#  $L^{2}$-Hodge theory on complete almost Kähler manifold and its application

Teng Huang and Qiang Tan

###### Abstract

Let $(X,J,\omega)$ be a complete $2n$-dimensional almost Kähler manifold.
First part of this article, we construct some identities of various
Laplacians, generalized Hodge and Serre dualities, a generalized hard
Lefschetz duality, and a Lefschetz decomposition, all on the space of
$\ker{\Delta_{\partial}}\cap\ker{\Delta_{\bar{\partial}}}$ on pure bidegree.
In the second part, as some applications of those identities, we establish
some vanishing theorems on the spaces of $L^{2}$-harmonic $(p,q)$-forms on $X$
under some growth assumptions on the Käher form $\omega$. We also give some
$L^{2}$-estimates to sharpen the vanishing theorems in two specific cases. At
last of the article, as an application, we study the topology and geometry of
the compact almost Kähler manifold with negative sectional curvature.

††T. Huang: School of Mathematical Sciences, University of Science and
Technology of China; CAS Key Laboratory of Wu Wen-Tsun Mathematics, University
of Science and Technology of China, Hefei, Anhui, 230026, P. R. China; e-mail:
htmath<EMAIL_ADDRESS>Tan: School of Mathematical Sciences,
Jiangsu University, Zhenjiang, Jiangsu 212013, People’s Republic of China;
e-mail<EMAIL_ADDRESS>
##  1 Introduction

In complex geometry the Dolbeault cohomology plays a fundamental role in the
study of complex manifolds and a classical way to compute it on compact
complex manifolds is through the use of the associated spaces of harmonic
forms. More precisely, if $X$ is a complex manifold, then the exterior
derivative $d$ splits as $\partial+\bar{\partial}$ and such operators satisfy
$\bar{\partial}^{2}=\partial^{2}=[\partial,\bar{\partial}]=0$. Hence one can
define the Dolbeault cohomology and its conjugate as

$H^{p,q}_{\bar{\partial}}:=\frac{\ker\bar{\partial}}{{\rm{Im}}\bar{\partial}},\
H^{p,q}_{\partial}:=\frac{\ker\partial}{{\rm{Im}}\partial}.$

If $X$ is compact, then it turns out there spaces are isometric to the kernel
of two elliptic operators $\Delta_{\bar{\partial}}$ and $\Delta_{\partial}$.

But in a non-complex Hermitian manifold $X$, i.e., the almost complex
structure $J$ is non-integrable on $X$, the exterior derivative splits as
$\partial+\mu+\bar{\partial}+\bar{\mu}$ and in particular
$\bar{\partial}^{2}\neq 0$. Hence, the standard Dolbeault cohomology and its
conjugate are not well-defined. Recently, Cirici and Wilson in [9] gave a
definition for the Dolbeault cohomology in the non-integrable setting
considering also the operator $\bar{\mu}$ together with $\bar{\partial}$.
Similar to the integrable case, one can develop an Hodge theory for harmonic
forms on $(X,J,\omega)$ without a cohomological counterpart (see [11, 37, 38,
41, 42]). We define two elliptic selfadjoint differential operators

$\Delta_{\bar{\partial}}=\bar{\partial}\bar{\partial}^{\ast}+\bar{\partial}^{\ast}\bar{\partial},\
\Delta_{\partial}=\partial\partial^{\ast}+\partial^{\ast}\partial.$

Cirici and Wilson recently proved a generalized Lefschetz decomposition
theorem for compact almost Kähler manifold. Denote by
$\Delta_{d}=dd^{\ast}+d^{\ast}d$ the Hodge Laplacian. The space of harmonic
$(p,q)$-forms $\ker\Delta_{d}\cap\Omega^{p,q}$ will be indicated by
$\mathcal{H}^{p,q}_{d}$. In [8, Corollary 5.4]) they showed that if
$(X,J,\omega)$ is a compact almost Kähler manifold, then

$\mathcal{H}^{p,q}_{d}=\bigoplus_{r\geq\max(p+q-n,0)}L^{r}(\mathcal{H}^{p-r,q-r}_{d}\cap
P^{p-r,q-r}).$

In this article we will consider the spaces of
$(\bar{\partial}+\partial)$-harmonic forms given by the intersections

$\mathcal{H}^{p,q}_{(2);\bar{\partial}}\cap\mathcal{H}^{p,q}_{(2);\partial}.$

These are identified with the kernel of the self-adjoint elliptic operator
given by
$\Delta_{\bar{\partial}+\partial}=[\bar{\partial}+\partial,\bar{\partial}^{\ast}+\partial^{\ast}]$
in $\Omega^{p,q}_{(2)}(X)$. We will denote by

$l^{p,q}_{(2)}:=\dim(\mathcal{H}^{p,q}_{(2);\bar{\partial}}\cap\mathcal{H}^{p,q}_{(2);\partial})=\dim\ker\Delta_{\bar{\partial}+\partial}\cap\Omega^{p,q}_{(2)}=\dim\ker(\Delta_{\bar{\partial}}+\Delta_{\partial})\cap\Omega^{p,q}_{(2)}$

the dimensions of these spaces. In the integrable case, since
$\Delta_{\partial}=\Delta_{\bar{\partial}}=\frac{1}{2}\Delta_{d}$, these are
just the $L^{2}$-Hodge numbers of the Kähler manifold.

######  Theorem 1.1.

(Generalized Hard Lefschetz Duality) For any complete almost Kähler manifold
of dimension $2n$, the operators $\\{L,\Lambda,H=[L,\Lambda]\\}$ define a
finite dimensional representation of $\mathfrak{sl}(2,\mathbb{C})$ on

$\bigoplus_{p,q\geq
0}\ker{\Delta_{\partial}}\cap\ker{\Delta_{\bar{\partial}}}\cap\Omega^{p,q}_{(2)}(X).$

Moreover, for all $0\leq p\leq k\leq n$,

$L^{n-k}:\ker{\Delta_{\partial}}\cap\ker{\Delta_{\bar{\partial}}}\cap\Omega^{p,k-p}_{(2)}(X)\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\ker{\Delta_{\partial}}\cap\ker{\Delta_{\bar{\partial}}}\cap\Omega^{p+n-k,n-p}_{(2)}(X)$

are isomorphisms. Furthermore, for any $p,q$ we have an orthogonal direct sum
decomposition

$\ker{\Delta_{\partial}}\cap\ker{\Delta_{\bar{\partial}}}\cap\Omega^{p,q}_{(2)}(X)=\bigoplus_{j\geq
0}L^{j}(\ker{\Delta_{\partial}}\cap\ker{\Delta_{\bar{\partial}}}\cap
P^{p-j,q-j}_{(2)}(X)),$

where

$P^{r,s}_{(2)}(X)=\ker\Lambda\cap\Omega^{r,s}_{(2)}(X).$

######  Remark 1.2.

For any compact almost Kähler manifold $X$, Cirici1 and Wilson [8] defined the
$\delta$-Laplacian by letting

$\Delta_{\delta}=\delta\delta^{\ast}+\delta^{\ast}\delta.$

For all $p,q$, they also denoted by

$\mathcal{H}^{p,q}_{\delta}=\ker\Delta_{\delta}\cap\Omega^{p,q}=\ker\delta\cap\ker\delta^{\ast}\cap\Omega^{p,q}$
(1.1)

the space of $\delta$-harmonic forms in bidegree $(p,q)$. But in non-compact,
for any $\alpha\in\Omega^{p,q}_{(2)}$, $\delta\alpha$ and
$\delta^{\ast}\alpha$ always not in $L^{2}$ when $\delta=\mu,\bar{\mu}$.
Therefore the definition of the space $\mathcal{H}^{p,q}_{(2);\delta}$ is not
a well-definition and $\mathcal{H}^{p,q}_{(2);\delta}$ always can’t be
satisfied (1.1).

Let $(X,J,\omega)$ be a complete $2n$-dimensional almost Kähler manifold. A
basic question, pertaining both to the topology and function theory of $X$,
is: when are there non-trivial harmonic forms on $X$, in the various bidegree
$(p,q)$ determined by the almost complex structure? When $X$ is not compact,
we denote by $\Omega^{p,q}_{(2)}(X)$ the $L^{2}$-forms of type $(p,q)$ on $X$
and $\mathcal{H}^{p,q}_{(2);d}(X)$ the harmonic forms in
$\Omega^{p,q}_{(2)}(X)$. One version of this basic question is: what is the
structure of $\mathcal{H}^{p,q}_{(2);d}(X)$?

The Hodge theorem for compact manifolds states that every de Rahm cohomology
class of a compact manifold $X$ is represented by a unique harmonic form. That
is, the space of solutions to the differential equation $\Delta_{d}\alpha=0$
on smooth forms over $X$ is a space that depends on the metric on $X$. This
space is canonically isomorphic to the purely topological de Rahm cohomology
space of $X$. The study of $\mathcal{H}^{p,q}_{(2);d}(X)$, a question of so-
called $L^{2}$-cohomology of $X$, is rooted in the attempt to extend Hodge
theory to non-compact manifolds. The study of the $L^{2}$-harmonic forms on a
complete Riemannian manifold is a very fascinating and important subject.
There are numerous partial results about the $L^{2}$-cohomology of non-compact
manifold (see [1, 7, 15]), but this extension is not yet complete. When $J$ is
integrable in $X$, then $X$ is Kählerian. The study of
$\mathcal{H}^{p,q}_{(2);d}(X)$ is one of the focal points in complex geometry
[10, 16, 17, 18, 23, 34, 35] and the references therein provide a good view on
the subject.

The second purpose of this paper is to prove some vanishing results on
$\mathcal{H}^{p,q}_{(2);d}(X)$ when $p+q\neq n$, under a growth assumption on
a primitive of $\omega$ does not grow to fast at infinity.

######  Theorem 1.3.

Let $(X,J,\omega)$ be a complete almost Kähler manifold which is
$(L,c)$-vanishingly exhaustible. There exists a uniform positive constant
$C(n)$ depends only on $n$ such that if $c\leq C(n)$, then for any $p+q\neq
n$,

$\ker{\Delta_{\bar{\partial}}}\cap\ker{\Delta_{\partial}}\cap\Omega^{p,q}_{(2)}(X)=0.$

In particular,

$\mathcal{H}^{p,q}_{(2);d}(X)=0.$

Next, we also give some $L^{2}$-estimates to sharpen the vanishing theorem 1.3
in two specific cases.

######  Theorem 1.4 (=Theorem 5.2+Corollary 5.3+Corollary 5.4).

Let $(X,J,\omega)$ be a complete almost Kähler manifold. Suppose that there
exists a bounded $1$-form $\theta$ such that

$\sup|\omega-(d\theta)^{1,1}|\leq c,$

then for any $\alpha\in\Omega^{k}_{0}(X)$, $(k\neq n)$, we have

$\begin{split}&\|\alpha\|(1-c(n,k)c)\leq
c(n,k)\|\theta\|_{L^{\infty}(X)}((\Delta_{\partial}+\Delta_{\bar{\partial}})\alpha,\alpha)^{\frac{1}{2}}\\\
&\|\alpha\|^{2}(1-c(n,k)c)(1-c(n,k)c-4c(n,k)\|\theta\|_{L^{\infty}(X)}\sup|N_{J}|)\leq
c^{2}(n,k)\|\theta\|^{2}_{L^{\infty}(X)}(\Delta_{d}\alpha,\alpha)\\\
&\|\alpha\|^{2}((1-c(n,k)c)^{2}-2c(n,k)^{2}\|\theta\|^{2}_{L^{\infty}(X)}\sup|N_{J}|^{2}))\leq
2c^{2}(n,k)\|\theta\|^{2}_{L^{\infty}(X)}(\Delta_{\bullet}\alpha,\alpha),\\\
\end{split}$

where $\bullet=\partial,\bar{\partial}$. Furthermore,
(1) if $c$ is small enough such $c(n,k)c<1$, then

$\ker(\Delta_{\partial}+\Delta_{\bar{\partial}})\cap\Omega^{k}_{(2)}(X)=\\{0\\}.$

(2) if $c(n,k)(c+4\|\theta\|_{L^{\infty}(X)}\sup|N_{J}|)<1$, then for any
$k\neq n$,

$\mathcal{H}^{k}_{(2);\delta}(X)=\\{0\\},$

where $\delta=d,\partial,\bar{\partial}$.

######  Theorem 1.5.

Let $(X,J,\omega)$ be a complete almost Kähler manifold. Suppose that there
exists a smooth function $f\geq 1$ on $X$. Also assume that the function $f$
satisfies the convexity condition on $X$, i.e., for some $A,B\geq 0$,
$|df|^{2}\leq A+Bf$. If

$c:=1-B-C\sup|\tilde{\omega}|>0,$

where $\tilde{\omega}:=\omega-(dJdf)^{1,1}$ and $C$ is an uniform positive
constant which depends only on $p,q,n$, then there exist constants $m$, $M$
depending only on universal constants and the constants $A,B,c$ such that for
any $p+q\neq n$,

$m\int_{X}\frac{1}{f+M}|\alpha|^{2}\leq(\|\partial\alpha\|^{2}+\|\partial^{\ast}\alpha\|^{2}+\|\bar{\partial}\alpha\|^{2}+\|\bar{\partial}^{\ast}\alpha\|^{2}),\
\forall\alpha\in\Omega^{p,q}_{0}(X).$ (1.2)

In particular,

$\mathcal{H}^{p,q}_{(2);d}(X)=0.$

At next part of the article, we will study the topology and geometry of the
compact almost Kähler manifold with negative sectional curvature. Let us start
the last part by recalling one well-known conjecture related to the negativity
of Riemannian sectional curvature.

######  Conjecture 1.

(Hopf Conjecture) Let $X$ be a closed 2n-dimensional Riemannian manifold $X$
with sectional curvature $sec$. Then

$\left\\{\begin{aligned} (-1)^{n}\chi(X)>0,&\ if\ sec<0\\\ (-1)^{n}\chi(X)\geq
0,&\ if\ sec\leq 0.\\\ \end{aligned}\right.$

This is true for $n=1$ and $2$ as the Gauss–Bonnet integrands in these two low
dimensional cases have the desired sign. However, in higher dimensions, it is
known that the sign of the sectional curvature does not determine the sign of
the Gauss-Bonnet-Chern integrand. Let $(X,g)$ be a Riemannian manifold and
$\pi:(\tilde{X},\tilde{g})\rightarrow(X,g)$ be the universal covering with
$\tilde{g}=\pi^{\ast}g$. A form $\alpha$ on $X$ is called $\tilde{d}$(bounded)
if $\pi^{\ast}\alpha$ is a $d$(bounded) form on $(X,g)$. Gromov pointed out
that if the Riemannian manifold $(X,g)$ is a complete simply-connected
manifold with strictly negative sectional curvature, then every smooth bounded
closed form of degree $k\geq 2$ is $d$(bounded). Then he proved the Hopf
conjecture by Kähler identities in the Kähler case. For symplectic case,
inspired by Kähler geometry, we also can give the definition of symplectic
hyperbolic manifold. A compact almost Kähler manifold $(X,\omega)$ is called
symplectic hyperbolic if the lift $\tilde{\omega}$ of $\omega$ to the
universal covering $(\tilde{X},\tilde{\omega})\rightarrow(X,\omega)$ is
$d$(bounded) on $(\tilde{X},\tilde{\omega})$. Hind and Tomassini [29]
constructed a $d$(bounded) complete almost Kähler manifold $X$ satisfies
$\mathcal{H}^{1}_{(2)}(X)\neq\\{0\\}$ by using methods of contact geometry.

The first application in our article is that we can confirm that the Hopf
conjecture is correct in the case of almost Kähler manifold $X$ with small
Nijenhuis tensor. A special case is that Nijenhuis tensor vanishes, i.e., the
manifold $X$ is Käherian (see [23]).

######  Theorem 1.6.

Let $(X,J,\omega)$ be a compact almost Kähler manifold with negative sectional
curvature, i.e., there exists a constant $K>0$ such that

$sec\leq-K.$

Let $\pi:(\tilde{X},\tilde{J},\tilde{\omega})\rightarrow(X,J,\omega)$ the
universal covering map for $X$. If the Nijenhuis tensor of $X$ satisfies

$|\nabla J|^{2}\leq c(n)K,$

where $c(n)$ is an uniform positive constant, then

$\left\\{\begin{aligned} \mathcal{H}^{k}_{(2);d}(\tilde{X})=\\{0\\},&k\neq
n\\\ \mathcal{H}^{n}_{(2);d}(\tilde{X})\neq\\{0\\},&\\\ \end{aligned}\right.$

is equivalent to

$\left\\{\begin{aligned} h^{k}_{(2)}(X)=0,&k\neq n\\\ h^{n}_{(2)}(X)>0.&\\\
\end{aligned}\right.$

In particular,

$(-1)^{n}\chi(X)>0.$

If $(X,g)$ is a closed $n$-dimensional Riemannian manifold with sectional
curvature $-1<sec<0$, then $Vol(X)>C^{-1}_{n}$,
$C_{n}\leq\exp{\exp{\exp{\exp{n}}}}$ [22]. In [21], Gromov proved that for any
$n\geq 8$ and $0>sec>-1$ the volume of closed $n$-dimensional manifold $X$
satisfies $Vol(X)\geq C(1+D)$, where constant $C$ depends only on $n$. The
second application is that we can prove that the volume of almost Kähler
manifold with pinched negative sectional curvature has a lower bounded in
another way. For some compact Riemannian manifolds $X$ of dimension $2n$ with
some suitable pinched negative sectional curvature, the Euler number of these
manifolds had been studied by many authors [19, 33].

######  Theorem 1.7.

Let $(X,J,\omega)$ be a compact almost Kähler manifold with pinched negative
sectional curvature, i.e., there are constants $K$ and $\delta\geq 1$ such
that

$-\delta K\leq sec\leq-K.$

Then there exists a constant $C(n)>0$ dependents on $n$ such that

$\delta^{2}K[Vol(X)]^{\frac{1}{n}}>C(n).$

######  Remark 1.8.

In fact, we can also show that $C(n)^{-1}$ is a polynomial function on $n$.
The lower bounded in Theorem 1.7 is sharper than the constant in [22].

##  2 Almost Kähler manifold

We recall some definitions and results on the differential forms for almost
complex and almost Hermitian manifolds. Let $X$ be a $2n$-dimensional manifold
(without boundary) and $J$ be a smooth almost complex structure on $X$. There
is a natural action of $J$ on the space
$\Omega^{k}(X,\mathbb{C}):=\Omega^{k}(X)\otimes\mathbb{C}$, which induces a
topological type decomposition

$\Omega^{k}(X,\mathbb{C})=\bigoplus_{p+q=k}\Omega^{p,q}(X),$

where $\Omega^{p,q}(X)$ denotes the space of complex forms of type $(p,q)$
with respect to $J$ [28]. Then $J$ acts as an isomorphism on $\Omega^{p,q}$ by
$J(\bullet)=\sqrt{-1}^{q-p}(\bullet)$. If $k$ is even, $J$ also acts on
$\Omega^{k}(X,\mathbb{C})$ as an involution. We have that

$d:\Omega^{p,q}\rightarrow\Omega^{p+2,q-1}\oplus\Omega^{p+1,q}\oplus\Omega^{p,q+1}\oplus\Omega^{p-1,q+2}$

and so the operator $d$ splits according as

$d=\mu+\partial+\bar{\partial}+\bar{\mu},$

where all the pieces are graded algebra derivations, $\mu$, $\bar{\mu}$ are
$0$-order differential operators. Note that each component of $d$ is a
derivation, with bi-degrees given by

$|\mu|=(2,-1),\ |\partial|=(1,0),\ |\bar{\partial}|=(0,1),\
|\bar{\mu}|=(-1,2).$

Expanding the equation $d^{2}=0$ we obtain the following set of equations:

$\begin{split}&\mu^{2}=0,\\\ &\mu\partial+\partial\mu=0,\\\
&\partial^{2}+\mu\bar{\partial}+\bar{\partial}\mu=0,\\\
&\partial\bar{\partial}+\bar{\partial}\partial+\mu\bar{\mu}+\bar{\mu}\mu=0,\\\
&\bar{\partial}^{2}+\bar{\mu}\partial+\partial\bar{\mu}=0,\\\
&\bar{\mu}\bar{\partial}+\bar{\partial}\bar{\mu}=0,\\\ &\bar{\mu}^{2}=0.\\\
\end{split}$ (2.1)

The integrability theorem of Newlander and Nirenberg states that the almost
complex structure $J$ is integrable if and only if $N_{J}=0$, where

$N_{J}:TX\otimes TX\rightarrow TX,$

denotes the Nijenhuis tensor

$N_{J}(X,Y):=[X,Y]+J[X,JY]+J[JX,Y]-[JX,JY].$

One can show that $\mu+\bar{\mu}$ is equal, up to a scalar, to the dual of the
Nijenhuis tensor (cf. [8]). In fact,

$\mu+\bar{\mu}=-\frac{1}{4}(N_{J}\otimes{\rm{id}_{\mathbb{C}}})^{\ast}.$

where the right hand side has been extended over all forms as a derivation. In
particular, $J$ is integrable if only if $N_{J}=0$, i.e, $\mu=0$ [8, 9].

Let $(X,J,\omega)$ be a compact almost Kähler manifold. We denote by
$\mathfrak{X}(X)$ the Lie algebra of all smooth vector fields on $x$. It is
known that the Nijenhuis tensor $N_{J}$ of $X$ is expressed by

$g(N_{J}(X,Y),JZ)=2g((\nabla_{Z}J)X,Y),$

for $X,Y,Z\in\mathfrak{X}(X)$. We denote by $\\{Z_{i}\\}$ the orthonormal
basis of $T_{p}^{1,0}$, $p\in X$. We extend the curvature operator
$R:\wedge^{2}T_{p}X\rightarrow\wedge^{2}T_{p}X$ to a complex linear
transformation
$R^{\mathbb{C}}:\wedge^{2}T_{p}X\otimes\mathbb{C}\rightarrow\wedge^{2}T_{p}X\otimes\mathbb{C}$
[26]. Given a nonzero decomposition
$\Pi\in\Lambda^{2}T_{p}X\otimes\mathbb{C}$, its complex sectional curvature is
the real number

$K^{\mathbb{C}}(\Pi)=\frac{(R^{\mathbb{C}}(\Pi),\overline{\Pi})}{(\Pi,\overline{\Pi})}$

(one also can see [26, Definition 2.2]). Then following [27, Lemma 3.3] and
[39, Lemma 2.4 ] or [40, Equation (3.1)–(3.3)]), we have

$\begin{split}|\nabla{J}|^{2}&=\frac{1}{4}|N_{J}|^{2}\\\
&=-8\sum_{i,j=1}^{n}(R^{\mathbb{C}}(Z_{i}\wedge Z_{j}),\overline{Z_{i}\wedge
Z_{j}}).\end{split}$ (2.2)

######  Proposition 2.1.

(cf. [26, Theorem 2.5]) Let $(X,J,\omega)$ be a complete almost Kähler
manifold with pinched negative sectional curvature, i.e., there are constants
$K$ and $\delta\geq 1$ such that

$-\delta K\leq sec\leq-K.$

Then complex sectional curvature of $X$ satisfies

$|\nabla J|^{2}\leq 10n^{2}\delta K.$

###### Proof.

Let $\Pi\in\Lambda^{2}T_{p}X\otimes\mathbb{C}$ be decomposable.
First case, $\Pi$ contains real vectors.
Let $0\neq x\in\Pi\cap\bar{\Pi}$, then $\Pi=x\wedge z$ for some
$z=a+\sqrt{-1}b$, $a,b\in T_{p}X$; then

$(R^{\mathbb{C}}(\Pi),\overline{\Pi})=\langle R(x\wedge a),x\wedge
a\rangle+\langle R(x\wedge b),x\wedge b\rangle.$

Therefore,

$-\delta K\leq K^{\mathbb{C}}(\Pi)\leq-K.$

Second case, $\Pi$ contains no real vectors.
Let $\Pi=Z\wedge Y$, $Z=u+\sqrt{-1}v$ and $Y=x+\sqrt{-1}w$, we can assume that
$u=Ue_{1}$, $v=Ve_{2}$, $x=Xe_{3}$, $w=We_{4}$, where $\langle
e_{i},e_{j}\rangle=\delta_{ij}$ and $U^{2}+V^{2}=X^{2}+W^{2}=1$. If $K_{ij}$
denotes the sectional curvature of $e_{i}\wedge e_{j}$, then

$\begin{split}(R^{\mathbb{C}}(Z\wedge Y),\overline{Z\wedge Y})&=\langle
R(u\wedge x),u\wedge x\rangle+\langle R(u\wedge w),u\wedge w\rangle\\\
&+\langle R(v\wedge x),v\wedge x\rangle+\langle R(v\wedge w),v\wedge
w\rangle\\\ &-2\langle R(u\wedge v),x\wedge w\rangle\\\
&=U^{2}X^{2}K_{13}+U^{2}W^{2}K_{14}+V^{2}X^{2}K_{23}+V^{2}W^{2}K_{24}\\\
&-2UVWX\langle R(e_{1}\wedge e_{2}),e_{3}\wedge e_{4}\rangle.\\\ \end{split}$

Therefore,

$\begin{split}-\delta K-\frac{1}{2}|\langle R(e_{1}\wedge e_{2}),e_{3}\wedge
e_{4}\rangle|\leq(R^{\mathbb{C}}(Z\wedge
Y),\overline{Z}\wedge\overline{Y})\leq-K+\frac{1}{2}|\langle R(e_{1}\wedge
e_{2}),e_{3}\wedge e_{4}\rangle|\end{split}$

Noting that $|\langle R(e_{1}\wedge e_{2}),e_{3}\wedge e_{4}\rangle|\leq\delta
K.$ (cf. [6] and [26]). Hence, we get

$-\frac{5}{4}\delta K\leq K^{\mathbb{C}}(\Pi)\leq(\frac{\delta}{2}-1)K.$

Combining the preceding inequalities with (2.2) yields

$|\nabla{J}|^{2}\leq 8n^{2}\max|K^{\mathbb{C}}|\leq 10n^{2}\delta K.$

∎

On an almost Kähler manifold, Cirici and Wilson constructed the almost Kähler
identities, involving the differential operators $\partial$ and
$\bar{\partial}$, the operators $\mu$ and $\bar{\mu}$, the Lefschetz operator
$L$, and their complex conjugates and adjoints. We will recall some identities
which will be used in this article.

If $A$ and $B$ are operators on forms, defined the graded commutator

$[A,B]=AB-(-1)^{degA\cdot degB}BA,$

where $degT$ is the integer $l$ for which

$T:\bigoplus_{p+q=k}\Omega^{p,q}(X)\rightarrow\bigoplus_{p+q=k+l}\Omega^{p,q}(X).$

######  Proposition 2.2.

(cf. [8]) For any almost Kähler manifold the following identities hold:
(1)
$[\partial,\bar{\partial}^{\ast}]=[\bar{\mu}^{\ast},\bar{\partial}]+[\mu,\partial^{\ast}]$
and
$[\bar{\partial},\partial^{\ast}]=[\mu^{\ast},\partial]+[\bar{\mu},\bar{\partial}^{\ast}]$.
(2) $[L,\bar{\mu}^{\ast}]=\sqrt{-1}\mu$, $[L,\mu^{\ast}]=-\sqrt{-1}\bar{\mu}$
and $[\Lambda,\bar{\mu}]=\sqrt{-1}\mu^{\ast}$,
$[\Lambda,\mu]=-\sqrt{-1}\bar{\mu}^{\ast}$.
(3) $[L,\bar{\partial}^{\ast}]=-\sqrt{-1}\partial$,
$[L,\partial^{\ast}]=\sqrt{-1}\bar{\partial}$ and
$[\Lambda,\bar{\partial}]=-\sqrt{-1}\partial^{\ast}$,
$[\Lambda,\partial]=\sqrt{-1}\bar{\partial}^{\ast}$.

Another commutation result that we shall need is the following.

######  Proposition 2.3.

([34, Propostion 1.8] and [14]) Let $r\in\Omega^{1,1}(X)$ be a real
$(1,1)$-from and let $\Xi(r):\Omega^{p,q}(X)\rightarrow\Omega^{p+1,q+1}(X)$ be
the operator $\Xi(r)=r\wedge u$.

At each $x\in X$, there exists an orthonormal basis $(dz_{1},\cdots,dz_{n})$
for $T^{\ast(1,0)}_{x}(X)$ and real numbers $\\{e_{1},\cdots,e_{n}\\}$, such
that if $u\in\Omega^{p,q}(X)$, $u(x)=\sum_{I,J}u_{IJ}(x)dz^{I}\wedge
d\bar{z}^{J}$, then

$[\Xi(r),\Lambda]u(x)=\sum_{|I|=p,|J|=q}\big{(}\sum_{j\in I}e_{j}+\sum_{j\in
J}e_{j}-\sum_{j=1}^{n}e_{j}\big{)}u_{IJ}(x)dz^{I}\wedge d\bar{z}^{J}.$

##  3 $L^{2}$-Hodge theory

###  3.1 de Rham harmonic $(p,q)$-forms

Throughout, $(X,J,\omega)$ denotes a complete almost Kähler manifold of
complex dimension $n$. The global inner product is defined

$(u,v)=\int_{X}\langle u,v\rangle dV,$

where $dV=\frac{\omega^{n}}{n!}$ is the volume form determined by $\omega$. We
also write $|u|^{2}=\langle u,u\rangle$, $\|u\|^{2}=\int_{X}|u|^{2}dV$. We
denote by

$\mathcal{H}^{p,q}_{(2);d}(X):=\\{\alpha\in\Omega^{p,q}_{(2)}(X):\Delta_{d}\alpha=0\\}$

the space of $L^{2}$-harmonic forms of bi-degree $(p,q)$. Here

$\Omega^{p,q}_{(2)}(X):=\\{\alpha\in\Omega^{p,q}(X):\|\alpha\|_{L^{2}(X)}<\infty\\}.$

######  Lemma 3.1.

([31, Lemma3.3]) If an $L^{2}$ (p, q)-form $\alpha$ on $X$ is
$\Delta_{d}$-harmonic form, then $d\alpha=0,d^{\ast}\alpha=0$.

The Lefschetz operator $L:\Omega^{p,q}\rightarrow\Omega^{p+1,q+1}$ defined by

$L(\alpha)=\omega\wedge\alpha.$

The dual Lefschetz operator $\Lambda$ is then just the adjoint of $L$,
$\Lambda=(-1)^{k}\ast L\ast$. The $d^{\Lambda}$ operator is related via the
Hodge star operator defined with respect to the compatible metric $g$ by the
relation, see [Tseng-Yau, Lemma 2.9],

$d^{\Lambda}=(-1)^{k+1}\ast J^{-1}d\ast J^{-1}=-{\ast}J^{-1}dJ\ast.$

######  Lemma 3.2.

([41, Theorem 6.7] and [31, Lemma 3.2])

$\ker d\cap\ker d^{\ast}\cap\Omega^{p,q}(X)=\ker d^{\Lambda}\cap\ker
d^{\Lambda_{\ast}}\cap\Omega^{p,q}(X)$

###### Proof.

Noting that $J^{2}=(-1)^{k}$ acting on a $k$-form, $k=p+q$. We then have

$d\ast
J^{-1}\alpha_{p,q}=d\ast(-1)^{k}J\alpha_{p,q}=(-1)^{k}(\sqrt{-1})^{q-p}d\ast\alpha_{p,q},$

and

$dJ\alpha_{p,q}=(\sqrt{-1})^{q-p}d\alpha_{p,q}.$

Therefore,

$|d^{\Lambda}\alpha_{p,q}|=|J^{-1}d\ast J^{-1}\alpha_{p,q}|=|d\ast
J^{-1}\alpha_{p,q}|=|d\ast\alpha_{p,q}|=|d^{\ast}\alpha_{p,q}|,$

and

$|d^{\Lambda_{\ast}}\alpha_{p,q}|=|J^{-1}dJ\alpha_{p,q}|=|d\alpha_{p,q}|.$

Therefore, $d^{\Lambda}\alpha_{p,q}=d^{\Lambda_{\ast}}\alpha_{p,q}=0$ if only
if $d^{\ast}\alpha_{p,q}=d\alpha_{p,q}=0$. ∎

######  Proposition 3.3.

([8, Lemma 2.2]) If $\alpha_{p,q}\in\mathcal{H}^{p,q}_{(2);d}(X)$, the
$L(\alpha_{p,q})=\omega\wedge\alpha_{p,q}\in\mathcal{H}^{p+1,q+1}_{(2);d}(X)$.

###  3.2 Dolbeault harmonic $(p,q)$-forms

We denote by

$\mathcal{H}^{p,q}_{(2);\bullet}(X):=\\{\alpha\in\Omega^{p,q}_{(2)}(X):\Delta_{\bullet}\alpha=0\\}$

the space of $L^{2}$ $\Delta_{\bullet}$-harmonic forms of bi-degree $(p,q)$,
where $\bullet=\partial,\bar{\partial}$.

######  Lemma 3.4.

([31, Lemma3.3]) If an $L^{2}$ $(p,q)$-form $\alpha$ on $X$ is
$\Delta_{\partial}$\- (resp. $\Delta_{\bar{\partial}}$-) harmonic form, then
$\partial\alpha=0,\partial^{\ast}\alpha=0$ (resp.
$\bar{\partial}\alpha=0,\bar{\partial}^{\ast}\alpha=0$).

######  Theorem 3.5.

For any complete almost Kähler manifold of dimension $2n$ and for all $(p,q)$,
the following dualities hold:
(1) (Complex conjugation). We have equalities

$\ker{\Delta_{\partial}}\cap\ker{\Delta_{\bar{\partial}}}\cap\Omega^{p,q}_{(2)}(X)=\ker{\Delta_{\partial}}\cap\ker{\Delta_{\bar{\partial}}}\cap\Omega^{q,p}_{(2)}(X).$

(2) (Hodge duality). The Hodge $\ast$-operator induces isomorphisms

$\ast:\ker{\Delta_{\partial}}\cap\ker{\Delta_{\bar{\partial}}}\cap\Omega^{p,q}_{(2)}(X)\rightarrow\ker{\Delta_{\partial}}\cap\ker{\Delta_{\bar{\partial}}}\cap\Omega^{n-q,n-p}_{(2)}(X).$

(3) (Serre duality). There are isomorphisms:

$\ker{\Delta_{\partial}}\cap\ker{\Delta_{\bar{\partial}}}\cap\Omega^{p,q}_{(2)}(X)\cong\ker{\Delta_{\partial}}\cap\ker{\Delta_{\bar{\partial}}}\cap\Omega^{n-q,n-p}_{(2)}(X).$

###### Proof of Theorem 1.1.

For any almost Hermitian manifold of dimension $2n$ there are isomorphisms

$L^{n-k}:\Omega^{p,k-p}_{(2)}(X)\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\Omega^{p+n-k,n-p}_{(2)}(X)$

for every $0\leq p\leq n$ and all $p\leq k\leq n$. By [8, Corollary 3.5], we
get

$[L,\Delta_{\partial}+\Delta_{\bar{\partial}}]=0$

and

$[\Lambda,\Delta_{\partial}+\Delta_{\bar{\partial}}]=0,$

so $L$ and $\Lambda$ preserve
$\ker{\Delta_{\partial}}\cap\ker{\Delta_{\bar{\partial}}}$. It follows the
maps

$L^{n-k}:\ker{\Delta_{\partial}}\cap\ker{\Delta_{\bar{\partial}}}\cap\Omega^{p,q}_{(2)}(X)\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\ker{\Delta_{\partial}}\cap\ker{\Delta_{\bar{\partial}}}\cap\Omega^{n-p,n-q}_{(2)}(X)$

are well defined, and are injective, since they are isomorphisms before
restricting the domain. By Hodge duality of Theorem 3.5, the domain and
codomain have the same dimension, so the map is an isomorphism. ∎

######  Proposition 3.6.

If
$\alpha_{p,q}\in\ker{\Delta_{\bar{\partial}}}\cap\ker{\Delta_{\partial}}\cap\Omega^{p,q}_{(2)}(X)$,
the
$L(\alpha_{p,q})=\omega\wedge\alpha_{p,q}\in\ker{\Delta_{\bar{\partial}}}\cap\ker{\Delta_{\partial}}\cap\Omega^{p+1,q+1}_{(2)}(X)$.

###### Proof.

It follows from Theorem 1.1. ∎

##  4 Vanishing theorem for $L^{2}$-harmonic forms

###  4.1 Vanishingly exhaustible $k$-form

Let $(X,g)$ be a complete Riemannian manifold. Recall that a function
$E:X\rightarrow\mathbb{R}$ is called exhaustion function if

$X_{k}=\\{x\in X:E(x)<k\\}\subset X$

for any $k\in\mathbb{R}$ [35]. In this article, we only consider $C^{1}$
exhaustion function as follows.

######  Definition 4.1.

([35, Definition 1]) Let $f:\mathbb{R}\rightarrow\mathbb{R}^{+}$ be continuous
and $E$ be a $C^{1}$ exhaustion function. We say that a $k$-form $\theta$ on
$X$ is $f(E)$-bounded, if

$|\theta(x)|\leq f(E(x)),\ for\ all\ x\in X.$

Note that the distance function $\rho$ associated to the metric $g$ on a
complete manifold $X$ has the property that its differential $d\rho$ is
$f(E)$-bounded for $f\equiv constant$ and for any exhaustion function $E$.
Following the idea of McNeal in [35], we consider some smooth differential
forms as following.

######  Definition 4.2.

The smooth $k$-form $\omega$, ($k\geq 1$), on a complete Riemannian manifold
$X$ is vanishingly exhaustible if there exist $C^{1}$ exhaustion functions $E$
on $X$, continuous, nondecreasing functions
$f,g:\mathbb{R}\rightarrow\mathbb{R}^{+}$, and $C^{1}$ $(k-1)$-forms $\theta$,
on $X$ such that
(1) $\omega$ is bounded,
(2) $\omega=d\theta$,
(3) $\theta$ is $f(E)$-bounded, $dE$ is $g(E)$-bounded,
(4) the series

$\sum_{k=N}^{\infty}\frac{1}{f(k)g(k)}$

diverges.

######  Definition 4.3.

The complete almost Kähler manifold $(X,J,\omega)$ is $(L,c)$-vanishingly
exhaustible if there exist a sequence of vanishing exhaustible $2$-forms,
$\\{\omega_{1},\cdots,\omega_{L}\\}$ and a uniform positive constant $c$ such
that

$\sup|\omega-\sum_{i=1}^{L}\omega_{i}^{1,1}|\leq c,$

where $\omega_{i}^{1,1}$ is the $(1,1)$-part of $\omega_{i}$.

######  Remark 4.4.

(1) Let $(X,g)$ be a simply-connected $n$-dimensional complete Riemannian
manifold with sectional curvature bounded from above by a negative constant,
i.e.

$sec\leq-K$

for some $K>0$. We have the following classical fact pointed by Gromov [23]
(one also can see [3, Proposition 8.4] and [12, Lemma 3.2]).
For any bounded and closed $k$-form $\omega$ on $X$, where $k>1$, there exists
a bounded $(k-1)$-form $\theta$ on $X$ such that

$\omega=d\theta$

and

$\|\theta\|_{L^{\infty}(X)}\leq K^{-\frac{1}{2}}\|\omega\|_{L^{\infty}(X)}.$

Following above statement, we only need take $f(x)=g(x)=E(x)=1$.
(2) The author in [34] considered a complete Kähler manifold $(X,\omega)$
which given by a global potential, i.e.,
$\omega=\sqrt{-1}\partial\bar{\partial}\lambda=\frac{1}{2}dJd\lambda$ for some
smooth function $f$. The hypotheses of the function $f$ on Theorem 2.6 of [34]
follows from the Definition 4.3 by taking $f(x)=g(x)=\sqrt{A+Bx}$ and
$E(x)=\lambda(x)$. The hypotheses of [10, Theorem 2] similarly follows from
Definition 4.3 by taking $f(x)=c(1+x)$, $g(x)=1$, and $E(x)=\rho(x,x_{0})$
where $\rho(x,x_{0})$ denotes the Riemannian distance between $x$ and a fixed
base point $x_{0}\in X$.

###  4.2 Example

In this section, we let $(X,g)$ be a complete manifold of finite volume with
pinched negative sectional curvature, i.e., there are positive constants $a,b$
such that $-b^{2}\leq sec_{g}\leq-a^{2}$. We now recall some standard facts
about the topology and geometry of this manifold (see [20, 25, 43]). At First,
$X$ has a finite number of ends, it means that $X=X_{0}\cup E_{i}$, where
$X_{0}$ is a compact manifold with boundary $\partial X_{0}$, and the
boundaries of $E_{i}$, $\partial E_{i}$, are the components of $\partial
X_{0}$. For each end $E_{i}$, it is $C^{2}$-diffeomorphic to
$\mathbb{R}\times\partial E_{i}$, and the metric on $E_{i}$ is as follows:

$g=d^{2}\rho_{i}+h_{\rho_{i}},$

where $\rho_{i}$ is the Busemann function, $h_{\rho_{i}}$ is a family of
metrics on the compact manifold $\partial E_{i}$, and satisfies
$e^{-b\rho_{i}}h_{0}\leq h_{\rho_{i}}\leq e^{-a\rho_{i}}h_{0}$.

On a complete almost Kähler manifold of finite volume and with pinched
negative sectional curvature, the Kähler form $\omega$ cannot be $d$(bounded).
In fact, for any bounded nonzero harmonic smooth $k$-form $\alpha$, $k\geq 1$,
on a complete Riemannian manifold of finite volume, $\alpha$ cannot be
$d$(bounded). If not, there exists a $(k-1)$-form $\beta$ such that
$\alpha=d\beta$ and $\beta$ is bounded, then $d^{\ast}d\beta=0$. As the volume
is finite, one can see that $\beta\in L^{2}$. Therefore,
$0=(d^{\ast}d\beta,\beta)=\|d\beta\|^{2}$, i.e., $\alpha=d\beta=0$. However,
we have the following crucial lemma.

######  Proposition 4.5.

(cf. [43, Lemma 3.2]) Let $(X,J,\omega)$ be a complete almost Kähler manifold
of finite volume and with pinched negative curvature $-b^{2}\leq
sec\leq-a^{2}<0$. Then outside a compact subset, its Kähler form is
$d$(bounded). More specifically, there exist a bounded open subset $D\subset
X$ and a bounded and continuous $1$-form $\theta$ such that we have
$\omega=d\theta$ in the weak sense on $X\backslash D$.

###### Proof.

The proof is the same as [43, Lemma 3.2]. ∎

###  4.3 Key lemma

Now we will prove a lemma which extend the Stokes formula to complete manifold
under some conditions.

######  Lemma 4.6.

Let $X$ be a complete Riemannian manifold. Suppose that $\omega$ is a
vanishingly exhaustible $k$-form, $k\geq 1$. For any
$\alpha\in\Omega^{k+l}_{(2)}(X)\cap\ker d^{\ast}$, and
$\beta\in\Omega^{l}_{(2)}(X)\cap\ker d$, we have

$(\alpha,\omega\wedge\beta)=0.$

###### Proof.

Let $h:\mathbb{R}\rightarrow\mathbb{R}$ be smooth, $0\leq h\leq 1$,

$h(t)=\left\\{\begin{aligned} 1,&&t\geq 1\\\ 0,&&t\leq 0\end{aligned}\right.$

and consider the compactly supported function

$h_{k}(x)=h(k-E(x)),$

where $k$ is a positive integer. Note that ${\rm{supp}}h_{k}\subset X_{k}$ and
$h_{k}=1$ on $X_{k-1}$.

We denote $\omega:=d\theta$. Let $\gamma=\theta\wedge\beta$. Since
$h_{k}\cdot\gamma$ has compact support, and $d^{\ast}\alpha=0$, an integration
by parts gives

$\begin{split}(\alpha,d(h_{k}\cdot\gamma))&=(d^{\ast}\alpha,h_{k}\cdot\gamma)\\\
&=0.\\\ \end{split}$ (4.1)

Since $d\beta=0$, we also have

$\begin{split}d(h_{k}\cdot\gamma)=h^{\prime}(k-E)&\cdot
dE\wedge\theta\wedge\beta\\\ &+h_{k}\cdot\omega\wedge\beta.\\\ \end{split}$
(4.2)

We now substitute (4.2) into (4.1) and consider the two terms coming from the
right-hand side of (4.2) separately. For the first term, the fact that
$supph^{\prime}_{k}\subset X_{k}\backslash X_{k-1}$ and the fact that $\omega$
is bounded imply

$\begin{split}|(\alpha,h^{\prime}_{k}\cdot
dE\wedge\theta\wedge\beta)|&\leq\int_{X_{k}\backslash
X_{k-1}}|dE\wedge\theta|\cdot|\alpha|\cdot|\beta|\\\
&\leq\int_{X_{k}\backslash X_{k-1}}(f(E)g(E))\cdot|\alpha|\cdot|\beta|\\\
&\leq(f(k)\cdot g(k))\int_{X_{k}\backslash X_{k-1}}|\alpha|\cdot|\beta|,\\\
\end{split}$ (4.3)

for the functions $f,g$ in Definition 4.3. The second inequality follows from
our hypotheses on $E$ and $\theta$. The assumption that $\alpha,\beta\in
L^{2}$ implies that $|\alpha|\cdot|\beta|\in L^{1}$, then there exists a
subsequence $\\{k_{i}\\}$ such that

$(f(k_{i})\cdot g(k_{i}))\int_{X_{k_{i}}\backslash
X_{k_{i}-1}}|\alpha|\cdot|\beta|\rightarrow 0,\ as\ i\rightarrow\infty.$ (4.4)

Otherwise, for some $c>0$,

$\begin{split}\int_{X}|\alpha|\cdot|\beta|&=\sum_{k=1}^{\infty}\int_{X_{k}\backslash
X_{k-1}}|\alpha|\cdot|\beta|\\\ &\geq
c\sum_{k=1}^{\infty}\frac{1}{f(k)g(k)}\\\ &=\infty.\\\ \end{split}$

a contradiction.

So, for the sequence gives by (4.4), it follows from (4.3) that

$\lim_{i\rightarrow\infty}(\alpha,h^{\prime}_{k_{i}}\cdot
dE\wedge\theta\wedge\beta)=0.$ (4.5)

For the term coming from the second term on the right-hand side of (4.2), the
dominated convergences theorem implies

$\lim_{k\rightarrow\infty}(\alpha,h_{k}\cdot\omega\wedge\beta)=(\alpha,\omega\wedge\beta).$
(4.6)

Substituting (4.5) and (4.6) into (4.1), it follows that
$(\alpha,\omega\wedge\beta)=0$. ∎

######  Proposition 4.7.

If the $(1,1)$-form $\tilde{\omega}$ on a complete almost Kähler manifold $X$
is exact, then there exists a $1$-form $\theta$ such that

$\tilde{\omega}=\partial\theta^{0,1}+\bar{\partial}\theta^{1,0},$

where $\theta^{1,0}$ (resp. $\theta^{0,1}$) is the $(1,0)$ (resp.$(0,1)$) part
of $\theta$.

###### Proof.

By the hypothesis, one can see that

$\begin{split}\tilde{\omega}&=d\theta\\\
&=(\partial+\mu+\bar{\partial}+\bar{\mu})(\theta^{1,0}+\theta^{0,1})\\\
&=(\partial\theta^{1,0}+\mu\theta^{0,1})+(\partial\theta^{0,1}+\bar{\partial}\theta^{1,0})+(\bar{\partial}\theta^{0,1}+\bar{\mu}\theta^{1,0}).\\\
\end{split}$

Noting that $\tilde{\omega}$ is a $(1,1)$-form. We get

$\partial\theta^{1,0}+\mu\theta^{0,1}=\bar{\partial}\theta^{0,1}+\bar{\mu}\theta^{1,0}=0\
and\ \tilde{\omega}=\partial\theta^{0,1}+\bar{\partial}\theta^{1,0}.$

∎

Following Proposition 4.7 and Lemma 4.6, we then have

######  Lemma 4.8.

Let $(X,J,\omega)$ be a complete almost Käher manifold. Suppose that
$\tilde{\omega}$ is a vanishingly exhaustible $(1,1)$-form. For any
$\alpha\in\Omega^{p,q}_{(2)}(X)\cap\ker\partial^{\ast}\cap\ker\bar{\partial}^{\ast}$,
and $\beta\in\Omega^{p-1,q-1}_{(2)}(X)\cap\ker\partial\cap\ker\bar{\partial}$,
we have

$(\alpha,\tilde{\omega}\wedge\beta)=0.$

###### Proof.

Following Proposition 4.7, we have
$\tilde{\omega}=\partial\theta^{0,1}+\bar{\partial}\theta^{1,0}$. Let
$\gamma_{1}=\theta^{1,0}\wedge\beta$ and $\gamma_{2}=\theta^{0,1}\wedge\beta$.
We denote by $h_{k}$ the compactly supported function on Lemma 4.6. Noting
that

$(h^{\prime}(k-E)\cdot
dE\wedge\theta\wedge\beta)^{p,q}=h^{\prime}(k-E)\cdot\bar{\partial}E\wedge\theta^{1,0}\wedge\beta+h^{\prime}(k-E)\cdot\partial
E\wedge\theta^{0,1}\wedge\beta.$

Since $h_{k}\cdot\gamma$ has compact support, and
$\partial^{\ast}\alpha=\bar{\partial}^{\ast}\alpha=0$, an integration by parts
gives

$\begin{split}(\alpha,\bar{\partial}(h_{k}\cdot\gamma_{1})+\partial(h_{k}\cdot\gamma_{2}))&=(\alpha,h^{\prime}(k-E)\cdot\bar{\partial}E\wedge\theta^{1,0}\wedge\beta+h_{k}\cdot\bar{\partial}\theta^{1,0}\wedge\beta)\\\
&+(\alpha,h^{\prime}(k-E)\cdot\partial
E\wedge\theta^{0,1}\wedge\beta+h_{k}\cdot\partial\theta^{0,1}\wedge\beta)\\\
&=(\alpha,h^{\prime}(k-E)\cdot
dE\wedge\theta\wedge\beta)+(\alpha,h_{k}\cdot\tilde{\omega}\wedge\beta)\\\
&=(\bar{\partial}^{\ast}\alpha,h_{k}\cdot\gamma_{1})+(\partial^{\ast}\alpha,h_{k}\cdot\gamma_{2})\\\
&=0.\\\ \end{split}$ (4.7)

Following the idea in Lemma 4.6, there exists a subsequence $\\{k_{i}\\}$ such
that

$\lim_{i\rightarrow\infty}(\alpha,h^{\prime}_{k_{i}}\cdot
dE\wedge\theta\wedge\beta)=0.$ (4.8)

and

$\lim_{k\rightarrow\infty}(\alpha,h_{k}\cdot\tilde{\omega}\wedge\beta)=(\alpha,\tilde{\omega}\wedge\beta).$
(4.9)

Substituting (4.8) and (4.9) into (4.7), it follows that
$(\alpha,\tilde{\omega}\wedge\beta)=0$. ∎

###  4.4 Vanishing theorem

We begin to establish the vanishing theorem of $L^{2}$-harmonic forms on
complete almost Kähler manifold. At first, following Lemma 4.8, we have the
following estimate on $(\partial+\bar{\partial})$-harmonic forms.

######  Corollary 4.9.

Let $(X,J,\omega)$ be a complete almost Kähler manifold. Suppose that
$\\{\omega_{i}\\}$, $i=1,\cdots,L$ is a sequence of vanishing exhaustible
$2$-forms. Then for any
$\alpha\in\ker{\Delta_{\bar{\partial}}}\cap\ker{\Delta_{\partial}}\cap\Omega^{p,q}_{(2)}(X)$,
we have

$\|\omega\wedge\alpha\|\leq\|(\omega-\sum_{i=1}^{L}\omega^{1,1}_{i})\wedge\alpha\|.$

###### Proof.

Suppose that
$\alpha\in\ker{\Delta_{\bar{\partial}}}\cap\ker{\Delta_{\partial}}\cap\Omega^{p,q}_{(2)}(X)$.
By Proposition 3.6,
$\omega\wedge\alpha\in\ker{\Delta_{\bar{\partial}}}\cap\ker{\Delta_{\partial}}\cap\Omega^{p+1,q+1}_{(2)}(X)$
and so Lemma 3.1 implies that

$(\partial^{\ast}+\bar{\partial}^{\ast})(\omega\wedge\alpha)=0.$

Noting that $\omega_{i}^{(0,2)}\wedge\alpha\in\Omega^{p,q+2}(X)$,
$\omega_{i}^{(2,0)}\wedge\alpha\in\Omega^{p+2,q}(X)$ and
$\omega\wedge\alpha\in\Omega^{p+1,q+1}(X)$. Therefore, we have

$(\omega\wedge\alpha,\omega_{i}^{1,1}\wedge\alpha)=(\omega\wedge\alpha,\omega_{i}\wedge\alpha).$

Following Lemma 4.8, we obtain

$(\omega\wedge\alpha,\omega_{i}\wedge\alpha)=0.$ (4.10)

By (4.10), one can see that

$\begin{split}\|\omega\wedge\alpha\|^{2}&=(\omega\wedge\alpha,\sum_{i=1}^{L}\omega^{1,1}_{i}\wedge\alpha)+(\omega\wedge\alpha,(\omega-\sum_{i=1}^{L}\omega^{1,1}_{i})\wedge\alpha)\\\
&=(\omega\wedge\alpha,(\omega-\sum_{i=1}^{L}\omega^{1,1}_{i})\wedge\alpha).\\\
\end{split}$

Hence,

$\|\omega\wedge\alpha\|\leq\|(\omega-\sum_{i=1}^{L}\omega^{1,1}_{i})\wedge\alpha\|.$

∎

###### Proof of Theorem 1.3.

By the hypotheses, there exists a sequence of $(1,1)$-forms
$\\{\omega_{1},\cdots,\omega_{L}\\}$ such that

$\sup|\omega-\sum_{i=1}^{L}(\omega_{i})^{1,1}|\leq c.$

Support that $p+q<n$ and
$\alpha\in\ker{\Delta_{\bar{\partial}}}\cap\ker{\Delta_{\partial}}\cap\Omega^{p,q}_{(2)}(X)$.
By Corollary 4.9, we have

$\begin{split}\|\omega\wedge\alpha\|&\leq\|(\omega-\sum_{i=1}^{L}\omega^{1,1}_{i})\wedge\alpha)\|\\\
&\leq C_{1}(p,q)\sup|\omega-\sum_{i=1}^{L}\omega^{1,1}_{i}|\cdot\|\alpha\|.\\\
\end{split}$

Here we use the inequality

$\langle\alpha\wedge\beta,\alpha\wedge\beta\rangle\leq\binom{r+s}{r}\langle\alpha,\alpha\rangle\langle\beta,\beta\rangle,$

where $\alpha\in\Omega^{r}$ and $\beta\in\Omega^{s}$. Following [32, Corollary
1.2.28], for any $\alpha\in\Omega^{k}(X)$ we have

$[\Lambda,L]\alpha=(n-k)\alpha.$

Therefore,

$\begin{split}(\omega\wedge\alpha,\omega\wedge\alpha)&=([\Lambda,L]\alpha+\omega\wedge(\Lambda\alpha),\alpha)\\\
&=(n-k)\|\alpha\|^{2}+\|\Lambda\alpha\|^{2}.\\\ \end{split}$

We then have

$\|\alpha\|\leq C_{2}(p,q,n)\|\omega\wedge\alpha\|.$

Combining the preceding inequalities yields

$\|\alpha\|\leq
C_{1}C_{2}\sup|\omega-\sum_{i=1}^{L}\omega^{1,1}_{i}|\cdot\|\alpha\|\leq
cC_{1}C_{2}\|\alpha\|.$

We can choose $c$ small enough to such that $cC_{1}C_{2}<1$, then $\alpha=0$.
Finally, Poincaré duality extends the argument just given to the case when
$p+q>n$. ∎

##  5 The $L^{2}$-estimates

Throughout this section, we write $\alpha\lesssim\beta$ to mean that
$\alpha\leq C\beta$ for some positive constant $C$ independent of certain
parameters on which $\alpha$ and $\beta$ depend. The parameters on which $C$
is independent will be clear or specified at each occurrence. We also use
$\beta\lesssim\alpha$ and $\alpha\approx\beta$ analogously. We also denote by
$\Omega^{k}_{0}(X)$ (resp. $\Omega^{p,q}_{0}(X)$) the smooth $k$\- (resp.
$(p,q)$-) forms with compact support on $X$.

###  5.1 Case one

A differential form $\alpha$ in a Riemannian manifold $(X,g)$ is called
bounded with respect to the metric $g$ if the $L^{\infty}$-norm of $\alpha$ is
finite, namely,

$\|\alpha\|_{L^{\infty}(X)}=\sup_{x\in X}|\alpha(x)|<\infty.$

By definition, a $k$-form $\alpha$ is said to be $d$(bounded) if
$\alpha=d\beta$, where $\beta$ is a bounded $(k-1)$-form. It is obvious that
if $X$ is compact, then every exact form is $d$(bounded).

######  Proposition 5.1.

Let $(X,J,\omega)$ be a complete almost Kähler manifold. If $\theta$ is a
bounded $1$-form, then for any $\alpha\in\Omega^{k}_{0}(X)$, $(k<n)$,

$|((d\theta)^{1,1}\wedge\alpha,\omega\wedge\alpha)|\leq
c(n,k)\|\theta\|_{L^{\infty}(X)}\|\alpha\|_{L^{2}(X)}((\Delta_{\partial}+\Delta_{\bar{\partial}})\alpha,\alpha)^{\frac{1}{2}},$
(5.1)

where $c(n,k)$ is a positive constant which depends only $n,k$.

###### Proof.

An integration by parts gives

$\begin{split}((d\theta)^{1,1}\wedge\alpha,\omega\wedge\alpha)&=((\partial\theta^{0,1}+\bar{\partial}\theta^{1,0})\wedge\alpha,\omega\wedge\alpha)\\\
&=(\partial(\theta^{0,1}\wedge\alpha)+\theta^{0,1}\wedge\partial\alpha,\omega\wedge\alpha)\\\
&+(\bar{\partial}(\theta^{1,0}\wedge\alpha)+\theta^{1,0}\wedge\bar{\partial}\alpha,\omega\wedge\alpha)\\\
&=(\theta^{0,1}\wedge\alpha,[\partial^{\ast},L]\alpha)+(\theta^{0,1}\wedge\alpha,L(\partial^{\ast}\alpha))+(\theta^{0,1}\wedge\partial\alpha,\omega\wedge\alpha)\\\
&+(\theta^{1,0}\wedge\alpha,[\bar{\partial}^{\ast},L]\alpha)+(\theta^{1,0}\wedge\alpha,L(\bar{\partial}^{\ast}\alpha))+(\theta^{1,0}\wedge\bar{\partial}\alpha,\omega\wedge\alpha)\\\
&=(\theta^{0,1}\wedge\alpha,-\sqrt{-1}\bar{\partial}\alpha)+(\theta^{0,1}\wedge\alpha,L(\partial^{\ast}\alpha))+(\theta^{0,1}\wedge\partial\alpha,\omega\wedge\alpha)\\\
&+(\theta^{1,0}\wedge\alpha,\sqrt{-1}\partial\alpha)+(\theta^{1,0}\wedge\alpha,L(\bar{\partial}^{\ast}\alpha))+(\theta^{1,0}\wedge\bar{\partial}\alpha,\omega\wedge\alpha).\\\
\end{split}$

Here we used the almost Kähler identities
$[\partial^{\ast},L]=-\sqrt{-1}\bar{\partial}$,
$[\bar{\partial}^{\ast},L]=\sqrt{-1}\partial$. Therefore, we get

$\begin{split}&\quad((d\theta)^{1,1}\wedge\alpha,\omega\wedge\alpha)\\\
&\lesssim\|\theta^{0,1}\|_{L^{\infty}(X)}\|\alpha\|_{L^{2}(X)}\|\bar{\partial}\alpha\|_{L^{2}(X)}+\|\theta^{0,1}\|_{L^{\infty}(X)}\|\alpha\|_{L^{2}(X)}\|\partial^{\ast}\alpha\|_{L^{2}(X)}+\|\theta^{0,1}\|_{L^{\infty}(X)}\|\alpha\|_{L^{2}(X)}\|\partial\alpha\|_{L^{2}(X)}\\\
&+\|\theta^{1,0}\|_{L^{\infty}(X)}\|\alpha\|_{L^{2}(X)}\|\partial\alpha\|_{L^{2}(X)}+\|\theta^{1,0}\|_{L^{\infty}(X)}\|\alpha\|_{L^{2}(X)}\|\bar{\partial}^{\ast}\alpha\|_{L^{2}(X)}+\|\theta^{1,0}\|_{L^{\infty}(X)}\|\alpha\|_{L^{2}(X)}\|\bar{\partial}\alpha\|_{L^{2}(X)}\\\
&\leq
c(n,k)\|\theta\|_{L^{\infty}(X)}\|\alpha\|_{L^{2}(X)}((\Delta_{\partial}+\Delta_{\bar{\partial}})\alpha,\alpha)^{\frac{1}{2}}.\end{split}$

∎

######  Theorem 5.2.

Let $(X,J,\omega)$ be a complete almost Kähler manifold. Suppose that there
exists a bounded $1$-form $\theta$ such that

$\sup|\omega-(d\theta)^{1,1}|\leq c,$

then for any $\alpha\in\Omega^{k}_{0}(X)$, $(k\neq n)$,

$\|\alpha\|(1-c(n,k)c)\leq
c(n,k)\|\theta\|_{L^{\infty}(X)}((\Delta_{\partial}+\Delta_{\bar{\partial}})\alpha,\alpha)^{\frac{1}{2}}.$
(5.2)

Furthermore, if $c$ is small enough such $c(n,k)c<1$, then

$\ker(\Delta_{\partial}+\Delta_{\bar{\partial}})\cap\Omega^{k}_{(2)}(X)=\\{0\\}.$

###### Proof.

Suppose that $k<n$ and $\alpha\in\Omega^{k}_{0}(X)$. Following (5.1), we get

$\begin{split}\|\alpha\|^{2}&\lesssim\|\omega\wedge\alpha\|^{2}\\\
&=(\omega\wedge\alpha,(d\theta)^{1,1}\wedge\alpha)+(\omega\wedge\alpha,(\omega-(d\theta)^{1,1})\wedge\alpha)\\\
&\leq
c(n,k)c\|\alpha\|^{2}+c(n,k)\|\theta\|_{L^{\infty}(X)}\|\alpha\|_{L^{2}(X)}((\Delta_{\partial}+\Delta_{\bar{\partial}})\alpha,\alpha)^{\frac{1}{2}}.\\\
\end{split}$

Rearrangement gives (5.2). ∎

######  Corollary 5.3.

Let $(X,J,\omega)$ be a complete almost Kähler manifold. Suppose that there
exists a bounded $1$-form $\theta$ such that

$\sup|\omega-(d\theta)^{1,1}|\leq c,$

then for any $\alpha\in\Omega^{k}_{0}(X)$, $(k\neq n)$,

$\|\alpha\|^{2}(1-c(n,k))(1-c(n,k)c-4c(n,k)\|\theta\|_{L^{\infty}(X)}\sup|N_{J}|)\leq
c^{2}(n,k)\|\theta\|^{2}_{L^{\infty}(X)}(\Delta_{d}\alpha,\alpha).$ (5.3)

Furthermore, if $c(n,k)(c+4\|\theta\|_{L^{\infty}(X)}\sup|N_{J}|)<1$, then for
any $k\neq n$,

$\mathcal{H}^{k}_{(2);d}(X)=\\{0\\}.$

###### Proof.

Firstly, expanding $\Delta_{d}=[d,d^{\ast}]$ and using
$d=\partial+\mu+\bar{\partial}+\bar{\mu}$, we have

$\begin{split}\Delta_{d}&=\Delta_{\partial}+\Delta_{\bar{\partial}}+\Delta_{\mu}+\Delta_{\bar{\mu}}\\\
&+[\bar{\partial},\partial^{\ast}]+[\partial,\bar{\partial}^{\ast}]\\\
&+[\partial+\bar{\partial},\mu^{\ast}+\bar{\mu}^{\ast}]+[\mu+\bar{\mu},\partial^{\ast}+\bar{\partial}^{\ast}].\\\
\end{split}$

We observe that

$\begin{split}I&=(([\bar{\partial},\partial^{\ast}]+[\partial,\bar{\partial}^{\ast}])\alpha,\alpha)\\\
&=2([\mu^{\ast},\partial]\alpha,\alpha)+2([\bar{\mu},\bar{\partial}^{\ast}]\alpha,\alpha)\\\
&=2(\partial\alpha,\mu\alpha)+2(\mu^{\ast}\alpha,\partial^{\ast}\alpha)+2(\bar{\partial}^{\ast}\alpha,\alpha)+2(\bar{\mu}\alpha,\bar{\partial}\alpha).\end{split}$

Here we use the identities in [8, Proposition 3.3] as follows

$[\partial,\bar{\partial}^{\ast}]=[\bar{\mu}^{\ast},\bar{\partial}]+[\mu,\partial^{\ast}]\
and\
[\bar{\partial},\partial^{\ast}]=[\mu^{\ast},\partial]+[\bar{\mu},\bar{\partial}^{\ast}].$

Therefore, we have

$\begin{split}|I|&\leq
2\sup|N_{J}|\cdot\|\alpha\|(\|\partial\alpha\|+\|\partial^{\ast}\alpha\|+\|\bar{\partial}\alpha\|+\|\bar{\partial}^{\ast}\alpha\|)\\\
&\leq
2\sup|N_{J}|\cdot\|\alpha\|((\Delta_{\partial}+\Delta_{\bar{\partial}})\alpha,\alpha)^{\frac{1}{2}}.\\\
\end{split}$

We also observe that

$\begin{split}|II|&=|(([\partial+\bar{\partial},\mu^{\ast}+\bar{\mu}^{\ast}]+[\mu+\bar{\mu},\partial^{\ast}+\bar{\partial}^{\ast}])\alpha,\alpha)|\\\
&=2|((\mu^{\ast}+\bar{\mu}^{\ast})\alpha,(\partial^{\ast}+\bar{\partial}^{\ast})\alpha)+((\mu+\bar{\mu})\alpha,(\partial+\bar{\partial})\alpha)|\\\
&\leq
2\sup|N_{J}|\cdot\|\alpha\|(\|\partial\alpha\|+\|\partial^{\ast}\alpha\|+\|\bar{\partial}\alpha\|+\|\bar{\partial}^{\ast}\alpha\|)\\\
&\leq
2\sup|N_{J}|\cdot\|\alpha\|((\Delta_{\partial}+\Delta_{\bar{\partial}})\alpha,\alpha)^{\frac{1}{2}}.\\\
\end{split}$

Combining the preceding inequalities with estimate (5.2) yields

$\begin{split}(\Delta_{d}\alpha,\alpha)&\geq((\Delta_{\partial}+\Delta_{\bar{\partial}})\alpha,\alpha)-4\sup|N_{J}|\cdot\|\alpha\|((\Delta_{\partial}+\Delta_{\bar{\partial}})\alpha,\alpha)^{\frac{1}{2}}.\\\
&\geq((\Delta_{\partial}+\Delta_{\bar{\partial}})\alpha,\alpha)^{\frac{1}{2}}(c(n,k)^{-1}\|\theta\|^{-1}_{L^{\infty}(X)}(1-c(n,k)c)-4\sup|N_{J}|)\|\alpha\|\\\
&\geq
c(n,k)^{-1}\|\theta\|^{-1}_{L^{\infty}(X)}(1-c(n,k)c)(c(n,k)^{-1}\|\theta\|^{-1}_{L^{\infty}(X)}(1-c(n,k)c)-4\sup|N_{J}|)\|\alpha\|^{2}.\end{split}$

Rearrangement gives (5.3). ∎

Recall that by [8] (cf. also [41]) on complete almost-Kähler manifolds,

$\Delta_{\bar{\partial}}+\Delta_{\mu}=\Delta_{\partial}+\Delta_{\bar{\mu}}.$

######  Corollary 5.4.

Let $(X,J,\omega)$ be a complete almost Kähler manifold. Suppose that there
exists a bounded $1$-form $\theta$ such that

$\sup|\omega-(d\theta)^{1,1}|\leq c,$

then for any $\alpha\in\Omega^{k}_{0}(X)$, $(k\neq n)$,

$\|\alpha\|^{2}((1-c(n,k)c)^{2}-2c(n,k)^{2}\|\theta\|^{2}_{L^{\infty}(X)}\sup|N_{J}|^{2})\leq
2c^{2}(n,k)\|\theta\|^{2}_{L^{\infty}(X)}(\Delta_{\bullet}\alpha,\alpha),$
(5.4)

where $\bullet=\partial,\bar{\partial}$. Furthermore, if
$c(n,k)(c+\sqrt{2}\|\theta\|_{L^{\infty}(X)}\sup|N_{J}|)<1$, then for any
$k\neq n$,

$\mathcal{H}^{k}_{(2);\bullet}(X)=\\{0\\}.$

###### Proof.

For any $\alpha\in\Omega^{k}_{0}(X)$, we have

$\begin{split}((\Delta_{\mu}+\Delta_{\bar{\mu}})\alpha,\alpha)&=\|\mu\alpha\|^{2}+\|\bar{\mu}\alpha\|+\|\mu^{\ast}\alpha\|^{2}+\|\bar{\mu}^{\ast}\alpha\|\\\
&\leq 2\sup|N_{J}|^{2}\cdot\|\alpha\|^{2}.\\\ \end{split}$

We observe that

$\begin{split}&\Delta_{\partial}=\frac{1}{2}(\Delta_{\partial}+\Delta_{\bar{\partial}}+\Delta_{\mu}-\Delta_{\bar{\mu}}),\\\
&\Delta_{\bar{\partial}}=\frac{1}{2}(\Delta_{\partial}+\Delta_{\bar{\partial}}-\Delta_{\mu}+\Delta_{\bar{\mu}}).\\\
\end{split}$

Combining the preceding inequalities with estimate (5.2) yields

$\begin{split}(\Delta_{\bullet}\alpha,\alpha)&\geq\frac{1}{2}((\Delta_{\partial}+\Delta_{\bar{\partial}})\alpha,\alpha)-\sup|N_{J}|^{2}\cdot\|\alpha\|^{2}.\\\
&\geq(\frac{1}{2}(1-c(n,k)c)^{2}c(n,k)^{-2}\|\theta\|^{-2}_{L^{\infty}(X)}-\sup|N_{J}|^{2})\|\alpha\|^{2}.\end{split}$

Rearrangement gives (5.4). ∎

###  5.2 Case two

In almost Kählerian case, we could not to defined a almost Kähler manifold
$(X,J,\omega)$ which given by global potential function $f$ since $dJdf$
possible has the $(2,0)+(0,2)$ part.

######  Definition 5.5.

([30, Definition 3.2]) Let $(X,J,\omega)$ be a complete almost Kähler
manifold. If there is a function $f\in C^{2}(X)$ such that

$\tilde{\omega}:=\omega-(dJdf)^{1,1}$

is sufficiently small in $L^{\infty}$-norm, we call $(X,J,\omega)$ a complete
manifold given by a global perturbation potential.

######  Lemma 5.6.

For any smooth function on complete almost Kähler manifold $X$, we have

$\|([dJdf,\Lambda]\alpha,\alpha)\|\leq
2(\|\partial\alpha\|^{2}+\|\partial^{\ast}\alpha\|^{2}+\|\bar{\partial}\alpha\|^{2}+\|\bar{\partial}^{\ast}\alpha\|^{2})+C\sup|df|^{2}\cdot\|\alpha\|^{2},\
\alpha\in\Omega^{p,q}_{0}(X).$ (5.5)

###### Proof.

Noting that

$\begin{split}dJdf&=dJ(\partial f+\bar{\partial}f)\\\
&=(\partial+\mu+\bar{\partial}+\bar{\mu})(\sqrt{-1}\bar{\partial}f-\sqrt{-1}\partial
f)\\\
&=-\sqrt{-1}(\partial^{2}f-\mu\bar{\partial}f)-\sqrt{-1}(\bar{\partial}\partial
f-\partial\bar{\partial}f)-\sqrt{-1}(-\bar{\partial}^{2}f+\bar{\mu}\partial
f)\\\
&=2\sqrt{-1}\mu\bar{\partial}f+2\sqrt{-1}\partial\bar{\partial}f-2\sqrt{-1}\bar{\mu}\partial
f.\\\ \end{split}$

Here we use the identities in (2.1). For a $(p,q)$-form $\alpha$, we get

$\begin{split}([(dJdf),\Lambda]\alpha,\alpha)&=([(dJdf)^{1,1},\Lambda]\alpha,\alpha)\\\
&=([2\sqrt{-1}\partial\bar{\partial}f,\Lambda]\alpha,\alpha)\\\
&=(2\sqrt{-1}\partial\bar{\partial}f\wedge(\Lambda\alpha),\alpha)-(\Lambda(2\sqrt{-1}\partial\bar{\partial}f\wedge\alpha),\alpha).\\\
\end{split}$ (5.6)

We observe that

$\begin{split}I:&=(2\sqrt{-1}\partial\bar{\partial}f\wedge(\Lambda\alpha),\alpha)\\\
&=(2\sqrt{-1}\partial(\bar{\partial}f\wedge\Lambda\alpha),\alpha)+(2\sqrt{-1}\bar{\partial}f\wedge\partial(\Lambda\alpha),\alpha)\\\
&=(2\sqrt{-1}\partial(\bar{\partial}f\wedge\Lambda\alpha),\alpha)+(2\sqrt{-1}\bar{\partial}f\wedge[\partial,\Lambda]\alpha,\alpha)+(2\sqrt{-1}\bar{\partial}f\wedge\Lambda(\partial\alpha),\alpha)\\\
&=(2\sqrt{-1}(\bar{\partial}f\wedge\Lambda\alpha),\partial^{\ast}\alpha)+(-2\bar{\partial}f\wedge\bar{\partial}^{\ast}\alpha,\alpha)+(2\sqrt{-1}\bar{\partial}f\wedge\Lambda(\partial\alpha),\alpha).\\\
\end{split}$ (5.7)

Here we use the almost Kähler identity
$[\partial,\Lambda]=-\sqrt{-1}\bar{\partial}^{\ast}$. Therefore,

$\begin{split}|I|&\leq
C(n,k)\sup|df|\cdot\|\alpha\|\cdot(\|\partial^{\ast}\alpha\|+\|\bar{\partial}^{\ast}\alpha\|+\|\partial\alpha\|)\\\
&\leq(\|\partial^{\ast}\alpha\|^{2}+\|\bar{\partial}^{\ast}\alpha\|^{2}+\|\partial\alpha\|^{2})+C\sup|df|^{2}\cdot\|\alpha\|^{2}.\\\
\end{split}$ (5.8)

We also observe that

$\begin{split}II:&=-(\Lambda(2\sqrt{-1}\partial\bar{\partial}f\wedge\alpha),\alpha)\\\
&=-(2\sqrt{-1}\partial\bar{\partial}f\wedge\alpha,L\alpha)\\\
&=-(2\sqrt{-1}\partial(\bar{\partial}f\wedge\alpha)+2\sqrt{-1}\bar{\partial}f\wedge\partial\alpha,L\alpha)\\\
&=-(2\sqrt{-1}\bar{\partial}f\wedge\alpha,\partial^{\ast}(L\alpha))-(2\sqrt{-1}\bar{\partial}f\wedge\partial\alpha,L\alpha)\\\
&=-(2\sqrt{-1}\bar{\partial}f\wedge\alpha,[\partial^{\ast},L]\alpha)-(2\sqrt{-1}\bar{\partial}f\wedge\alpha,L(\partial^{\ast}\alpha))-(2\sqrt{-1}\bar{\partial}f\wedge\partial\alpha,L\alpha)\\\
&=(2\bar{\partial}f\wedge\alpha,\bar{\partial}\alpha)-(2\sqrt{-1}\bar{\partial}f\wedge\alpha,L(\partial^{\ast}\alpha))-(2\sqrt{-1}\bar{\partial}f\wedge\partial\alpha,L\alpha).\\\
\end{split}$ (5.9)

Here we use the almost Kähler identity
$[\partial^{\ast},L]=-\sqrt{-1}\bar{\partial}$. Therefore,

$\begin{split}|II|&\leq
C(n,k)\sup|df|\cdot\|\alpha\|\cdot(\|\partial^{\ast}\alpha\|+\|\partial\alpha\|+\|\bar{\partial}\alpha\|)\\\
&\leq(\|\partial^{\ast}\alpha\|^{2}+\|\partial\alpha\|^{2}+\|\bar{\partial}\alpha\|^{2})+C\sup|df|^{2}\cdot\|\alpha\|^{2}.\\\
\end{split}$ (5.10)

It’s easy to that (5.5) follows from (5.8) and (5.10). ∎

######  Definition 5.7.

([34] and [30, Definiton 3.4]) Let $f\in C^{2}(X)$ be a function on $X$,
$f\geq 1$. We say that $f$ dominates its gradient, or $f$ dominates $df$, if
there exist constants $A>0$ and $B\geq 0$ such that

$|df|^{2}(x)\leq A+Bf(x),\ \forall x\in X.$ (5.11)

###### Proof of Theorem 1.5.

Suppose now that $f$ dominates $df$. Replace $f$ by $\tilde{f}=tf+1$, $t>0$
and small, we may assume
(i) $\tilde{f}\geq 1$, $x\in X$,
(ii) $|d\tilde{f}|^{2}\leq B\tilde{f}(x)$, $x\in X$, where $B$ in (ii) above
is the constant appearing in Definition 5.7. Fix a $t$ such that the
conditions (i) and (ii) hold. For notational convenience, we will continue to
denote $\tilde{f}$ as just $f$.

Fix a form level $\Omega^{p,q}(X)$ with $p+q\neq n$. For $\varepsilon>0$ to be
determined, let $F=\varepsilon\sigma\log f$, where
$\sigma={\rm{sign}}(p+q-n)$. Note

$\begin{split}(dJdF)^{1,1}&=\sigma\varepsilon(\frac{(dJdf)^{1,1}}{f}-\frac{df\wedge
Jdf}{f^{2}})\\\ &=\sigma\varepsilon(\frac{\omega}{f}-\frac{df\wedge
Jdf}{f^{2}}+\frac{(dJdf)^{1,1}-\omega}{f})\\\
&=r^{\prime}+r^{\prime\prime}+r^{\prime\prime\prime},\\\ \end{split}$

where the last equality defines the $(1,1)$-forms $r^{\prime\prime}$ and
$r^{\prime\prime\prime}$. It follows from Proposition 2.3 that

$([\Xi(r^{\prime}),\Lambda]\alpha,\alpha)=\frac{\sigma\varepsilon}{f}([\omega,\Lambda]\alpha,\alpha)=\varepsilon|p+q-n|\int_{X}\frac{1}{f}|\alpha|^{2}.$
(5.12)

Let $e_{j}$ be associated to $df\wedge Jdf=2\sqrt{-1}\partial
f\wedge\bar{\partial}f\in\Omega^{1,1}(X)$ by Proposition 2.3. Since each
$|e_{j}|\leq Bf$, by (ii) above, we also obtain from Proposition 2.3

$\begin{split}([\Xi(r^{\prime\prime}),\Lambda]\alpha,\alpha)&=\varepsilon\sigma\big{(}\frac{1}{f^{2}}\big{(}\sum_{j\in
I}e_{j}+\sum_{j\in
J}e_{j}-\sum_{j=1}^{n}e_{j}\big{)}\alpha_{IJ}(x)dz^{I}\wedge
d\bar{z}^{J},\alpha\big{)}\\\
&\leq\varepsilon|p+q-n|\int_{X}\frac{B}{f}|\alpha|^{2}.\end{split}$ (5.13)

We also observe that

$\begin{split}([\Xi(r^{\prime\prime\prime}),\Lambda]\alpha,\alpha)&=\sigma\varepsilon([\frac{(dJdf)^{1,1}-\omega}{f},\Lambda]\alpha,\alpha)\\\
&\leq
C\varepsilon\sup|(dJdf)^{1,1}-\omega|\cdot\int_{X}\frac{1}{f}|\alpha|^{2}.\\\
\end{split}$ (5.14)

Noting that

$|dF|^{2}\leq\frac{\varepsilon^{2}|df|^{2}}{f^{2}}\leq\frac{\varepsilon^{2}B}{f}.$
(5.15)

Let $N=|p+q-n|$. Substituting (5.12)–(5.15) into (5.5), we obtain

$\varepsilon N\int_{X}[\frac{1}{f}-\frac{B}{f}-\frac{\varepsilon
C}{N}\frac{B}{f}-\frac{C\sup|\tilde{\omega}|}{N}\frac{1}{f}]|\alpha|^{2}\leq
2(\|\partial\alpha\|^{2}+\|\partial^{\ast}\alpha\|^{2}+\|\bar{\partial}\alpha\|^{2}+\|\bar{\partial}^{\ast}\alpha\|^{2}),\
\alpha\in\Omega^{p,q}_{0}(X).$ (5.16)

As $1-B-\frac{C}{N}\sup|\tilde{\omega}|>0$, choose $\varepsilon$ so that
$1-B-\frac{C}{N}\sup|\tilde{\omega}|-\frac{CB}{N}\varepsilon:=c>0$. It follows
from (5.16) that (1.2) holds with $\tilde{f}$ in place of $f$ when
$m=\frac{c\varepsilon}{2}$ and $M=0$. Recalling that $\tilde{f}=tf+1$, it
follows that (1.2) holds for $f$ with $m=\frac{c\varepsilon}{2}$ and
$M=\frac{1}{t}$, which completes the proof. ∎

Noting that

$\|\partial\alpha\|^{2}+\|\partial^{\ast}\alpha\|^{2}+\|\bar{\partial}\alpha\|^{2}+\|\bar{\partial}^{\ast}\alpha\|^{2}\leq\|d\alpha\|^{2}+\|d^{\ast}\alpha\|^{2},\alpha\in\Omega^{p,q}_{0}(X).$

If $f$ is not bounded on $X$, Theorem 1.5 does not imply

$\|d\alpha\|^{2}+\|d^{\ast}\alpha\|^{2}\geq L\|\alpha\|^{2},\ for\ all\
\alpha\in\Omega^{p,q}_{0}(X),p+q\neq n,$ (5.17)

for a positive constant $L$. However, when $(X,J,\omega)$ is actually
symplectic hyperbolic, i.e, $\omega=d\theta$ and $\theta$ is bounded, the
author extended Gromov’s idea [23] to symplectic case, then (5.17) does hold
(see [31, Theorem 3.15]). The constant he obtained was $L=L(n)/\sup|\theta|$,
for some constant $L(n)$ depending on $n=\dim_{\mathbb{C}}(X)$. In Kählerian
case, when $\omega$ is given by a global potential, Berndtsson obtained a
different proof of inequality (5.17), along with a good estimate on $L$ (see
[5]).

It is of some interest to know the size of $L$, the so-called “spectral gap”.
The next proposition gives two situations where we can obtain (5.17) with a
reasonable estimate on $L$.

######  Proposition 5.8.

(cf. [34, Proposition 2.2]) Let $(X,J,\omega)$ be a complete almost Kähler
manifold. Suppose that there exists a smooth function $f\geq 1$ on $X$. Also
assume that the function $f$ satisfies the convexity condition on $X$, i.e.,
for some $A,B\geq 0$, $|df|^{2}\leq A+Bf$. Set $N=|p+q-n|$ when $p+q\neq n$.
(i) If $|f|\leq M$ on $X$ for any constant $M$ and
$c:=1-B-\frac{C}{N}\sup|(dJdf)^{1,1}-\omega|>0$, then (5.17) holds with

$L=\frac{c^{2}N^{2}}{8BC}.$ (5.18)

(ii) If $B=0$ and $\sup|(dJdf)^{1,1}-\omega|\leq\frac{1}{4C}$, then (5.17)
holds with

$L=\frac{(N-\frac{3}{4})}{4AC}.$ (5.19)

###### Proof.

For (i), it follows directly from the proof of Theorem 1.5 that

$\|d\alpha\|^{2}+\|d^{\ast}\alpha\|^{2}\geq\frac{c\varepsilon
N}{2t}\frac{1}{M+\frac{1}{t}}\|\alpha\|^{2},\ \alpha\in\Omega^{p,q}_{0}(X),$

for any $\varepsilon>0$, such that
$1-B-\frac{C}{N}\sup|(dJdf)^{1,1}-\omega|-\frac{CB}{N}\varepsilon>0$ and for
any $t>0$ such that $t(t+1)\leq\frac{B}{A}$. Letting
$\varepsilon=\frac{1-B}{2B}\frac{N}{C}$ and making sure that
$t\leq\frac{1}{M}$ gives (5.18).

For (ii), we can choose $F=\frac{\sigma}{2AC}f$, where
$\sigma={\rm{sign}}(p+q-n)$, it follows from (5.5) that

$\begin{split}2(\|d\alpha\|^{2}+\|d^{\ast}\alpha\|^{2})+\frac{1}{4AC}\|\alpha\|^{2}&\geq\frac{\sigma}{2AC}([(dJdJ)^{1,1}-\omega,\Lambda]\alpha,\alpha)+\frac{\sigma}{2AC}([\omega,\Lambda]\alpha,\alpha)\\\
&\geq\frac{N}{2AC}\|\alpha\|^{2}-C\sup|(dJdf)^{1,1}-\omega|\cdot\frac{1}{2AC}\|\alpha\|^{2}.\\\
\end{split}$ (5.20)

Here we use the fact $([\omega,\Lambda]\alpha,\alpha)=(p+q-n)\|\alpha\|^{2}$,
therefore, (5.19) follows directly from (5.20). ∎

The inequalities (1.2) on differential forms have an important application in
the following problem:
The $L^{2}$-existence theorem and $L^{2}$-estimate of the Cartan-De Rham
equation

$d\beta=\alpha$

where $\alpha\in\Omega^{k}_{(2)}(X)$ is a given $(k+1)$-form satisfying

$d\alpha=0.$

######  Proposition 5.9.

(cf. [34, Proposition 2.4]) Assume the hypotheses of Theorem 1.5. Suppose that
$f$ dominates $df$ and that the constant
$c:=1-B-\frac{C}{N}\sup|(dJdf)^{1,1}-\omega|>0$. Then for any
$\alpha\in\Omega^{p,q}(X)$ with $p+q\neq n$ such that (i) $d\alpha=0$ and (ii)
$f\alpha\in\Omega^{p,q}_{(2)}(X)$, there exists a solution to $d\beta=\alpha$
which satisfies the estimate

$\|\beta\|^{2}\leq C\int_{X}|\alpha|^{2}\cdot(f+M),$

where the positive constant $C$ depends only on $A,B$.

###### Proof.

Note that $|\alpha|^{2}\leq f|\alpha|^{2}\leq f^{2}|\alpha|^{2}$ since $f\geq
1$. Hence

$\int_{X}|\alpha|^{2}\leq\int_{X}f|\alpha|^{2}\leq\int_{X}f^{2}|\alpha|^{2}.$

Our proof here use McNeal’s argument in [34] for the
$\bar{\partial}$-equation. Let
$N=\\{\alpha\in\Omega^{k}_{(2)}(X):d\alpha=0\\}$ and
$S=\\{d^{\ast}\beta:\beta\in\Omega^{k}_{0}\cap N\\}$. On $S$ consider the
linear functional

$d^{\ast}\beta\rightarrow(\beta,u).$

Using (1.2), we obtain

$\begin{split}|(\gamma,\alpha)|&=\big{|}(\frac{1}{\sqrt{f+M}}\gamma,\sqrt{f+M}\alpha)\big{|}\\\
&\leq\big{(}\int_{X}\frac{1}{f+M}|\gamma|^{2}\big{)}^{\frac{1}{2}}\cdot\big{(}\int_{X}(f+M)|\alpha|^{2}\big{)}^{\frac{1}{2}}\\\
&\lesssim\|d^{\ast}\gamma\|\big{(}\int_{X}(f+M)|\alpha|^{2}\big{)}^{\frac{1}{2}}.\end{split}$
(5.21)

Thus the functional is bounded on $S$. However we also have $(\gamma,u)=0$ if
$\gamma\in S^{\bot}$ since $d\alpha=0$, so (5.21) actually holds for all
$\beta\in\Omega^{k}_{0}(X)$. Since $\Omega^{k}_{0}(X)$ is dense in

$Dom(d^{\ast}):=\\{u\in\Omega^{k}_{(2)}(X):d^{\ast}u\in\Omega^{k-1}_{(2)}(X)\\}$

in the norm $\|u\|^{2}+\|d^{\ast}u\|^{2}$, (5.21) holds for all $\gamma\in
Dom(d^{\ast})$. The Hahn-Banach theorem extends the functional to all of
$\Omega^{k}_{(2)}(X)$ and then the Riesz representation theorem gives a
$\beta\in\Lambda^{k-1}_{(2)}(X)$ such that

$(d^{\ast}\gamma,\beta)=(\gamma,\alpha),\forall\gamma\in Dom(d^{\ast}).$

This is equivalent to $d\beta=\alpha$, and

$\|\beta\|\lesssim\big{(}\int_{X}|\alpha|^{2}\cdot(f+M)\big{)}^{\frac{1}{2}},$

which is the claimed norm estimate. ∎

##  6 Application

###  6.1 $L^{2}$-Hodge number

We assume throughout this subsection that $(X,J,\omega)$ is a compact almost
Kähler $2n$-dimensional manifold with a Hermitian metric $g$, and
$\pi:(\tilde{X},\tilde{J},\tilde{\omega})\rightarrow(X,J,\omega)$ its
universal covering with $\Gamma$ as an isometric group of deck
transformations. Denote by $\mathcal{H}^{k}_{(2);d}(\tilde{X})$ the spaces of
$L^{2}$-harmonic $k$-forms on $\Omega^{k}_{(2)}(\tilde{X})$, where
$\Omega^{k}_{(2)}(\tilde{X})$ is space of the squared integrable $k$-forms on
$(\tilde{X},\tilde{J},\tilde{\omega})$, and denote by
$\dim_{\Gamma}\mathcal{H}^{k}_{(2);d}(\tilde{X})$ the Von Neumann dimension of
$\mathcal{H}^{k}_{(2);d}(\tilde{X})$ with respect to $\Gamma$ [2, 36]. We
denote by $h_{(2)}^{k}(X)$ the $L^{2}$-Hodge numbers of $X$, which are defined
to be

$h_{(2)}^{k}(X):=\dim_{\Gamma}\mathcal{H}_{(2)}^{k}(\tilde{X}),\ (0\leq k\leq
2n).$

It turns out that $h^{k}_{(2)}(X)$ are independent of the Hermitian metric $g$
and depend only on $X$ and $J$. We recall the following two basic facts.

######  Lemma 6.1.

$\dim_{\Gamma}\mathcal{H}_{(2);d}^{k}(X)=0\Leftrightarrow\mathcal{H}_{(2);d}^{k}(X)=\\{0\\},$

and $\dim_{\Gamma}\mathcal{H}$ is additive: Given

$0\rightarrow\mathcal{H}_{1}\rightarrow\mathcal{H}_{2}\rightarrow\mathcal{H}_{3}\rightarrow
0,$

one have

$\dim_{\Gamma}\mathcal{H}_{2}=\dim_{\Gamma}\mathcal{H}_{1}+\dim_{\Gamma}\mathcal{H}_{3}.$

By the $L^{2}$-index theorem of Atiyah [2], we have the following crucial
identities between $\chi(X)$ and the $L^{2}$-Hodge numbers $h_{(2)}^{k}(X)$:

$\chi(X)=\sum_{k=0}^{2n}(-1)^{k}h_{(2)}^{k}(X).$

###  6.2 Small Nijenhuis tensor

Let $E_{1}$ and $E_{2}$ be $C^{\infty}$-vector bundles over a smooth manifold
$X$, and $\mathcal{D}:C^{\infty}(E_{1})\rightarrow C^{\infty}(E_{2})$ be a
differential operator between $C^{\infty}$-sections of these bundle. We also
suppose that $X$ is a smooth Riemannian manifold and $\Gamma$ is a discrete
group of isometrics of $X$, such that the differential operator $\mathcal{D}$
commutes with the action of $\Gamma$. We consider a $\Gamma$-invariant
Hermitian line bundle $(L,\nabla)$ on $X$ we assume $X/\Gamma$ is compact, and
we state Atiyah’s $L^{2}$-index theorem for $\mathcal{D}\otimes\nabla$.

######  Theorem 6.2.

[23, Theorem 2.3.A] Let $\mathcal{D}$ be a first-order elliptic operator. Then
there exists a closed nonhomogeneous form

$I_{D}=I^{0}+I^{1}+\cdots+I^{n}\in\Omega^{\ast}(X)=\Omega^{0}\oplus\Omega^{1}\oplus\cdots\oplus\Omega^{n}$

invariant under $\Gamma$, such that the $L^{2}$-index of the twisted operator
$\mathcal{D}\otimes\nabla$ satisfies

$L^{2}Index_{\Gamma}(\mathcal{D}\otimes\nabla)=\int_{X/\Gamma}I_{\mathcal{D}}\wedge\exp{[\omega]},$

where $[\omega]$ is the Chern form of $\nabla$, and

$\exp{[\omega]}=1+[\omega]+\frac{[\omega]\wedge[\omega]}{2!}+\frac{[\omega]\wedge[\omega]\wedge[\omega]}{3!}+\cdots.$

######  Remark 6.3.

(1) $L^{2}Index_{\Gamma}(\mathcal{D}\otimes\nabla)\neq 0$ implies that either
$\mathcal{D}\otimes\nabla$ or its adjoint has a non-trivial $L^{2}$-kernel.
(2) The operator $\mathcal{D}$ used in our article is $d+d^{\ast}$. In this
case the $I^{0}$-component of $I_{\mathcal{D}}$ is nonzero. Hence
$\int_{X/\Gamma}I_{\mathcal{D}}\wedge\exp{\alpha[\omega]}$, for almost all
$\alpha$, provided the curvature form $\omega$ is “homologically nonsingular
$\int_{X/\Gamma}\omega^{n}\neq 0$, for $n=\dim_{\mathbb{C}}X$.

Gromov defined the lower spectral bound
$\lambda_{0}=\lambda_{0}(\mathcal{D})\geq 0$ as the upper bound of the
negative numbers $\lambda$, such that

$\|\mathcal{D}s\|_{L^{2}}\geq\lambda\|s\|_{L^{2}}$

for those sections $e$ of $E$ where $\mathcal{D}s$ in $L^{2}$. Let
$\mathcal{D}$ be a $\Gamma$-invariant elliptic operator on $X$ of the first
order, and let $I_{D}=I^{0}+I^{1}+\cdots+I^{n}\in\Omega^{\ast}(X)$ be the
corresponding index form on $X$. Let $\omega$ be a closed $\Gamma$-invariant
$2$-form on $X$ and denote by $I_{\alpha}^{n}$ the top component of product
$I_{\mathcal{D}}\wedge\exp{\alpha\omega}$, for $\alpha\in\mathbb{R}$. Hence
$I_{\alpha}^{n}$ is an $\Gamma$-invariant $n$-form on $X$, $\dim X=n$
depending on parameter $\alpha$.

######  Theorem 6.4.

([23, 2.4.A. Theorem]) Let $H^{1}_{dR}(X)=0$ and let $X/\Gamma$ be compact and
$\int_{X/\Gamma}I_{\alpha}^{n}\neq 0$, for some $\alpha\in\mathbb{R}$. If the
form $\omega$ is $d$(bounded), then either $\lambda_{0}(\mathcal{D})=0$ or
$\lambda_{0}(\mathcal{D}^{\ast})=0$, where $\mathcal{D}^{\ast}$ is the adjoint
operator.

Let $(X,J,\omega)$ be a compact almost Kähler manifold, with exact symplectic
form $\tilde{\omega}=d\theta$ on $\tilde{X}$. Let $\Gamma=\pi_{1}(X)$. For
each $\varepsilon$, $\nabla_{\varepsilon}=d+\sqrt{-1}\varepsilon\theta$ is a
unitary connection on the trivial line bundle $L=\tilde{X}\times\mathbb{C}$.
One can try made it $\Gamma$-invariant by changing to a non-trivial action of
$\Gamma$ on $\tilde{X}\times\mathbb{C}$, i.e., setting, for $\gamma\in\Gamma$,

$\gamma^{\ast}(\tilde{x},z)=(\gamma\tilde{x},\exp^{\sqrt{-1}u(\gamma,\tilde{x})}z).$

We want $\gamma^{\ast}\nabla_{\varepsilon}=\nabla_{\varepsilon}$, i.e.,
$du=-(\gamma^{\ast}\theta-\theta)$. Since
$d(\gamma^{\ast}\theta-\theta)=\gamma^{\ast}\tilde{\omega}-\tilde{\omega}=0$,
there always exists a solution $u(\gamma,\cdot)$, well defined up a constant.

However, one cannot adjust the constant $\varepsilon\omega$ to obtain an
action (if so, one would get a line bundle on $X$ with curvature
$\varepsilon\omega$ and first Chern class $\frac{\varepsilon}{2\pi}[\omega]$).
This means that the action only defined on a central extension, we call this
projective representation (see [36, Charp 9]).

######  Definition 6.5.

([36, Definition 9.2]) Let $G_{\varepsilon}$ be the subgroup of
${\textit{Diff}}(\tilde{X}\times\mathbb{C})$ formed by maps $g$ which are
linear unitary on fibers, preserve the connection $\nabla_{\varepsilon}$ and
cover an element of $\Gamma$.

Let $L\rightarrow X$ be a vector bundle equipped with a Hermitian metric and
Hermitian connection $\nabla$. Then there is an induced exterior differential
$d^{\nabla}$ on $\Omega^{\ast}(X)\otimes L$. If
$\mathcal{D}=d^{\nabla}+(d^{\ast})^{\nabla}$, then Atiyah-Singer’s index
theorem states

$Index(\mathcal{D})=\int_{X}\mathcal{L}_{X}\wedge Ch(L).$

Here $\mathcal{L}_{X}$ is Hizebruch’s class,

$\mathcal{L}_{X}=1+\cdots+e(X)$

where $1\in H^{0}(X)$ and $e(X)\in H^{\dim X}(X)$ is the Euler class. For each
$\varepsilon$, $\nabla_{\varepsilon}=d+\sqrt{-1}\varepsilon\theta$ is a
unitary connection on the trivial line bundle $L=\tilde{X}\times\mathbb{C}$.
The operator
$\mathcal{D}_{\varepsilon}:=d^{\nabla_{\varepsilon}}+(d^{\ast})^{\nabla_{\varepsilon}}$
can be view as a $G_{\varepsilon}$ operator on the Hilbert space $H$ of $U(1)$
equivalent basis $L^{2}$ differential forms on $\tilde{X}\times U(1)$ [36].

######  Theorem 6.6.

([36, Theorem 9.3]) The operator $\tilde{D}_{\varepsilon}$ has a finite
projective $L^{2}$ index give by

$L^{2}Index_{G_{\varepsilon}}(\tilde{D}_{\varepsilon})=\int_{X}\mathcal{L}_{X}\wedge\exp(\frac{\varepsilon}{2\pi}[\omega]).$

###### Proof of Theorem 1.6.

Noting that $|\tilde{\omega}|^{2}=n$. Then there exists a $1$-form $\theta$
such that

$\tilde{\omega}=d\theta\ and\ \|\theta\|_{L^{\infty(\tilde{X})}}\leq
K^{-\frac{1}{2}}\sqrt{n}.$

Following the second $L^{2}$-estimate in Theorem 1.4, for any
$\alpha\in\Omega^{k}_{0}(\tilde{X})$, $(k\neq n)$, we then have

$\|\alpha\|^{2}(1-4c(n)\sqrt{n}K^{-\frac{1}{2}}\sup|N_{J}|))\leq
c^{2}(n)nK^{-1}(\Delta_{d}\alpha,\alpha).$

Provided $4c(n)\sqrt{n}K^{-\frac{1}{2}}\sup|N_{J}|\leq\frac{1}{2}$, i.e.,

$\sup|N_{J}|\leq\frac{1}{4c(n)\sqrt{n}}K^{\frac{1}{2}},$

then

$(\Delta_{d}\alpha,\alpha)\geq\lambda\|\alpha\|^{2}_{L^{2}(\tilde{X})},$

where $\lambda=\lambda(n,K)$ is a positive constant. Following Theorem 6.6,
the number $L^{2}Index_{G_{\varepsilon}}(\tilde{D}_{\varepsilon})$ is a
polynomial in $\varepsilon$ whose highest degree term is
$\int_{X}(\frac{\omega}{2\pi})^{n}\neq 0$ thus for $\varepsilon$ small enough,
$\tilde{D}_{\varepsilon}$ has a non-zero $L^{2}$ kernel. By construction,
$\tilde{D}_{\varepsilon}$ is an $\varepsilon$-small perturbation of
$d+d^{\ast}$, so

$d+d^{\ast}:\Omega^{even}_{(2)}(\tilde{X})\rightarrow\Omega^{odd}_{(2)}(\tilde{X})$

is not invertible (see Theorem 6.4). It implies that either

$\ker(d+d^{\ast})\cap\Omega^{even}_{(2)}(\tilde{X})=\bigoplus_{k=even}\mathcal{H}^{k}_{(2);d}(\tilde{X})\neq\\{0\\}$

or

${\rm{coker}}(d+d^{\ast})\cap\Omega^{odd}_{(2)}(\tilde{X})=\bigoplus_{k=odd}\mathcal{H}^{k}_{(2);d}(\tilde{X})\neq\\{0\\}.$

For any $k\neq n$,

$\mathcal{H}^{k}_{(2);d}(\tilde{X})=\\{0\\},$

i.e., $h_{(2)}^{k}(X)=0$. Therefore, we get

$\mathcal{H}^{n}_{(2);d}(\tilde{X})\neq\\{0\\},$

i.e, $h_{(2)}^{n}(X)>0$. Hence

$\begin{split}(-1)^{n}\chi(X)&=(-1)^{n}\sum_{k=0}^{2n}(-1)^{k}h^{k}_{(2)}(X)\\\
&=h^{n}_{(2)}(X)>0.\end{split}$

∎

###  6.3 Volume of compact negatively curved almost Kähler manifold

######  Lemma 6.7.

Let $(X,J,\omega)$ be a compact $2n$-dimensional almost Kähler manifold with
negative sectional curvature. Let
$\pi:(\tilde{X},\tilde{J},\tilde{\omega})\rightarrow(X,J,\omega)$ the
universal covering map for $X$. Then for any
$\alpha\in\Omega^{k}_{0}(\tilde{X})$, we have

$\|\tilde{\mu}\alpha\|_{L^{2}(\tilde{X})}^{2}\leq
C_{n}\|\mu\|^{2}_{L^{2n}(X)}\|\nabla\alpha\|^{2}_{L^{2}(\tilde{X})}.$ (6.1)

where $\tilde{\mu}$ is the $0$-order differential operator on covering space
$\tilde{X}$. Furthermore, suppose that $X$ has pinched negative sectional
curvature, i.e., there are constants $K$ and $\delta\geq 1$ such that

$-\delta K\leq sec\leq-K.$

Then

$\|\tilde{\mu}\alpha\|_{L^{2}(\tilde{X})}^{2}\leq c_{2}(n)\delta
K[Vol(X)]^{\frac{1}{n}}C_{n}((\Delta_{d}\alpha,\alpha)_{L^{2}(\tilde{X})}+c_{1}(n)\delta
K\|\alpha\|_{L^{2}(\tilde{X})}^{2}).$ (6.2)

###### Proof.

We call an open set $U\subset\tilde{X}$ a fundamental domain of the action of
$\tilde{\Gamma}$ on $\tilde{X}$ if the following conditions are satisfied:
(1) $\tilde{X}=\cup_{\gamma\in\Gamma}\gamma(\bar{U})$,
(2) $\gamma_{1}(U)\cap\gamma_{2}(U)=\emptyset$ for
$\gamma_{1},\gamma_{2}\in\Gamma$, $\gamma_{1}\neq\gamma_{2}$ and
(3) $\bar{U}\backslash U$ has zero measure.
For notational convenience, we will continue to denote $\mu,\bar{\mu}$ as just
the $0$-order differential operator on covering space $\tilde{X}$. Hence for
any $\alpha\in\Omega^{k}_{0}(\tilde{X})$ we have

$\begin{split}\|\mu\alpha\|_{L^{2}(\tilde{X})}^{2}&=\sum_{\gamma\in\Gamma}\int_{\gamma(\bar{U})}|\mu\alpha|^{2}\\\
&\leq\sum_{\gamma\in\Gamma}(\int_{\gamma(\bar{U})}|\mu|^{2n})^{\frac{1}{n}}(\int_{\gamma(\bar{U})}|\alpha|^{\frac{2n}{n-1}})^{\frac{n-1}{n}}.\end{split}$
(6.3)

Here we use the Hölder inequality.

Since $\tilde{X}$ is a complete simply connected manifold with negative
sectional curvature, by Sobolev inequality in [24, Theorem 8.3], there exists
positive constant $C_{n}>0$, such that for any $u\in C^{\infty}_{0}(X)$,

$(\int_{\tilde{X}}|u|^{\frac{2n}{n-1}})^{\frac{n-1}{n}}\leq
C_{n}\int_{\tilde{X}}|\nabla u|^{2},$ (6.4)

Furthermore, following Croke’s result ([13] and [24, Theorem 8.3]), for any
$u\in C^{\infty}_{0}(X)$,

$\int_{\tilde{X}}|u|^{\frac{2n}{2n-1}}\leq\tilde{C}^{\frac{1}{2n}}_{n}\int_{\tilde{X}}|\nabla
u|.$ (6.5)

Here the constant

$\tilde{C}_{n}=\frac{[Vol(S^{2n-2})]^{2n-2}}{[Vol(S^{2n-1})]^{2n-1}}(\int_{0}^{\frac{\pi}{2}}\cos^{\frac{n}{n-1}}(t)\sin^{2n-2}(t)dt)^{2n-2},$

and $Vol(S^{n})$ denote the volumeof the standard unit sphere $S^{n}$. We can
denote $C_{n}=(\frac{2n-1}{n-1})^{2}\tilde{C}^{\frac{1}{n}}_{n}$.

Noting that $\gamma^{\ast}\mu=\mu$ and $\pi$ is local isometry, one can see
that

$\int_{\gamma(\bar{U})}|\mu|^{2n}=\int_{\gamma(\bar{U})}|\gamma^{\ast}\mu|^{2n}=\int_{\bar{U}}|\mu|^{2n}=\int_{X}|\mu|^{2n}.$

We then have

$\begin{split}\|\mu\alpha\|_{L^{2}(\tilde{X})}^{2}&\leq(\int_{X}|\mu|^{2n})^{\frac{1}{n}}\sum_{\gamma\in\Gamma}(\int_{\gamma(\bar{U})}|\alpha|^{\frac{2n}{n-1}})^{\frac{n-1}{n}}\\\
&\leq(\int_{X}|\mu|^{2n})^{\frac{1}{n}}\int_{\tilde{X}}|\alpha|^{\frac{2n}{n-1}})^{\frac{n-1}{n}}\\\
&\leq
C_{n}\|\mu\|^{2}_{L^{2n}(X)}\|\nabla\alpha\|^{2}_{L^{2}(\tilde{X})}.\end{split}$

Here we used the Sobolev embedding (6.4). We recall the general Weitzenböck
formula for differential forms:

$\Delta_{d}\alpha=\nabla^{\ast}\nabla\alpha+Riem\circ\alpha.$ (6.6)

Here Riem denotes the Riemann curvature tensor. Since $\pi$ is local isometry,
the Riemann curvatures of $\tilde{X}$ also have bounded. For notational
convenience, we will continue to denote Riem as just the Riemann curvature
tensor on $\tilde{X}$. Following [6, Lemma 3.7], it’s easy to see

$\sup_{x\in\tilde{X}}|Riem(x)|\leq c_{1}(n)\delta K.$

We also have (see Proposition 3.6)

$\sup_{x\in X}|\mu(x)|^{2}\leq c_{2}(n)\delta K.$

Taking $L^{2}$-inner product of (6.6) with $\alpha$ and integrating by parts,
we then have

$\begin{split}\|\nabla\alpha\|_{L^{2}(\tilde{X})}^{2}&\leq(\Delta_{d}\alpha,\alpha)_{L^{2}(\tilde{X})}+\sup|Riem|\cdot\|\alpha\|_{L^{2}(\tilde{X})}^{2}\\\
&\leq(\Delta_{d}\alpha,\alpha)_{L^{2}(\tilde{X})}+c_{1}(n)\delta
K\|\alpha\|_{L^{2}(\tilde{X})}^{2}.\\\ \end{split}$

Combining the preceding inequalities with estimate (6.1) yields

$\begin{split}\|\mu\alpha\|_{L^{2}(\tilde{X})}^{2}&\leq c_{2}(n)\delta
K[Vol(X)]^{\frac{1}{n}}C_{n}\|\nabla\alpha\|^{2}_{L^{2}(\tilde{X})}\\\ &\leq
c_{2}(n)\delta
K[Vol(X)]^{\frac{1}{n}}C_{n}((\Delta_{d}\alpha,\alpha)_{L^{2}(\tilde{X})}+c_{1}(n)\delta
K\|\alpha\|_{L^{2}(\tilde{X})}^{2}).\\\ \end{split}$

For $\bar{\mu}\alpha$, $\mu^{\ast}\alpha$ and $\bar{\mu}^{\ast}\alpha$, we
also have the similarly estimates. ∎

###### Proof of Theorem 1.7.

First, following the estimates in Corollary 5.3, we have

$\begin{split}|I|&=|2(\partial\alpha,\mu\alpha)+2(\mu^{\ast}\alpha,\partial^{\ast}\alpha)+2(\bar{\partial}^{\ast}\alpha,\bar{\mu}^{\ast}\alpha)+2(\bar{\mu}\alpha,\bar{\partial}\alpha)|\\\
&\leq\varepsilon((\Delta_{\partial}+\Delta_{\bar{\partial}})\alpha,\alpha)_{L^{2}(\tilde{X})}\\\
&+\frac{1}{\varepsilon}(\|\mu\alpha\|_{L^{2}(\tilde{X})}^{2}+\|\mu^{\ast}\alpha\|_{L^{2}(\tilde{X})}^{2}+\|\bar{\mu}\alpha\|_{L^{2}(\tilde{X})}^{2}+\|\bar{\mu}^{\ast}\alpha\|_{L^{2}(\tilde{X})}^{2}),\\\
\end{split}$ (6.7)
$\begin{split}|II|&=|(([\partial+\bar{\partial},\mu^{\ast}+\bar{\mu}^{\ast}]+[\mu+\bar{\mu},\partial^{\ast}+\bar{\partial}^{\ast}])\alpha,\alpha)|\\\
&\leq\varepsilon((\Delta_{\partial}+\Delta_{\bar{\partial}})\alpha,\alpha)_{L^{2}(\tilde{X})}\\\
&+\frac{1}{\varepsilon}(\|\mu\alpha\|_{L^{2}(\tilde{X})}^{2}+\|\mu^{\ast}\alpha\|_{L^{2}(\tilde{X})}^{2}+\|\bar{\mu}\alpha\|_{L^{2}(\tilde{X})}^{2}+\|\bar{\mu}^{\ast}\alpha\|_{L^{2}(\tilde{X})}^{2}),\\\
\end{split}$ (6.8)

where $\varepsilon\in(0,\infty)$ is a positive constant.

We take $\varepsilon=\frac{1}{3}$. Combining the preceding inequalities with
estimate (5.2) yields

$\begin{split}&\quad(\Delta_{d}\alpha,\alpha)_{L^{2}(\tilde{X})}\\\
&\geq((\Delta_{\partial}+\Delta_{\bar{\partial}})\alpha,\alpha)_{L^{2}(\tilde{X})}+((\Delta_{\mu}+\Delta_{\bar{\mu}})\alpha,\alpha)_{L^{2}(\tilde{X})}-|I|-|II|\\\
&\geq(1-2\varepsilon)((\Delta_{\partial}+\Delta_{\bar{\partial}})\alpha,\alpha)_{L^{2}(\tilde{X})}+(1-\frac{2}{\varepsilon})((\Delta_{\mu}+\Delta_{\bar{\mu}})\alpha,\alpha)_{L^{2}(\tilde{X})}\\\
&\geq\frac{1}{3}((\Delta_{\partial}+\Delta_{\bar{\partial}})\alpha,\alpha)_{L^{2}(\tilde{X})}-5((\Delta_{\mu}+\Delta_{\bar{\mu}})\alpha,\alpha)_{L^{2}(\tilde{X})}\\\
&\geq\frac{1}{3}((\Delta_{\partial}+\Delta_{\bar{\partial}})\alpha,\alpha)_{L^{2}(\tilde{X})}\\\
&-20c_{2}(n)\delta
K[Vol(X)]^{\frac{1}{n}}C_{n}((\Delta_{d}\alpha,\alpha)_{L^{2}(\tilde{X})}+c_{1}(n)\delta
K\|\alpha\|_{L^{2}(\tilde{X})}^{2})\\\
&\geq\frac{1}{3}c(n,k)K\|\alpha\|_{L^{2}(\tilde{X})}^{2}\\\ &-20c_{2}(n)\delta
K[Vol(X)]^{\frac{1}{n}}C_{n}(\Delta_{d}\alpha,\alpha)_{L^{2}(\tilde{X})}\\\
&-20c_{1}(n)c_{2}(n)\delta^{2}K^{2}[Vol(X)]^{\frac{1}{n}}C_{n}\|\alpha\|_{L^{2}(\tilde{X})}^{2}).\\\
\end{split}$

Then rearrangement gives

$\begin{split}(\Delta_{d}\alpha,\alpha)_{L^{2}(\tilde{X})}(1&+20c_{2}(n)\delta
K[Vol(X)]^{\frac{1}{n}}C_{n})\\\
&\geq\frac{1}{3}K(c(n,k)-60c_{1}(n)c_{2}(n)\delta^{2}K[Vol(X)]^{\frac{1}{n}})\|\alpha\|_{L^{2}(\tilde{X})}^{2}.\\\
\end{split}$

Provided

$\delta^{2}K[Vol(X)]^{\frac{1}{n}}\leq c_{3}(n)<\min_{k\neq
n}\frac{c(n,k)}{60c_{1}(n)c_{2}(n)C(n)},$

then

$(\Delta_{d}\alpha,\alpha)\geq\lambda\|\alpha\|^{2}_{L^{2}(\tilde{X})},$

where $\lambda=\lambda(X,n,k,\delta)$ is a positive constant. By the similar
way in Theorem 1.6, we get $\mathcal{H}^{n}_{(2);d}(\tilde{X})\neq\\{0\\}$,
i.e, $h_{(2)}^{n}(X)>0$. For any $k\neq n$,
$\mathcal{H}^{k}_{(2);d}(\tilde{X})=\\{0\\}$, i.e., $h_{(2)}^{k}(X)=0.$ Hence

$(-1)^{n}\chi(X)=(-1)^{n}\sum_{k=0}^{2n}(-1)^{k}h^{k}_{(2)}(X)=h^{n}_{(2)}(X)>0.$

On the other hand, following Gauss-Bonnet-Chern formula,

$\chi(X)=\int_{X}\chi(\Omega),$

where $\chi(\Omega)$ is a polynomial in the components of the curvature
tensor. Therefore

$1\leq|\chi(X)|\leq c_{4}(n)\delta^{n}K^{n}Vol(X).$

One also can see [4, Lemma 1]. It implies that there exists a constant $c$
only dependents on $n$ such that

$\delta K[Vol(X)]^{\frac{1}{n}}\geq[c_{4}(n)]^{-\frac{1}{n}}.$

Therefore,

$\delta^{2}K[Vol(X)]^{\frac{1}{n}}\geq\min\\{c_{3}(n),[c_{4}(n)]^{-\frac{1}{n}}\\}.$

∎

## Acknowledgements

This work is supported by the National Natural Science Foundation of China
Nos. 12271496, 11801539 (Huang), 11701226 (Tan) and the Youth Innovation
Promotion Association CAS, the Fundamental Research Funds of the Central
Universities, the USTC Research Funds of the Double First-Class Initiative.

My manuscript has no associated data.

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