_{\mathbb{R}}(x,1)\times K)\leq C,$

where the constant $C$ depends on $K$. Note that every nonnegative almost
periodic measure is translation bounded. To prove this we should take a
nonnegative test function $\varphi(z)\in D(T_{\Omega})$ such that
$\varphi(z)=1$ on $B_{\mathbb{R}}(0,1)\times K$, where $K$ is a compact subset
of $\Omega$ (for the case $T_{{\mathbb{R}}^{d}}$ we should take nonnegative
$\varphi\in D(R^{d}),\ \varphi(x)=1$ on $B_{\mathbb{R}}(0,1)$). The function

(2) $\int{\varphi(z-t)\mu(dz)}$

is almost periodic, hence it is bounded in $t\in{\mathbb{R}}^{d}$. On the
other hand, for all $t\in{\mathbb{R}}^{d}$

$\mu(B_{\mathbb{R}}(t,1)\times K)\leq\int\varphi(z-t)\mu(dz).$

If a measure $\mu\in D^{*}(T_{\Omega})$ is translation bounded, then we can
use any continuous function with compact support as test functions in
Definition 5. This follows from the fact that any such a function can be
uniformly approximated by $C^{\infty}$-functions supported on a fixed compact
set. On the other hand, there are signed almost periodic measures for which
(2) are not almost periodic for an appropriate continuous compactly supported
$\varphi$ ([4]). Note that if (2) is bounded for all continuous $\varphi$ with
compact support, then the complex measure $\mu$ is translation bounded ([9]).

Let $D=\\{a,p\\},p\in{\mathbb{N}}$, be a discrete multiset in $T_{\Omega}$ or
in ${\mathbb{R}}^{d}$. It can be identified with a sequence $\\{a_{n}\\}$
without condensation points in $T_{\Omega}$ (or in ${\mathbb{R}}^{d}$) such
that each point from $T_{\Omega}$ or in ${\mathbb{R}}^{d}$ can occur in this
sequence at most a finite number of times. In the case of
$T_{\Omega}\subset{\mathbb{C}}$ a discrete multiset is also called a divisor
(see [6]).

######  Definition 7.

([6]) A discrete multiset $D\subset{\mathbb{R}}^{d}$ is called almost periodic
if for all $\varepsilon>0$ there is a relatively dense set
$E_{\varepsilon}=E_{\varepsilon}(D)\subset{\mathbb{R}}^{d}$ such that a
bijection $\sigma:{\mathbb{N}}\to{\mathbb{N}}$ corresponds to any $\tau\in
E_{\varepsilon}$ with the property

$\sup_{n\in{\mathbb{N}}}|a_{n}-\tau-a_{\sigma(n)}|<\varepsilon.$

A discrete multiset $D\subset T_{\Omega}$ is called almost periodic if for all
$\varepsilon>0$ and compact sets $K\subset\Omega$ there is a relatively dense
set $E_{\varepsilon,K}=E_{\varepsilon,K}(D)\subset{\mathbb{R}}^{d}$ such that
a bijection $\sigma:{\mathbb{N}}\to{\mathbb{N}}$ corresponds to any $\tau\in
E_{\varepsilon,K}$ with the property

$\sup|a_{n}-\tau-a_{\sigma(n)}|<\varepsilon,$

where supremum is taken over all $n\in{\mathbb{N}}$ such that either $a_{n}$,
or $a_{\sigma(n)}$ belongs to $T_{K}$.

We also need a notion of bounded density. For a discrete multiset
$D\subset{\mathbb{R}}^{d}$, $D=\\{a_{n}\\}$, this means that

$\sup_{x\in{\mathbb{R}}^{d}}\\#\\{n:\,a_{n}\in B_{\mathbb{R}}(x,1)\\}<\infty.$

Also, $D\subset T_{\Omega}$ is of bounded density if for every compact
$K\subset\Omega$

(3) $N(K):=\sup_{x\in{\mathbb{R}}^{d}}\\#\\{n:\,a_{n}\in
B_{\mathbb{R}}(x,1)\times K\\}<\infty.$

It is easy to check that each almost periodic multiset is of bounded density.
For $T_{\Omega}\subset{\mathbb{C}}^{d}$ the proof can be found in [6]. For
convenience, we present it here. The proof for $D\subset{\mathbb{R}}^{d}$
differs only in the corresponding simplifications.

Set $\eta=\frac{1}{2}\operatorname{dist}(K,\partial\Omega)$ (in the case
$\Omega={\mathbb{R}}^{d}$ set $\eta=\frac{1}{2}$). Take $R<\infty$ such that
every ball $B_{\mathbb{R}}(x,R)$ intersects with $E_{\eta,K}$. Fix $\tau\in
B_{\mathbb{R}}(x,R)\cap E_{\eta,K}$ and take the bijection
$\sigma:\,{\mathbb{N}}\to{\mathbb{N}}$ such that for each $a_{n}\in T_{K}$

$|a_{n}-\tau-a_{\sigma(n)}|<\eta.$

For $a_{n}\in B_{\mathbb{R}}(x,1)\times K$ we get

$|{\textsf{Re}}\ a_{\sigma(n)}|\leq|{\textsf{Re}}\
a_{\sigma(n)}-{\textsf{Re}}\ a_{n}+\tau|+|{\textsf{Re}}\
a_{n}-x|+|x-\tau|<\eta+1+R,$ ${\textsf{Im}}\ a_{\sigma(n)}={\textsf{Im}}\
a_{n}+({\textsf{Im}}\ a_{\sigma(n)}-{\textsf{Im}}\ a_{n}).$

Since ${\textsf{Im}}\ a_{n}\in K$ and $|{\textsf{Im}}\
a_{\sigma(n)}-{\textsf{Im}}\ a_{n}|<\eta$, we get ${\textsf{Im}}\
a_{\sigma(n)}\in K_{1}$, where
$K_{1}=\\{y:\,\operatorname{dist}(y,K)\leq\eta\\}$. Thus,

$\\#\\{n:a_{n}\in B_{\mathbb{R}}(x,1)\times K\\}\leq\\#\\{n:a_{\sigma}(n)\in
B_{\mathbb{R}}(0,1+\eta+R)\times K_{1}\\},$

and we obtain (3).

Note that the measure

$\mu_{D}=\sum_{n}\delta_{a_{n}},$

corresponds to each discrete multiset $D=\\{a_{n}\\}$, where $\delta_{a_{n}}$
is the unit mass at the point $a_{n}$.

######  Theorem 4.

A discrete multiset $D$ is almost periodic iff the measure $\mu_{D}$ is almost
periodic.

For $D\subset{\mathbb{C}}$ this theorem was proved in [6], and for
$D\subset\leavevmode\nobreak\ {\mathbb{R}}^{d}$ in [3]. Here we give a new,
much simpler proof for $D\subset T_{\Omega}$. The proof for
$D\subset{\mathbb{R}}^{d}$ differs only in the corresponding simplifications.

Proof. Let a discrete multiset $D$ be almost periodic and $K\subset\Omega$ be
a compact set. Take a function $\varphi\in C^{\infty}(T_{\Omega})$ such that
$\operatorname{supp}\varphi\subset B_{\mathbb{R}}(0,1/2)\times K$. Let
$\varepsilon>0$ be arbitrary and
$\delta<(1/2)\operatorname{dist}(K,\partial\Omega)$ such that for
$|z-z^{\prime}|<\delta$

$|\varphi(z)-\varphi(z^{\prime})|<\frac{\varepsilon}{2N(K)},$

where $N(K)$ is defined in (3). Pick $\tau\in E_{\delta,K}(D)$ and the
corresponding bijection $\sigma$. We have

$\int\varphi(z-\tau)\mu_{D}(dz)-\int\varphi(z)\mu_{D}(dz)=\sum_{n}\varphi(a_{n}-\tau)-\sum_{n}\varphi(a_{n})=$
$=\sum_{n}[\varphi(a_{n}-\tau)-\varphi(a_{\sigma(n)})].$

The number of terms in the letter sum does not exceed $2N(K)$, moreover,
$|a_{n}-\tau-a_{\sigma(n)}|<\delta$, hence the difference between integrals
does not exceed $\varepsilon$. Therefore the points of the set
$E_{\delta,K}(D)$ are $\varepsilon$-almost periods of the function
$(\mu_{D}(\varphi(\cdot-t))$. This reasoning is valid for every $\varphi$ with
compact support, therefore the measure $\mu_{D}$ is almost periodic.

On the other hand, let $\mu_{D}$ be the almost periodic measure on
$T_{\Omega}$, which corresponds to a discrete multiset $D=\\{a_{n}\\}$. Fix a
compact set $K\subset\Omega$ and
$\varepsilon<\frac{1}{4}\min\\{1,\operatorname{dist}(K,\Omega)\\}$. Put

$\widetilde{K}=\\{y\in\Omega:\operatorname{dist}(y,K)\leq\varepsilon\\}.$

Choosing a sufficiently large $K$, we can assume that either $D\subset K$, or
$D\setminus T_{\tilde{K}}\neq\emptyset$. Since $\mu_{D}$ is almost periodic we
get that it is translation bounded, hence for some $N<\infty$

$\mu_{D}(B_{\mathbb{R}}(x,1)\times\widetilde{K})\leq N,\quad\forall
x\in{\mathbb{R}}^{d},$

therefore,

(4) $\\#\\{n:a_{n}\in B_{\mathbb{R}}(x,1)\times\widetilde{K}\\}\leq
N,\,\forall x\in{\mathbb{R}}^{d}.$

Set $\delta=\varepsilon/(4N+1)$. Let $A$ be any connected component of the set
$\bigcup\limits_{n}B_{\mathbb{C}}(a_{n},2\delta)$ such that $A\cap
T_{K}\neq\emptyset$. There exists $a_{n^{\prime}}\in A$ such that
$B_{\mathbb{C}}(a_{n^{\prime}},2\delta)\cap T_{K}\neq\emptyset$. If
$A\cap\partial B_{\mathbb{C}}(a_{n^{\prime}},\varepsilon)\neq\emptyset$, then
the connected set $A\cap B_{\mathbb{C}}(a_{n^{\prime}},\varepsilon)$ contains
at least $\varepsilon/(4\delta)>N$ points of $D$, which contradicts (4).
Hence,

$A\subset B_{\mathbb{C}}(a_{n^{\prime}},\varepsilon)\subset
B_{\mathbb{R}}({\textsf{Re}}\,a_{n^{\prime}},1)\times\widetilde{K}$

and, by (4), $\\#\\{n:a_{n}\in A\\}\leq N$.

By $\varphi(z)$ denote any $C^{\infty}$-function on ${\mathbb{C}}^{d}$ such
that

$0\leq\varphi(z)\leq
1,\quad\varphi(0)=1,\quad\operatorname{supp}\varphi\subset
B_{\mathbb{C}}(0,1),$

Let $\alpha=\int\varphi(z)\omega(dz)$, where $\omega$ is the Lebesgue measure
on ${\mathbb{C}}^{d}$. Put

$\Psi(z):=\int\varphi\left(\frac{z-w}{\delta}\right)\mu_{D}(dw)=\sum_{n}\varphi\left(\frac{z-a_{n}}{\delta}\right).$

Since

$\operatorname{dist}({\widetilde{K}},\partial\Omega)\geq\operatorname{dist}(K,\partial\Omega)-\varepsilon\geq\varepsilon,$

we see that $\Psi(z)$ is defined and almost periodic on $T_{\widetilde{K}}$.
Let $\tau$ be $\rho$-almost period of $\Psi(z)$ with
$\rho<\min\\{1;2^{-2d}\alpha/(N\omega_{2d})\\}$, where
$\omega_{2d}=\omega(B_{\mathbb{C}}(0,1))$. We have

(5) $|\Psi(z+\tau)-\Psi(z)|<\rho,\quad\forall z\in T_{\widetilde{K}}.$

On the other hand,

$\Psi(z)=0\quad\mbox{for}\quad
z\notin\cup_{n}B_{\mathbb{C}}(a_{n},\delta)\quad\mbox{and}\quad\Psi(z+\tau)=\Psi(a_{n})\geq
1\quad\mbox{for}\ z=a_{n}-\tau.$

Therefore the set $A\setminus\cup_{n}B_{\mathbb{C}}(a_{n},\delta)$ does not
contain any point $a_{n}-\tau$. If $A^{\prime}$ is another connected component
of the set $\cup_{n}B_{\mathbb{C}}(a_{n},2\delta)$, then for the same reason
$A^{\prime}\setminus\cup_{n}B_{\mathbb{C}}(a_{n},\delta)$ does not contain any
point $a_{n}-\tau$. Thus the set $A$ contains all balls
$B_{\mathbb{C}}(a_{n},\delta)$, for which $a_{n}\in A$ and all balls
$B_{\mathbb{C}}(a_{n}-\tau,\delta)$, for which $a_{n}-\tau\in A$, and does not
intersects with balls $B_{\mathbb{C}}(a_{n},\delta)$ with $a_{n}\notin A$ and
balls $B_{\mathbb{C}}(a_{n}-\tau,\delta)$ with $a_{n}-\tau\notin A$. We get

$\alpha\delta^{2d}\\#\\{n:a_{n}\in A\\}=\sum_{n:a_{n}\in
A}\int\varphi\left(\frac{z-a_{n}}{\delta}\right)\omega(dz)=\int_{A}\Psi(z)\omega(dz),$
$\alpha\delta^{2d}\\#\\{n:a_{n}-\tau\in A\\}=\sum_{n:a_{n}-\tau\in
A}\int\varphi\left(\frac{z+\tau-
a_{n}}{\delta}\right)\omega(dz)=\int_{A}\Psi(z+\tau)\omega(dz),$

Note that

$\int_{A}\omega(dz)\leq\sum_{a_{n}\in
A}\int_{B_{\mathbb{C}}(a_{n},2\delta)}\omega(dz)=N\omega_{2d}(2\delta)^{2d}.$

By (5),

$|\\#\\{n:a_{n}\in A\\}-\\#\\{n:a_{n}-\tau\in
A\\}|\leq\frac{\int_{A}|\Psi(z)-\Psi(z+\tau)|\omega(dz)}{\delta^{2d}\alpha}<\frac{\rho
N\omega_{2d}2^{2d}}{\alpha}<1.$

Therefore,

$\\#\\{n:a_{n}\in A\\}=\\#\\{n:a_{n}-\tau\in A\\},$

which allows to construct a bijection $\sigma$ between the sets
$\\{n:a_{n}-\tau\in A\\}$ and $\\{n\leavevmode\nobreak\ :\leavevmode\nobreak\
a_{n}\in\leavevmode\nobreak\ A\\}$.

This construction works for every connected component of the set
$\cup_{n}B_{\mathbb{C}}(a_{n},2\delta)$, hence there exists a bijection
$\sigma$ of a part $S_{1}$ of ${\mathbb{N}}$ to the part $S_{2}$ of
${\mathbb{N}}$. It follows from the inequality $\operatorname{diam}A\leq
2\varepsilon$ that

(6) $|a_{n}-\tau-a_{\sigma(n)}|<2\varepsilon.$

If $D\subset T_{K}$, we have $S_{1}=S_{2}={\mathbb{N}}$, and theorem is
proved. If $D\setminus T_{\widetilde{K}}\neq\emptyset$, we have only

$\\{n:a_{n}\in T_{K}\\}\subset S_{1}\cup S_{2}\subset\\{n:\leavevmode\nobreak\
a_{n}\in T_{\widetilde{K}}\\}.$

For $a\in D\setminus T_{\tilde{K}}$ put

$\eta<\frac{1}{2}\min\\{\operatorname{dist}({\textsf{Im}}\
a,\widetilde{K}),\operatorname{dist}\\{{\textsf{Im}}\ a,\partial\Omega\\}\\}$

and consider the function

$\Psi(z)=\int\varphi\left(\frac{w-z}{\eta}\right)\mu_{D}(dw)=\sum_{n:a_{n}\in
D}\varphi\left(\frac{a_{n}-z}{\eta}\right).$

In view of the choice of $\eta$, this function is well-defined and almost
periodic on $T_{\omega}$ with
$\omega=\\{y\in\Omega:\,\operatorname{dist}(y,\partial\Omega)>\eta\\}$.
Furthermore, $\Psi(a)\geq\varphi(0)=1$, hence $\Psi(a+t)$ is strictly positive
for some large $t\in{\mathbb{R}}^{d}$. Therefore the set $\\{n:a_{n}\in
D\backslash\widetilde{K}\\}$ is unbounded and countable, as well as the sets
${\mathbb{N}}\setminus S_{1}$ and ${\mathbb{N}}\setminus S_{2}$. For points
$a_{n}$ with $n\notin S_{1}$ the condition (6) need not be required, therefore
the bijection $\sigma:S_{1}\to S_{2}$ can be extended to a bijection
${\mathbb{N}}\to{\mathbb{N}}$. The Theorem is proved.

##  3\. Uniqueness theorems for almost periodic distributions, measures,
multisets

######  Definition 8.

We shall say that distributions $f,g\in D^{*}({\mathbb{R}}^{d})$ converge
weakly at infinity, if for any $\varphi\in D({\mathbb{R}}^{d})$ the functions
$(f,\leavevmode\nobreak\ \varphi\leavevmode\nobreak\
(\cdot\leavevmode\nobreak\ -t))$ and $(g,\leavevmode\nobreak\
\varphi(\leavevmode\nobreak\ \cdot-t))$ of the variable $t\in{\mathbb{R}}^{d}$
converge weakly at infinity.

Also, we shall say that distributions $f,g\in D^{*}(T_{\Omega})$ converge
weakly at infinity, if for any $\varphi\in D(T_{\Omega})$ the functions
$(f,\varphi(\cdot-z))$ and $(g,\varphi(\cdot-z))$ of the variable $z\in
T_{\omega}$ converge weakly at infinity ($\omega\subset\Omega$ is defined in
Definition 6).

It follows from Theorem 3

######  Theorem 5.

If two almost periodic distributions or measures $f,g\in
D^{*}({\mathbb{R}}^{d})$ converge weakly at infinity, then $f\equiv g$. The
similar assertion is valid for $f,g\in D^{*}(T_{\Omega})$.

######  Definition 9.

We shall say that two discrete multisets
$F=\\{a_{n}\\},H=\\{b_{n}\\}\subset{\mathbb{R}}^{d}$ converge weakly at
infinity, if there is a set $G\subset{\mathbb{R}}^{d}$ satisfying (1) such
that under an appropriate numbering

$\lim_{n\rightarrow\infty,n\in N(G)}{a_{n}-b_{n}}=0,$

${\mathbb{N}}(G)=\\{n\in{\mathbb{N}}:a_{n}\ \mbox{ or }\ b_{n}\in G\\}$.

######  Definition 10.

We shall say that two discrete multisets $F=\\{a_{n}\\},H=\\{b_{n}\\}\subset
T_{\Omega}$ converge weakly at infinity, if for every $K\subset\Omega$ there
is a set $G=G(K)\subset{\mathbb{R}}^{d}$ satisfying (1) such that under an
appropriate numbering

$\lim_{n\rightarrow\infty,n\in{\mathbb{N}}(G,K)}{a_{n}-b_{n}}=0,$

where ${\mathbb{N}}(G,K)=\\{n\in{\mathbb{N}}:a_{n}\in G\times K\ \mbox{ or }\
b_{n}\in G\times K\\}$.

######  Theorem 6.

If two discrete multisets $F=\\{a_{n}\\},H=\\{b_{n}\\}$ converge weakly at
infinity, then they are identical.

Proof. It follows from theorems 4 and 5 that we have to check the weak
convergence of measures $\mu_{F}$ and $\mu_{H}$ at infinity. The latter means
that for any $\varphi\in D({\mathbb{R}}^{d})$ (or $\varphi\leavevmode\nobreak\
\in\leavevmode\nobreak\ D(T_{\Omega})$) the almost periodic functions of the
variable $t\in{\mathbb{R}}^{d}$

$\Psi_{F}(t)=(\mu_{F},\varphi(\cdot-t))=\sum\limits_{n}\varphi(a_{n}-t)$

and

$\Psi_{H}(t)=(\mu_{H},\varphi(\cdot-t))=\sum\limits_{n}\varphi(b_{n}-t)$

converge weakly at infinity. To be specific consider the case $F,H\subset
T_{\Omega}$. A proof is the same for $F,H\subset{\mathbb{R}}^{d}$.

Suppose that $\operatorname{supp}\varphi\subset B_{\mathbb{R}}(0,1)\times K$
for compact $K\subset\Omega$. Take $\varepsilon>0$ and then $\delta>0$ such
that $|\varphi(z)-\varphi(z^{\prime})|<\varepsilon/(N(K))$ for
$|z-z^{\prime}|<\delta$, where $N(K)$ is the constant from (3). Let a set
$G\subset{\mathbb{R}}^{d}$ satisfy (1) with balls
$B_{\mathbb{R}}(x_{k},R_{k}),\ k\in{\mathbb{N}}$. It is easy to see that
having reduced by 3 times the radii of these balls and changing the location
of their centers, we can assume that
$\operatorname{dist}(B(x_{k},R_{k}),0)\to\infty$. For sufficiently large $k$
and for $a_{n},\ b_{n}\in B_{\mathbb{R}}(x_{k},R_{k})\times K$ we have
$|a_{n}-b_{n}|<\delta$. Also assume that $R_{k}>2$.

Let $t\in B_{\mathbb{R}}(x_{k},R_{k}/2)$ and
$a_{n}-t\in\operatorname{supp}\varphi$. Then $a_{n}\in
B_{\mathbb{R}}(x_{k},R_{k})\times K$, and the same is valid for $b_{n}-t$.
Therefore if $a_{n}-t\in\operatorname{supp}\varphi$ or
$b_{n}-t\in\operatorname{supp}\varphi$, we get $|a_{n}-b_{n}|<\delta$ and

$|\Psi_{F}(t)-\Psi_{H}(t)|\leq\sum_{n}|\varphi(a_{n}-t)-\varphi(b_{n}-t)|<\frac{\varepsilon}{N(K)}\cdot
N(K)=\varepsilon.$

Hence the almost periodic functions $\Psi_{F}(t)$ $\Psi_{H}(t)$ converge
weakly at infinity.

##  4\. Uniqueness theorems for delta-subharmonic and meromorphic functions

It follows immediately from the definition that any partial derivative of an
almost periodic distribution from $D^{*}({\mathbb{R}}^{d})$ or
$D^{*}(T_{\Omega})$ is also an almost periodic distribution. Since any
subharmonic function on any region from ${\mathbb{R}}^{d}$ is locally
integrable, it can be considered as a distribution. Thus, if $u$ is a
subharmonic almost periodic function on $D^{*}({\mathbb{R}}^{d})$ or
$D^{*}(T_{\Omega})$, then its Riesz measure $\Delta u$ is also an almost
periodic distribution, and the same is true for the difference of subharmonic
functions, the so-called delta-subharmonic functions.

It follows from Theorem 5

######  Theorem 7.

If two delta-subharmonic functions $u,v$ on ${\mathbb{R}}^{d}$ or $T_{\Omega}$
have converging weakly at infinity Riesz measures $\Delta u$ and $\Delta v$,
then $u=v+h$ with a harmonic function $h$.

The last part of the proof uses the fact that the condition $\Delta h=0$ in
the sense of distributions implies that $h$ is an ordinary harmonic function.

######  Definition 11.

(. [10], [5]) A meromorphic function $f(z)$ on the strip
$S_{a,b}=\\{z\in{\mathbb{C}}:{\textsf{Re}}\ z\in{\mathbb{R}},a<{\textsf{Im}}\
z<b\\}$, $-\infty\leq a<b\leq+\infty$, is called almost periodic, if in any
smaller strip $S_{\alpha,\beta}$, $a<\alpha<\beta<b$, the function
$\rho_{S}(f(z+t),f(z))$, where $\rho_{S}$ is the spherical distance, is almost
periodic in the variable $t\in{\mathbb{R}}$.

In [5] the following properties of meromorphic almost periodic functions are
proved:

  * •

The distance between any pole and any zero of meromorphic almost periodic
functions is bounded from below by a strictly positive constant depending on
the strip in which this pole and zero lie,

  * •

Every meromorphic almost periodic function on $S_{a,b}$ is a ratio of two
holomorphic almost periodic functions in $S_{a,b}$; the converse assertion is
only valid if distances between poles and zeros of this ratio are uniformly
bounded from below by a strictly positive constant in any smaller strip. In
particular, every holomorphic almost periodic function in a strip is
simultaneously a meromorphic almost periodic function.

It was proved in [9] that for any almost periodic holomorphic function $f$ on
$S_{a,b}$ the function $\log|f|$ is an almost periodic distribution, hence the
measure $\mu_{Z}$ corresponding to the multiset of zeros $Z_{f}$ is almost
periodic. Also, if $f$ is an almost periodic meromorphic function, then the
measures $\mu_{Z}$ and $\mu_{P}$ corresponding to the multiset of zeros
$Z_{f}$ and the multiset of poles $P_{f}$ of $f$ are also almost periodic.
Therefore, Theorem 7 implies

######  Theorem 8.

If multisets of poles $P_{f}$ and $P_{g}$ of meromorphic almost periodic
functions $f,g$ in a strip $S_{a,b}$ converge weakly and the same is true for
multisets of zeros $Z_{f}$ and $Z_{g}$, then $P_{f}=P_{g}$, $Z_{f}=Z_{g}$,
hence, $f/g$ is a holomorphic almost periodic function on $S_{a,b}$ without
zeros.

If $f,g$ are holomorphic almost periodic functions on $S_{a,b}$ and multisets
of zeros $Z_{f},\ Z_{g}$ converge weakly at infinity, then $Z_{f}=Z_{g}$, and
we obtain Theorem 6 from [7].

Note that a linear-fractional mapping of a meromorphic almost periodic
function $f$ is a meromorphic almost periodic function too. Then instead of
zeros and poles one can consider $A_{1}$ -points and $A_{2}$ -points,
$A_{1}\neq A_{2}$, that is zeros of functions $f-A_{1}$ and $f-A_{2}$. Also,
for $T_{\Omega}={\mathbb{C}}$ we obtain the following theorem:

######  Theorem 9.

Let $f,g$ be meromorphic almost periodic functions on ${\mathbb{C}}$ and let
$A_{j}$-points of $f$ converge weakly at infinity to $A_{j}$-points of $g$ for
three pairwise distinct values $A_{1},\,A_{2},\,A_{3}$. Then either $f=g$, or
$f$ and $g$ have the forms

(7) $f=T\left(\frac{1-h_{1}}{h_{2}-h_{1}}\right),\qquad
g=T\left(\frac{h_{2}-h_{1}h_{2}}{h_{2}-h_{1}}\right),$

where $h_{1},\,h_{2}$ are distinct entire functions without zeros, and $T$ is
a linear-fractional mapping that moves the triple point $0,1,\infty$ to the
triple point $A_{1},\,A_{2},\,A_{3}$.

At the final stage of the proof we use the following theorem from [8]: If two
meromorphic functions on ${\mathbb{C}}$ have the same multisets of $A$ -points
for three distinct values of $A_{1},\,A_{2},\,A_{3}$, then these functions
either coincide, or have form (7).

## References

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