January 28, 2023
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 of the compact almost Kähler manifold with negative sectional curvature.
Keywords. Almost Kähler manifold, Hodge theory, Hard Lefschetz, Hodge decomposition, Vanishing theorem, Negative sectional curvature
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 [1] 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 [2]–[6]). We define two elliptic self-adjoint 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 [7]) 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 \(\Delta_{\partial}+\Delta_{\bar{\partial}}\) in \(\Omega^{p,q}_{(2)}(X)\). We will denote by \[l^{p,q}_{(2)}:= \dim (\mathcal{H}^{p,q}_{(2);\bar{\partial}}\cap\mathcal{H}^{p,q}_{(2);\partial})=\dim\ker(\Delta_{\bar{\partial}}+\Delta_{\partial})\cap\Omega^{p,q}_{(2)}\] the dimensions of these spaces. In the integrable case, since \(\Delta_{\partial}=\Delta_{\bar{\partial}}=\frac{1}{2}\Delta_{d}\), these are just the Hodge numbers of the compact Kähler manifold. Since \(\Delta_{\partial}+\Delta_{\bar{\partial}}:\Omega^{p,q}\rightarrow\Omega^{p,q}\), there is an orthogonal direct sum decomposition \[\ker(\Delta_{\partial}+\Delta_{\bar{\partial}})\cap\Omega^{k}_{(2)}=\bigoplus_{p+q=k}\ker(\Delta_{\partial}+\Delta_{\bar{\partial}})\cap\Omega^{p,q}_{(2)}.\]
Theorem 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 \[\bigoplus_{p,q\geq0}\ker{\Delta_{\partial}}\cap\ker{\Delta_{\bar{\partial}}}\cap\Omega^{p,q}_{(2)}(X).\] Moreover, for all \(0\leq p\leq k\leq n\), \[L^{n-k}:\ker{\Delta_{\partial}}\cap\ker{\Delta_{\bar{\partial}}}\cap\Omega^{p,k-p}_{(2)}(X)\stackrel{\cong}{\longrightarrow}\ker{\Delta_{\partial}}\cap\ker{\Delta_{\bar{\partial}}}\cap\Omega^{p+n-k,n-p}_{(2)}(X)\] are isomorphisms. Furthermore, for any \(p,q\) we have an orthogonal direct sum decomposition \[\ker{\Delta_{\partial}}\cap\ker{\Delta_{\bar{\partial}}}\cap\Omega^{p,q}_{(2)}(X)=\bigoplus_{r\geq\max(p+q-n,0)}L^{j}(\ker{\Delta_{\partial}}\cap\ker{\Delta_{\bar{\partial}}}\cap P^{p-j,q-j}_{(2)}(X)),\] where \[P^{r,s}_{(2)}(X)=\ker\Lambda\cap\Omega^{r,s}_{(2)}(X).\]
Remark 2. For any compact almost Kähler manifold \(X\), Cirici1 and Wilson [7] defined the \(\delta\)-Laplacian by letting \[\Delta_{\delta}=\delta\delta^{\ast}+\delta^{\ast}\delta.\] For all \(p,q\), they also denoted by \[\label{E28} \mathcal{H}^{p,q}_{\delta}=\ker\Delta_{\delta}\cap\Omega^{p,q}=\ker\delta\cap\ker\delta^{\ast}\cap\Omega^{p,q}\tag{1}\] 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 ).
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 [8]–[10]), 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 [11]–[17] 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 3. Let \((X,J,\omega)\) be a complete \(2n\)-dimensional 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 3 in two specific cases.
Theorem 4 (=Theorem 16+Corollary 3+Corollary 4). Let \((X,J,\omega)\) be a complete \(2n\)-dimensional 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 \[\begin{align} &\|\alpha \|(1-c(n,k)c)\leq c(n,k)\|\theta\|_{L^{\infty}(X)}((\Delta_{\partial}+\Delta_{\bar{\partial}})\alpha ,\alpha )^{\frac{1}{2}}\\ &\|\alpha
\|^{2}(1-c(n,k)c)(1-c(n,k)c-4c(n,k)\|\theta\|_{L^{\infty}(X)}\sup|N_{J}|)\leq c^{2}(n,k)\|\theta\|^{2}_{L^{\infty}(X)}(\Delta_{d}\alpha ,\alpha )\\ &\|\alpha \|^{2}((1-c(n,k)c)^{2}-2c(n,k)^{2}\|\theta\|^{2}_{L^{\infty}(X)}\sup|N_{J}|^{2}))\leq
2c^{2}(n,k)\|\theta\|^{2}_{L^{\infty}(X)}(\Delta_{\bullet}\alpha ,\alpha ),\\ \end{align}\] 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 5. Let \((X,J,\omega)\) be a complete \(2n\)-dimensional almost Kähler manifold. Suppose that there exists a smooth function \(f\geq1\) on \(X\). Also assume that the function \(f\) satisfies the convexity condition on \(X\), i.e., for some \(A,B\geq0\), \(|df|^{2}\leq A+Bf\). 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 \(p+q\neq n\), \[\label{E13} m\int_{X}\frac{1}{f+M}|\alpha |^{2}\leq (\|\partial\alpha \|^{2}+\|\partial^{\ast}\alpha \|^{2}+\|\bar{\partial}\alpha \|^{2}+\|\bar{\partial}^{\ast}\alpha \|^{2}),\;\forall \alpha \in \Omega^{p,q}_{0}(X).\tag{2}\] 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 with sectional curvature \(sec\). Then \[\left\{ \begin{align} (-1)^{n}\chi(X)>0, &\;if\;sec<0\\ (-1)^{n}\chi(X)\geq0,&\;if\;sec\leq0.\\ \end{align} \right.\]
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 \(k\geq2\) 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 [18] constructed a \(d\)(bounded) complete almost Kähler manifold \(X\) satisfies \(\mathcal{H}^{1}_{(2)}(X)\neq\{0\}\) by using methods of contact geometry. 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], [20].
We denote by \(h^{k}_{(2)}(X)\) the \(k\)-th \(L^{2}\)-Betti number of Riemannian manifold \(X\). The second conjecture which proposed by Singer ([10]) is
Conjecture 2. (Singer Conjecture) Let \(X\) be a closed 2n-dimensional Riemannian manifold with negative sectional curvature. Then \[\left\{ \begin{align} h^{k}_{(2)}(X)=0, & k\neq n\\ h^{n}_{(2)}(X)>0.& \\ \end{align} \right.\]
By the Euler-Poincaré formula \(\chi(X)=\sum_{k\geq0}(-1)^{k}h_{(2)}^{k}(X)\), when \(X\) has negative sectional curvature, the Singer conjecture implies the Hopf conjecture.
The main 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 [15]).
Theorem 6. (cf. [21]) Let \((X,J,\omega)\) be a compact \(2n\)-dimensional 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 \[\left\{ \begin{align} \mathcal{H}^{k}_{(2);d}(\tilde{X})=\{0\}, & k\neq n\\ \mathcal{H}^{n}_{(2);d}(\tilde{X})\neq\{0\}, & \\ \end{align} \right.\] is equivalent to \[\left\{ \begin{align} h^{k}_{(2)}(X)=0, & k\neq n\\ h^{n}_{(2)}(X)>0.& \\ \end{align} \right.\] In particular, \[(-1)^{n}\chi(X)>0.\]
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 \(J\) [22]. 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: \[\label{E27} \begin{align} &\mu^{2}=0,\\ &\mu\partial+\partial\mu=0,\\ &\partial^{2}+\mu\bar{\partial}+\bar{\partial}\mu=0,\\ &\partial\bar{\partial}+\bar{\partial}\partial+\mu\bar{\mu}+\bar{\mu}\mu=0,\\ &\bar{\partial}^{2}+\bar{\mu}\partial+\partial\bar{\mu}=0,\\ &\bar{\mu}\bar{\partial}+\bar{\partial}\bar{\mu}=0,\\ &\bar{\mu}^{2}=0.\\ \end{align}\tag{3}\] 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. [7]). 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, \(\mu=0\) [1], [7].
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 \(R^{\mathbb{C}}: \wedge^{2}T_{p}X\otimes\mathbb{C}\rightarrow\wedge^{2}T_{p}X\otimes\mathbb{C}\) [23]. 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 [23]). Then following [24] and [25] or [26]), we have \[\label{E32} \begin{align} |\nabla{J}|^{2}&=\frac{1}{4}|N_{J}|^{2}\\ &=-8\sum_{i,j=1}^{n}(R^{\mathbb{C}}(Z_{i}\wedge Z_{j}),\overline{Z_{i}\wedge Z_{j}}). \end{align}\tag{4}\]
Proposition 7. (cf. [23])Let \((X,J,\omega)\) be a complete \(2n\)-dimensional almost Kähler manifold with pinched negative sectional curvature, i.e., there are constants \(K>0\) and \(\delta\geq1\) such that \[-\delta K\leq sec\leq -K.\] Then complex sectional curvature of \(X\) satisfies \[|\nabla J|^{2}\leq 12n^{2}\delta K.\]
Proof. Our proof follows the argument in [23]. 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 \[\begin{align}
(R^{\mathbb{C}}(Z\wedge Y),\overline{Z\wedge Y})&=\langle R(u\wedge x),u\wedge x\rangle+\langle R(u\wedge w),u\wedge w\rangle\\ &+\langle R(v\wedge x),v\wedge x\rangle+\langle R(v\wedge w),v\wedge w\rangle\\ &-2\langle R(u\wedge v),x\wedge
w\rangle\\ &=U^{2}X^{2}K_{13}+U^{2}W^{2}K_{14}+V^{2}X^{2}K_{23}+V^{2}W^{2}K_{24}\\ &-2UVWX\langle R(e_{1}\wedge e_{2}),e_{3}\wedge e_{4}\rangle.\\ \end{align}\] Therefore, \[\begin{align} -\delta
K-\frac{1}{2}|\langle R(e_{1}\wedge e_{2}),e_{3}\wedge e_{4}\rangle|\leq (R^{\mathbb{C}}(Z\wedge Y),\overline{Z\wedge Y})\leq -K+\frac{1}{2}|\langle R(e_{1}\wedge e_{2}),e_{3}\wedge e_{4}\rangle| \end{align}\] Noting that \(|\langle R(e_{1}\wedge e_{2}),e_{3}\wedge e_{4}\rangle|\leq\delta K.\) (cf. [27] and [23]). Hence, we get \[-\frac{3}{2}\delta K\leq K^{\mathbb{C}}(\Pi)\leq (\frac{\delta}{2}-1)K.\] Combining the preceding inequalities with (4 ) yields
\[|\nabla{J}|^{2}\leq 8n^{2}\max|K^{\mathbb{C}}|\leq 12n^{2}\delta K.\] ◻
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 \[[A,B]=AB-(-1)^{deg A\cdot deg B}BA,\] 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 8. (cf. [7]) 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 9. ([16] and [28])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 \[[\Xi(r),\Lambda]u(x)=\sum_{|I|=p,|J|=q}\big{(} \sum_{j\in I}e_{j}+\sum_{j\in J}e_{j}-\sum_{j=1}^{n}e_{j} \big{)}u_{IJ}(x)dz^{I}\wedge d\bar{z}^{J}.\]
Throughout, \((X,J,\omega)\) denotes a complete almost Kähler manifold of complex dimension \(n\). The global inner product is defined \[(u,v)=\int_{X}\langle u,v\rangle dV=\int_{X}u\wedge\ast\bar{v},\] 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 1. (cf. [29])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 [30], \[d^{\Lambda}=(-1)^{k+1}\ast J^{-1}d\ast J^{-1}=-{\ast}J^{-1}dJ\ast.\]
Lemma 2. ([5] and [29])\[\ker d\cap\ker d^{\ast}\cap\Omega^{p,q}(X)=\ker d^{\Lambda}\cap\ker d^{\Lambda_{\ast} }\cap\Omega^{p,q}(X)\]
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 10. ([7]) If \(\alpha _{p,q}\in\mathcal{H}^{p,q}_{(2);d}(X)\), the \(L(\alpha _{p,q})=\omega\wedge\alpha _{p,q}\in\mathcal{H}^{p+1,q+1}_{(2);d}(X)\).
Proof. Following Lemma 1, we have \(d\alpha _{p,q}=0\) and \(d^{\ast}\alpha _{p,q}=0\). By Lemma 2, we then have \(d^{\Lambda\ast}\alpha _{p,q}=0\) and \(d^{\Lambda}\alpha _{p,q}=0\). Using the identities \([d^{\ast},L]=-d^{\Lambda\ast}\) and \(d\omega=0\), we get \(d^{\ast}(\omega\wedge\alpha _{p,q})=0\) and \(d(\omega\wedge\alpha _{p,q})=0\). ◻
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. (cf. [29])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. \(\bar{\partial}\alpha =0,\bar{\partial}^{\ast}\alpha =0\)).
Following Lemmas 1 and 3, we get
Corollary 1. If an \(L^{2}\) \((p,q)\)-form \(\alpha\) on \(X\) is \(\Delta_{d}\)-harmonic form, then \(d\alpha =0,d^{\ast}\alpha =0\), i.e., \(\partial\alpha =\bar{\partial}\alpha =0\), \(\partial^{\ast}\alpha =\bar{\partial}^{\ast}\alpha =0\) and \(\mu\alpha =\bar{\mu}\alpha =0\), \(\mu^{\ast}\alpha =\bar{\mu}^{\ast}\alpha =0\). In particular, \(\mathcal{H}^{p,q}_{(2);d}(X)\subset\mathcal{H}^{p,q}_{(2);\partial}(X)\cap\mathcal{H}^{p,q}_{(2);\bar{\partial}}(X)\).
Theorem 11. (cf. [7]) 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. The first duality follows from the identity \[\ker(\Delta_{\bullet})\cap\Omega^{p,q}=\ker(\Delta_{\bar{\bullet}})\cap\Omega^{q,p}\] for any \(\bullet=\partial,\bar{\partial}\). Hodge duality follows from this same identity together with the relation \(\ast\Delta_{\bar{\bullet}}=\pm\Delta_{\bullet}\ast\), which also proves the Serre duality. ◻
Proof of Theorem 1. For any almost Hermitian manifold of dimension \(2n\) there are isomorphisms \[L^{n-k}:\Omega^{p,k-p}_{(2)}(X)\stackrel{\cong}{\longrightarrow}\Omega^{p+n-k,n-p}_{(2)}(X)\] for every \(0\leq p\leq n\) and all \(p\leq k\leq n\). By [7], 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 \[L^{n-k}:\ker{\Delta_{\partial}}\cap\ker{\Delta_{\bar{\partial}}}\cap\Omega^{p,q}_{(2)}(X)\stackrel{\cong}{\longrightarrow}\ker{\Delta_{\partial}}\cap\ker{\Delta_{\bar{\partial}}}\cap\Omega^{n-p,n-q}_{(2)}(X)\] are well defined, and are injective, since they are isomorphisms before restricting the domain. By Hodge duality of Theorem 11, the domain and codomain have the same dimension, so the map is an isomorphism. ◻
Proposition 12. 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. ◻
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 \(k\in\mathbb{R}\) [17]. In this article, we only consider \(C^{1}\) exhaustion function as follows.
Definition 1. ([17]) 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 [17], we consider some smooth differential forms as following.
Definition 2. The smooth \(k\)-form \(\omega\), (\(k\geq1\)), 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.
We return to almost Kähler manifold setting.
Definition 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 \(\omega_{i}\).
Example 1. (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 [15] (one also can see [31] and [32]).
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 [16] 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
[16] follows from the Definition 3 by taking \(f(x)=g(x)=\sqrt{A+Bx}\) and \(E(x)=\lambda(x)\). The hypotheses of [11] similarly follows from Definition
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 \(x_{0}\in X\).
(3) 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 \(-b^{2}\leq sec\leq -a^{2}\). We now recall some standard facts about the topology and geometry of this manifold (see [33]–[35]). 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 \(\alpha\), \(k\geq1\), 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 result.
Proposition 13. (cf. [35]) 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 [35]. ◻
Now we will prove a lemma which extend the Stokes formula to complete manifold under some conditions.
Lemma 4. Let \(X\) be a complete Riemannian manifold. Suppose that \(\omega\) is a vanishingly exhaustible \(k\)-form, \(k\geq1\). 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, \(0\leq h\leq1\), \[h(t)=\left\{ \begin{align} 1, & & t\geq1 \\ 0, & & t\leq0 \end{align} \right.\] 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 \[\label{E1} \begin{align} (\alpha ,d(h_{k}\cdot\gamma))&=(d^{\ast}\alpha ,h_{k}\cdot\gamma )\\ &=0.\\ \end{align}\tag{5}\] Since \(d\beta=0\), we also have \[\label{E2} \begin{align} d(h_{k}\cdot\gamma)=h'(k-E)&\cdot dE\wedge\theta\wedge\beta\\ &+h_{k}\cdot\omega\wedge\beta.\\ \end{align}\tag{6}\] We now substitute (6 ) into (5 ) and consider the two terms coming from the right-hand side of (6 ) separately. For the first term, the fact that \(supp h'_{k}\subset X_{k}\backslash X_{k-1}\) and the fact that \(\omega\) is bounded imply \[\label{E3} \begin{align} |(\alpha ,h'_{k}\cdot dE\wedge\theta\wedge\beta)|&\leq\int_{ X_{k}\backslash X_{k-1}}|dE\wedge\theta|\cdot|\alpha |\cdot|\beta|\\ &\leq\int_{ X_{k}\backslash X_{k-1}}(f(E)g(E))\cdot|\alpha |\cdot|\beta|\\ &\leq (f(k)\cdot g(k))\int_{ X_{k}\backslash X_{k-1}}|\alpha |\cdot|\beta|,\\ \end{align}\tag{7}\] for the functions \(f,g\) in Definition 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 \[\label{E4} (f(k_{i})\cdot g(k_{i}))\int_{ X_{k_{i}}\backslash X_{k_{i}-1}}|\alpha |\cdot|\beta|\rightarrow0,\;as\;i\rightarrow\infty.\tag{8}\] Otherwise, for some \(c>0\), \[\begin{align} \int_{X}|\alpha |\cdot|\beta|&=\sum_{k=1}^{\infty}\int_{ X_{k}\backslash X_{k-1}}|\alpha |\cdot|\beta|\\ &\geq c\sum_{k=1}^{\infty}\frac{1}{f(k)g(k)}\\ &=\infty.\\ \end{align}\] a contradiction.
So, for the sequence gives by (8 ), it follows from (7 ) that \[\label{E5} \lim_{i\rightarrow\infty}(\alpha ,h'_{k_{i}}\cdot dE\wedge\theta\wedge\beta)=0.\tag{9}\] For the term coming from the second term on the right-hand side of (6 ), the dominated convergences theorem implies \[\label{E6} \lim_{k\rightarrow\infty}(\alpha ,h_{k}\cdot\omega\wedge\beta)=(\alpha ,\omega\wedge\beta).\tag{10}\] Substituting (9 ) and (10 ) into (5 ), it follows that \((\alpha ,\omega\wedge\beta)=0\). ◻
Proposition 14. 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 \[\begin{align} \tilde{\omega}&=d\theta\\ &=(\partial+\mu+\bar{\partial}+\bar{\mu})(\theta^{1,0}+\theta^{0,1})\\ &=(\partial\theta^{1,0}+\mu\theta^{0,1})+(\partial\theta^{0,1}+\bar{\partial}\theta^{1,0})+(\bar{\partial}\theta^{0,1}+\bar{\mu}\theta^{1,0}).\\ \end{align}\] Noting that \(\tilde{\omega}\) is a \((1,1)\)-form. We get \[\partial\theta^{1,0}+\mu\theta^{0,1}=\bar{\partial}\theta^{0,1}+\bar{\mu}\theta^{1,0}=0\;and\;\tilde{\omega}=\partial\theta^{0,1}+\bar{\partial}\theta^{1,0}.\] We complete this proof. ◻
Following Proposition 14 and Lemma 4, we then have
Lemma 5. 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 14, 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. Noting that \[(h'(k-E)\cdot dE\wedge\theta\wedge\beta)^{p,q}=h'(k-E)\cdot \bar{\partial}E\wedge\theta^{1,0}\wedge\beta+h'(k-E)\cdot\partial E\wedge\theta^{0,1}\wedge\beta.\] Since \(h_{k}\cdot\gamma\) has compact support, and \(\partial^{\ast}\alpha =\bar{\partial}^{\ast}\alpha =0\), an integration by parts gives \[\label{E33} \begin{align} 0&=(\bar{\partial}^{\ast}\alpha ,h_{k}\cdot\gamma_{1})+(\partial^{\ast}\alpha ,h_{k}\cdot\gamma_{2})\\ &=(\alpha ,\bar{\partial}(h_{k}\cdot\gamma_{1})+\partial(h_{k}\cdot\gamma_{2}) )\\ &=(\alpha ,h'(k-E)\cdot \bar{\partial}E\wedge\theta^{1,0}\wedge\beta+h_{k}\cdot\bar{\partial}\theta^{1,0}\wedge\beta)\\ &+(\alpha ,h'(k-E)\cdot\partial E\wedge\theta^{0,1}\wedge\beta+h_{k}\cdot\partial\theta^{0,1}\wedge\beta)\\ &=(\alpha ,h'(k-E)\cdot dE\wedge\theta\wedge\beta)+(\alpha , h_{k}\cdot\tilde{\omega}\wedge\beta).\\ \end{align}\tag{11}\] Following the idea in Lemma 4, there exists a subsequence \(\{k_{i}\}\) such that \[\label{E34} \lim_{i\rightarrow\infty}(\alpha ,h'_{k_{i}}\cdot dE\wedge\theta\wedge\beta)=0.\tag{12}\] and \[\label{E35} \lim_{k\rightarrow\infty}(\alpha ,h_{k}\cdot\tilde{\omega}\wedge\beta)=(\alpha ,\tilde{\omega}\wedge\beta).\tag{13}\] Substituting (12 ) and (13 ) into (11 ), it follows that \((\alpha ,\tilde{\omega}\wedge\beta)=0\). ◻
We begin to establish the vanishing theorem of \(L^{2}\)-harmonic forms on complete almost Kähler manifold. At first, following Lemma [L4], we have an estimate on (\(\bar{\partial}\)-\(\partial\))-harmonic \((p,q)\)-forms as follows.
Corollary 2. 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 12, \(\omega\wedge\alpha \in\ker{\Delta_{\bar{\partial}}}\cap\ker{\Delta_{\partial}}\cap\Omega^{p+1,q+1}_{(2)}(X)\) and so Lemma 1 implies that \[\partial^{\ast}(\omega\wedge\alpha )=\bar{\partial}^{\ast}(\omega\wedge\alpha )=0.\] 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 5, we obtain \[\label{E22} (\omega\wedge\alpha ,\omega_{i}\wedge \alpha )=0.\tag{14}\] By (14 ), one can see that \[\begin{align} \|\omega\wedge\alpha \|^{2}&=(\omega\wedge\alpha , \sum_{i=1}^{L}\omega^{1,1}_{i}\wedge\alpha )+(\omega\wedge\alpha ,(\omega-\sum_{i=1}^{L}\omega^{1,1}_{i})\wedge \alpha )\\ &=(\omega\wedge\alpha ,(\omega-\sum_{i=1}^{L}\omega^{1,1}_{i})\wedge \alpha ).\\ \end{align}\] Hence, \[\|\omega\wedge\alpha \|\leq \|(\omega-\sum_{i=1}^{L}\omega^{1,1}_{i})\wedge \alpha \|.\] ◻
Proof of Theorem 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 2, we have \[\begin{align} \|\omega\wedge\alpha \|&\leq\|(\omega-\sum_{i=1}^{L}\omega^{1,1}_{i})\wedge\alpha )\|\\ &\leq C_{1}(p,q)\sup|\omega-\sum_{i=1}^{L}\omega^{1,1}_{i}|\cdot\|\alpha \|.\\ \end{align}\] 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 [36], for any \(\alpha \in\Omega^{k}(X)\) we have \[[\Lambda,L]\alpha =(n-k)\alpha .\] Therefore, \[\begin{align} (\omega\wedge\alpha ,\omega\wedge\alpha )&=([\Lambda,L]\alpha +\omega\wedge(\Lambda\alpha ),\alpha )\\ &=(n-k)\|\alpha \|^{2}+\|\Lambda\alpha \|^{2}.\\ \end{align}\] 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 \(p+q>n\). Following Corollary 1, it’s easy to see \(\mathcal{H}^{p,q}_{(2);d}(X)=0\) for all \(p+q\neq n\). ◻
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 \(X\).
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 15. Let \((X,J,\omega)\) be a complete \(2n\)-dimensional almost Kähler manifold. If \(\theta\) is a bounded \(1\)-form, then for any \(\alpha \in\Omega^{k}_{0}(X)\), \((k<n)\), \[\label{E23} |((d\theta)^{1,1}\wedge\alpha ,\omega\wedge\alpha )|\leq c(n,k)\|\theta\|_{L^{\infty}(X)}\|\alpha \|_{L^{2}(X)}((\Delta_{\partial}+\Delta_{\bar{\partial}})\alpha ,\alpha )^{\frac{1}{2}},\qquad{(1)}\] where \(c(n,k)\) is a positive constant which depends only \(n,k\).
Proof. An integration by parts gives \[\begin{align} ((d\theta)^{1,1}\wedge\alpha ,\omega\wedge\alpha )&=((\partial\theta^{0,1}+\bar{\partial}\theta^{1,0})\wedge\alpha ,\omega\wedge\alpha )\\ &=(\partial(\theta^{0,1}\wedge\alpha )+\theta^{0,1}\wedge\partial\alpha ,\omega\wedge\alpha )\\ &+(\bar{\partial}(\theta^{1,0}\wedge\alpha )+\theta^{1,0}\wedge\bar{\partial}\alpha ,\omega\wedge\alpha ) \\ &=(\theta^{0,1}\wedge\alpha , [\partial^{\ast},L]\alpha )+(\theta^{0,1}\wedge\alpha , L(\partial^{\ast}\alpha )) +(\theta^{0,1}\wedge\partial\alpha ,\omega\wedge\alpha )\\ &+(\theta^{1,0}\wedge\alpha ,[\bar{\partial}^{\ast},L]\alpha )+(\theta^{1,0}\wedge\alpha ,L(\bar{\partial}^{\ast}\alpha ))+(\theta^{1,0}\wedge\bar{\partial}\alpha ,\omega\wedge\alpha ) \\ &=(\theta^{0,1}\wedge\alpha , -\sqrt{-1}\bar{\partial}\alpha )+(\theta^{0,1}\wedge\alpha , L(\partial^{\ast}\alpha )) +(\theta^{0,1}\wedge\partial\alpha ,\omega\wedge\alpha )\\ &+(\theta^{1,0}\wedge\alpha ,\sqrt{-1}\partial\alpha )+(\theta^{1,0}\wedge\alpha ,L(\bar{\partial}^{\ast}\alpha ))+(\theta^{1,0}\wedge\bar{\partial}\alpha ,\omega\wedge\alpha ).\\ \end{align}\] Here we used the almost Kähler identities \([\partial^{\ast},L]=-\sqrt{-1}\bar{\partial}\), \([\bar{\partial}^{\ast},L]=\sqrt{-1}\partial\). Therefore, we get \[\begin{align} &\quad((d\theta)^{1,1}\wedge\alpha ,\omega\wedge\alpha )\\ &\lesssim \|\theta^{0,1}\|_{L^{\infty}(X)}\|\alpha \|_{L^{2}(X)}\|\bar{\partial}\alpha \|_{L^{2}(X)}+\|\theta^{0,1}\|_{L^{\infty}(X)}\|\alpha \|_{L^{2}(X)}\|\partial^{\ast}\alpha \|_{L^{2}(X)}+\|\theta^{0,1}\|_{L^{\infty}(X)}\|\alpha \|_{L^{2}(X)}\|\partial\alpha \|_{L^{2}(X)} \\ &+\|\theta^{1,0}\|_{L^{\infty}(X)}\|\alpha \|_{L^{2}(X)}\|\partial\alpha \|_{L^{2}(X)}+\|\theta^{1,0}\|_{L^{\infty}(X)}\|\alpha \|_{L^{2}(X)}\|\bar{\partial}^{\ast}\alpha \|_{L^{2}(X)}+\|\theta^{1,0}\|_{L^{\infty}(X)}\|\alpha \|_{L^{2}(X)}\|\bar{\partial}\alpha \|_{L^{2}(X)}\\ &\leq c(n,k)\|\theta\|_{L^{\infty}(X)}\|\alpha \|_{L^{2}(X)}((\Delta_{\partial}+\Delta_{\bar{\partial}})\alpha ,\alpha )^{\frac{1}{2}}. \end{align}\] ◻
Theorem 16. Let \((X,J,\omega)\) be a complete \(2n\)-dimensional 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 \(\alpha \in\Omega^{k}_{0}(X)\), \((k\neq n)\), \[\label{E24} \|\alpha \|(1-c(n,k)c)\leq c(n,k)\|\theta\|_{L^{\infty}(X)}((\Delta_{\partial}+\Delta_{\bar{\partial}})\alpha ,\alpha )^{\frac{1}{2}}.\tag{15}\] 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 (?? ), we get \[\begin{align} \|\alpha \|^{2}&\lesssim \|\omega\wedge\alpha \|^{2}\\ &=(\omega\wedge\alpha , (d\theta)^{1,1}\wedge\alpha )+(\omega\wedge\alpha ,(\omega-(d\theta)^{1,1})\wedge \alpha )\\ &\leq c(n,k)c\|\alpha \|^{2}+c(n,k)\|\theta\|_{L^{\infty}(X)}\|\alpha \|_{L^{2}(X)}((\Delta_{\partial}+\Delta_{\bar{\partial}})\alpha ,\alpha )^{\frac{1}{2}}.\\ \end{align}\] Rearrangement gives (15 ). ◻
Corollary 3. Let \((X,J,\omega)\) be a complete \(2n\)-dimensional 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 \(\alpha \in\Omega^{k}_{0}(X)\), \((k\neq n)\), \[\label{E25} \|\alpha \|^{2}(1-c(n,k))(1-c(n,k)c-4c(n,k)\|\theta\|_{L^{\infty}(X)}\sup|N_{J}|)\leq c^{2}(n,k)\|\theta\|^{2}_{L^{\infty}(X)}(\Delta_{d}\alpha ,\alpha ).\tag{16}\] 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 \[\begin{align} \Delta_{d}&=\Delta_{\partial}+\Delta_{\bar{\partial}}+\Delta_{\mu}+\Delta_{\bar{\mu}}\\ &+[\bar{\partial},\partial^{\ast}]+[\partial,\bar{\partial}^{\ast}]\\ &+[\partial+\bar{\partial},\mu^{\ast}+\bar{\mu}^{\ast}]+[\mu+\bar{\mu},\partial^{\ast}+\bar{\partial}^{\ast}].\\ \end{align}\] We observe that \[\begin{align} I&=(([\bar{\partial},\partial^{\ast}]+[\partial,\bar{\partial}^{\ast}])\alpha ,\alpha )\\ &=2([\mu^{\ast},\partial]\alpha ,\alpha )+2([\bar{\mu},\bar{\partial}^{\ast}]\alpha ,\alpha )\\ &=2(\partial\alpha ,\mu\alpha )+2(\mu^{\ast}\alpha ,\partial^{\ast}\alpha )+2(\bar{\partial}^{\ast}\alpha ,\alpha )+2(\bar{\mu}\alpha ,\bar{\partial}\alpha ). \end{align}\] Here we use the identities in [7] 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 \[\begin{align} |I|&\leq 2\sup|N_{J}|\cdot\|\alpha \|(\|\partial\alpha \|+\|\partial^{\ast}\alpha \|+\|\bar{\partial}\alpha \|+\|\bar{\partial}^{\ast}\alpha \|)\\ &\leq 2\sup|N_{J}|\cdot\|\alpha \|((\Delta_{\partial}+\Delta_{\bar{\partial}})\alpha ,\alpha )^{\frac{1}{2}}.\\ \end{align}\] We also observe that \[\begin{align} |II|&=|(([\partial+\bar{\partial},\mu^{\ast}+\bar{\mu}^{\ast}]+[\mu+\bar{\mu},\partial^{\ast}+\bar{\partial}^{\ast}])\alpha ,\alpha )|\\ &=2|((\mu^{\ast}+\bar{\mu}^{\ast})\alpha ,(\partial^{\ast}+\bar{\partial}^{\ast})\alpha )+((\mu+\bar{\mu})\alpha ,(\partial+\bar{\partial})\alpha )|\\ &\leq 2\sup|N_{J}|\cdot\|\alpha \|(\|\partial\alpha \|+\|\partial^{\ast}\alpha \|+\|\bar{\partial}\alpha \|+\|\bar{\partial}^{\ast}\alpha \|)\\ &\leq 2\sup|N_{J}|\cdot\|\alpha \|((\Delta_{\partial}+\Delta_{\bar{\partial}})\alpha ,\alpha )^{\frac{1}{2}}.\\ \end{align}\] Combining the preceding inequalities with estimate (15 ) yields \[\begin{align} (\Delta_{d}\alpha ,\alpha )&\geq((\Delta_{\partial}+\Delta_{\bar{\partial}})\alpha ,\alpha )-4\sup|N_{J}|\cdot\|\alpha \|((\Delta_{\partial}+\Delta_{\bar{\partial}})\alpha ,\alpha )^{\frac{1}{2}}.\\ &\geq((\Delta_{\partial}+\Delta_{\bar{\partial}})\alpha ,\alpha )^{\frac{1}{2}}(c(n,k)^{-1}\|\theta\|^{-1}_{L^{\infty}(X)}(1-c(n,k)c)-4\sup|N_{J}|)\|\alpha \| \\ &\geq c(n,k)^{-1}\|\theta\|^{-1}_{L^{\infty}(X)}(1-c(n,k)c)(c(n,k)^{-1}\|\theta\|^{-1}_{L^{\infty}(X)}(1-c(n,k)c)-4\sup|N_{J}|)\|\alpha \|^{2}. \end{align}\] Rearrangement gives (16 ). ◻
Recall that by [7] (cf. also [5]) on complete almost-Kähler manifolds, \[\Delta_{\bar{\partial}}+\Delta_{\mu}=\Delta_{\partial}+\Delta_{\bar{\mu}}.\]
Corollary 4. Let \((X,J,\omega)\) be a complete \(2n\)-dimensional 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 \(\alpha \in\Omega^{k}_{0}(X)\), \((k\neq n)\), \[\label{E26} \|\alpha \|^{2}((1-c(n,k)c)^{2}-2c(n,k)^{2}\|\theta\|^{2}_{L^{\infty}(X)}\sup|N_{J}|^{2})\leq 2c^{2}(n,k)\|\theta\|^{2}_{L^{\infty}(X)}(\Delta_{\bullet}\alpha ,\alpha ),\tag{17}\] 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 \[\begin{align} ((\Delta_{\mu}+\Delta_{\bar{\mu}})\alpha ,\alpha )&=\|\mu\alpha \|^{2}+\|\bar{\mu}\alpha \|+\|\mu^{\ast}\alpha \|^{2}+\|\bar{\mu}^{\ast}\alpha \|\\ &\leq 2\sup|N_{J}|^{2}\cdot\|\alpha \|^{2}.\\ \end{align}\] We observe that \[\begin{align} &\Delta_{\partial}=\frac{1}{2}(\Delta_{\partial}+\Delta_{\bar{\partial}}+\Delta_{\mu}-\Delta_{\bar{\mu}}),\\ &\Delta_{\bar{\partial}}=\frac{1}{2}(\Delta_{\partial}+\Delta_{\bar{\partial}}-\Delta_{\mu}+\Delta_{\bar{\mu}}).\\ \end{align}\] Combining the preceding inequalities with estimate (15 ) yields \[\begin{align} (\Delta_{\bullet}\alpha ,\alpha )&\geq\frac{1}{2}((\Delta_{\partial}+\Delta_{\bar{\partial}})\alpha ,\alpha )-\sup|N_{J}|^{2}\cdot\|\alpha \|^{2}.\\ &\geq(\frac{1}{2}(1-c(n,k)c)^{2}c(n,k)^{-2}\|\theta\|^{-2}_{L^{\infty}(X)}-\sup|N_{J}|^{2})\|\alpha \|^{2}. \end{align}\] Rearrangement gives (17 ). ◻
In almost Kählerian case, we could not to defined an almost Kähler manifold \((X,J,\omega)\) which given by global potential function \(f\) since \(dJdf\) possible has the \((2,0)+(0,2)\) part.
Definition 4. ([37]) 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 6. For any smooth function on complete almost Kähler manifold \(X\), we have \[\label{E7} \|([dJdf,\Lambda]\alpha ,\alpha )\|\leq 2(\|\partial\alpha \|^{2}+\|\partial^{\ast}\alpha \|^{2}+\|\bar{\partial}\alpha \|^{2}+\|\bar{\partial}^{\ast}\alpha \|^{2})+C\sup|df|^{2}\cdot\|\alpha \|^{2},\;\alpha \in\Omega^{p,q}_{0}(X).\tag{18}\]
Proof. Noting that \[\begin{align} dJdf&=dJ(\partial f+\bar{\partial}f)\\ &=(\partial+\mu+\bar{\partial}+\bar{\mu})(\sqrt{-1}\bar{\partial} f-\sqrt{-1}\partial f)\\ &=-\sqrt{-1}(\partial^{2}f-\mu\bar{\partial}f)-\sqrt{-1}(\bar{\partial}\partial f-\partial\bar{\partial}f)-\sqrt{-1}(-\bar{\partial}^{2}f+\bar{\mu}\partial f)\\ &=2\sqrt{-1}\mu\bar{\partial}f+2\sqrt{-1}\partial\bar{\partial}f-2\sqrt{-1}\bar{\mu}\partial f.\\ \end{align}\] Here we use the identities in (3 ). For a \((p,q)\)-form \(\alpha\), we get \[\begin{align} ([(dJdf),\Lambda]\alpha ,\alpha )&=([(dJdf)^{1,1},\Lambda]\alpha ,\alpha )\\ &=([2\sqrt{-1}\partial\bar{\partial}f,\Lambda]\alpha ,\alpha )\\ &=(2\sqrt{-1}\partial\bar{\partial}f\wedge(\Lambda\alpha ),\alpha )-(\Lambda(2\sqrt{-1}\partial\bar{\partial}f\wedge\alpha ),\alpha ).\\ \end{align}\] We observe that \[\begin{align} I:&=(2\sqrt{-1}\partial\bar{\partial}f\wedge(\Lambda\alpha ),\alpha )\\ &=(2\sqrt{-1}\partial(\bar{\partial}f\wedge\Lambda\alpha ),\alpha )+(2\sqrt{-1}\bar{\partial}f\wedge\partial(\Lambda\alpha ),\alpha )\\ &=(2\sqrt{-1}\partial(\bar{\partial}f\wedge\Lambda\alpha ),\alpha )+(2\sqrt{-1}\bar{\partial}f\wedge[\partial,\Lambda]\alpha ,\alpha )+(2\sqrt{-1}\bar{\partial}f\wedge\Lambda(\partial\alpha ),\alpha )\\ &=(2\sqrt{-1}(\bar{\partial}f\wedge\Lambda\alpha ),\partial^{\ast}\alpha )+(2\bar{\partial}f\wedge\bar{\partial}^{\ast}\alpha ,\alpha )+(2\sqrt{-1}\bar{\partial}f\wedge\Lambda(\partial\alpha ),\alpha ).\\ \end{align}\] Here we use the almost Kähler identity \([\partial,\Lambda]=-\sqrt{-1}\bar{\partial}^{\ast}\). Therefore, \[\label{E20} \begin{align} |I|&\leq C(n,k)\sup|df|\cdot\|\alpha \|\cdot(\|\partial^{\ast}\alpha \|+\|\bar{\partial}^{\ast}\alpha \|+\|\partial\alpha \|)\\ &\leq (\|\partial^{\ast}\alpha \|^{2}+\|\bar{\partial}^{\ast}\alpha \|^{2}+\|\partial\alpha \|^{2})+C\sup|df|^{2}\cdot\|\alpha \|^{2}.\\ \end{align}\tag{19}\] We also observe that \[\begin{align} II:&=-(\Lambda(2\sqrt{-1}\partial\bar{\partial}f\wedge\alpha ),\alpha )\\ &=-(2\sqrt{-1}\partial\bar{\partial}f\wedge\alpha ,L\alpha )\\ &=-(2\sqrt{-1}\partial(\bar{\partial}f\wedge\alpha )+2\sqrt{-1}\bar{\partial}f\wedge\partial\alpha ,L\alpha )\\ &=-(2\sqrt{-1}\bar{\partial}f\wedge\alpha ,\partial^{\ast}(L\alpha ))-(2\sqrt{-1}\bar{\partial}f\wedge\partial\alpha ,L\alpha )\\ &=-(2\sqrt{-1}\bar{\partial}f\wedge\alpha ,[\partial^{\ast},L]\alpha )-(2\sqrt{-1}\bar{\partial}f\wedge\alpha ,L(\partial^{\ast}\alpha ))-(2\sqrt{-1}\bar{\partial}f\wedge\partial\alpha ,L\alpha )\\ &=(2\bar{\partial}f\wedge\alpha ,\bar{\partial}\alpha )-(2\sqrt{-1}\bar{\partial}f\wedge\alpha ,L(\partial^{\ast}\alpha ))-(2\sqrt{-1}\bar{\partial}f\wedge\partial\alpha ,L\alpha ).\\ \end{align}\] Here we use the almost Kähler identity \([\partial^{\ast},L]=-\sqrt{-1}\bar{\partial}\). Therefore, \[\label{E21} \begin{align} |II|&\leq C(n,k)\sup|df|\cdot\|\alpha \|\cdot(\|\partial^{\ast}\alpha \|+\|\partial\alpha \|+\|\bar{\partial}\alpha \|)\\ &\leq (\|\partial^{\ast}\alpha \|^{2}+\|\partial\alpha \|^{2}+\|\bar{\partial}\alpha \|^{2})+C\sup|df|^{2}\cdot\|\alpha \|^{2}.\\ \end{align}\tag{20}\] It’s easy to that (18 ) follows from (19 ) and (20 ). ◻
Definition 5. ([16] and [37])Let \(f\in C^{2}(X)\) be a function on \(X\), \(f\geq1\). We say that \(f\) dominates its gradient, or \(f\) dominates \(df\), if there exist constants \(A>0\) and \(B\geq0\) such that \[\label{E14} |df|^{2}(x)\leq A+Bf(x),\;\forall x\in X.\tag{21}\]
Proof of Theorem 5. Suppose now that \(f\) dominates \(df\). Replace \(f\) by \(\tilde{f}=tf+1\), \(t>0\) and small, we may assume
(i) \(\tilde{f}\geq1\), \(x\in X\),
(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. 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 \[\begin{align} (dJdF)^{1,1}&=\sigma\varepsilon(\frac{(dJdf)^{1,1}}{f}-\frac{df\wedge Jdf}{f^{2}})\\ &=\sigma\varepsilon(\frac{\omega}{f}-\frac{df\wedge Jdf}{f^{2}}+\frac{(dJdf)^{1,1}-\omega}{f})\\ &=r'+r''+r''',\\ \end{align}\] where the last equality defines the \((1,1)\)-forms \(r''\) and \(r'''\). It follows from Proposition 9 that \[\label{E8} ([\Xi(r'),\Lambda]\alpha ,\alpha )=\frac{\sigma\varepsilon}{f}([\omega,\Lambda]\alpha ,\alpha )=\varepsilon|p+q-n|\int_{X}\frac{1}{f}|\alpha |^{2}.\tag{22}\] 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 9. Since each \(|e_{j}|\leq Bf\), by (ii) above, we also obtain from Proposition 9 \[\label{E9} \begin{align} ([\Xi(r''),\Lambda]\alpha ,\alpha )&=\varepsilon\sigma\big{(}\frac{1}{f^{2}}\big{(}\sum_{j\in I}e_{j}+\sum_{j\in J}e_{j}-\sum_{j=1}^{n}e_{j} \big{)}\alpha _{IJ}(x)dz^{I}\wedge d\bar{z}^{J},\alpha \big{)}\\ &\leq\varepsilon|p+q-n|\int_{X}\frac{B}{f}|\alpha |^{2}. \end{align}\tag{23}\] We also observe that \[\label{E10} \begin{align} ([\Xi(r'''),\Lambda]\alpha ,\alpha )&=\sigma\varepsilon([\frac{(dJdf)^{1,1}-\omega}{f},\Lambda]\alpha ,\alpha )\\ &\leq C\varepsilon\sup|(dJdf)^{1,1}-\omega|\cdot\int_{X}\frac{1}{f}|\alpha |^{2}.\\ \end{align}\tag{24}\] Noting that \[\label{E11} |dF|^{2}\leq\frac{\varepsilon^{2}|df|^{2} }{f^{2}}\leq \frac{\varepsilon^{2}B }{f}.\tag{25}\] Let \(N=|p+q-n|\). Substituting (22 )–(25 ) into (18 ), we obtain \[\label{E12} \varepsilon N\int_{X}[\frac{1}{f}-\frac{B}{f}-\frac{\varepsilon C}{N}\frac{B}{f}-\frac{C\sup|\tilde{\omega}|}{N}\frac{1}{f}]|\alpha |^{2} \leq 2(\|\partial\alpha \|^{2}+\|\partial^{\ast}\alpha \|^{2}+\|\bar{\partial}\alpha \|^{2}+\|\bar{\partial}^{\ast}\alpha \|^{2}),\;\alpha \in\Omega^{p,q}_{0}(X).\tag{26}\] 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 (26 ) that (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 (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 5 does not imply \[\label{E15} \|d\alpha \|^{2}+\|d^{\ast}\alpha \|^{2}\geq L\|\alpha \|^{2},\;for\;all\;\alpha \in\Omega^{p,q}_{0}(X), p+q\neq n,\tag{27}\] 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 [15] to symplectic case, then (27 ) does hold (see [29]). 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 (27 ), along with a good estimate on \(L\) (see [38]).
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 (27 ) with a reasonable estimate on \(L\).
Proposition 17. (cf. [16]) Let \((X,J,\omega)\) be a complete \(2n\)-dimensional almost Kähler manifold. Suppose that there exists a smooth function \(f\geq1\) on \(X\). Also assume that the function \(f\) satisfies the convexity condition on \(X\), i.e., for some \(A,B\geq0\), \(|df|^{2}\leq A+Bf\). 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 (27 ) holds with \[\label{E16} L=\frac{c^{2}N^{2}}{8BC}.\qquad{(2)}\] (ii) If \(B=0\) and \(\sup|(dJdf)^{1,1}-\omega|\leq\frac{1}{4C}\), then (27 ) holds with \[\label{E17} L=\frac{(N-\frac{3}{4})}{4AC}.\qquad{(3)}\]
Proof. For (i), it follows directly from the proof of Theorem 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 (?? ).
For (ii), we can choose \(F=\frac{\sigma}{2AC}f\), where \(\sigma={\rm{sign}}(p+q-n)\), it follows from (18 ) that \[\label{E18} \begin{align} 2(\|d\alpha \|^{2}+\|d^{\ast}\alpha \|^{2})+\frac{1}{4AC}\|\alpha \|^{2}&\geq\frac{\sigma}{2AC}([(dJdJ)^{1,1}-\omega,\Lambda]\alpha ,\alpha )+\frac{\sigma}{2AC}([\omega,\Lambda]\alpha ,\alpha )\\ &\geq\frac{N}{2AC}\|\alpha \|^{2}-C\sup|(dJdf)^{1,1}-\omega|\cdot\frac{1}{2AC}\|\alpha \|^{2}.\\ \end{align}\tag{28}\] Here we use the fact \(([\omega,\Lambda]\alpha ,\alpha )=(p+q-n)\|\alpha \|^{2}\), therefore, (?? ) follows directly from (28 ). ◻
The inequalities (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 18. (cf. [16]) Assume the hypotheses of Theorem 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 \(f\geq1\). 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 [16] 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 (2 ), we obtain \[\label{E19} \begin{align} |(\gamma,\alpha )|&=\big{|}(\frac{1}{\sqrt{f+M}}\gamma,\sqrt{f+M}\alpha )\big{|}\\ &\leq\big{(}\int_{X}\frac{1}{f+M}|\gamma|^{2}\big{)}^{\frac{1}{2}}\cdot\big{(}\int_{X}(f+M)|\alpha |^{2}\big{)}^{\frac{1}{2}}\\ &\lesssim\|d^{\ast}\gamma\|\big{(}\int_{X}(f+M)|\alpha |^{2}\big{)}^{\frac{1}{2}}. \end{align}\tag{29}\] Thus the functional is bounded on \(S\). However we also have \((\gamma,u)=0\) if \(\gamma\in S^{\bot}\) since \(d\alpha =0\), so (29 ) 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 \(\|u\|^{2}+\|d^{\ast}u\|^{2}\), (29 ) 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. ◻
We assume throughout this subsection that \((X,J,\omega)\) is a compact \(2n\)-dimensional almost Kähler 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 \(\Gamma\) [39], [40]. We denote by \(h_{(2)}^{k}(X)\) the \(L^{2}\)-Betti 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 7. \[\dim_{\Gamma}\mathcal{H}_{(2);d}^{k}(X)=0 \Leftrightarrow \mathcal{H}_{(2);d}^{k}(X)=\{0\},\] and \(\dim_{\Gamma}\mathcal{H}\) is additive. Given \[0\rightarrow\mathcal{H}_{1}\rightarrow\mathcal{H}_{2}\rightarrow \mathcal{H}_{3}\rightarrow 0,\] one has \[\dim_{\Gamma}\mathcal{H}_{2}=\dim_{\Gamma}\mathcal{H}_{1}+\dim_{\Gamma}\mathcal{H}_{3}.\]
By the \(L^{2}\)-index theorem of Atiyah [39], we have the following crucial identities between \(\chi(X)\) and the \(L^{2}\)-Betti numbers \(h_{(2)}^{k}(X)\): \[\chi(X)=\sum_{k=0}^{2n}(-1)^{k}h_{(2)}^{k}(X).\]
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 19. [15]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 20. (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 21. ([15])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 [40]).
Definition 6. ([40]) 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 \(\tilde{X}\times U(1)\) [40].
Theorem 22. ([40])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 6. Noting that \(|\tilde{\omega}|^{2}=n\). Then there exists a \(1\)-form \(\theta\) such that (cf. [32]) \[\tilde{\omega}=d\theta\;and\;\|\theta\|_{L^{\infty}(\tilde{X})}\leq K^{-\frac{1}{2}}\sqrt{n}.\] Following the second \(L^{2}\)-estimate in Theorem 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 22, 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 21). 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 \[\begin{align} (-1)^{n}\chi(X)&=(-1)^{n}\sum_{k=0}^{2n}(-1)^{k}h^{k}_{(2)}(X)\\ &=h^{n}_{(2)}(X)>0. \end{align}\] ◻
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.
T. Huang: School of Mathematical Sciences, CAS Key Laboratory of Wu Wen-Tsun Mathematics, University of Science and Technology of China, Hefei, Anhui, 230026, People’s Republic of China; e-mail: htmath@ustc.edu.cn;htustc@gmail.com↩︎
Q. Tan: School of Mathematical Sciences, Jiangsu University, Zhenjiang, Jiangsu 212013, People’s Republic of China; e-mail: tanqiang@ujs.edu.cn↩︎