Estimates for a function on almost Hermitian manifolds


 We study some estimates for a real-valued smooth function φ on almost Hermitian manifolds. In the present paper, we show that ∂∂∂̄ φ and ∂̄∂∂̄ φ can be estimated by the gradient of the function φ.


Introduction
Let (M n , J) be an almost complex manifold of real dimension n and g an almost Hermitian metric on M. Let {Zr} be an arbitrary local ( , )-frame around a xed point p ∈ M. We shall use the following notations: for a function φ, ∇ i ∇¯j φ := ∇ Z i ∇ Z¯j φ and φ ij : where ∇ is the Chern connection with respect to g, (see Section 2 for the de nition of Bs ij ). Since we have [Z i , Z¯j] ( , ) (φ) = Bs ij Zs(φ), we obtain that ∂ i ∂¯j φ = ∇ i ∇¯j φ.  ( , ) . An almost complex structure J is integrable if N = on M (Newlander-Nirenberg theorem). Giving a complex structure to a di erentiable manifold M is equivalent to giving an integrable almost complex structure to M. A Riemannian metric g on M is called J-invariant if J is compatible with g, i.e., for any X, Y ∈ Γ(TM), g(X, Y) = g(JX, JY). In this case, the pair (J, g) is called an almost Hermitian structure. Let {Zr} be a local ( , )-frame on (M, J) with an almost Hermitian metric g and let {ζ r } be a local associated coframe with respect to {Zr}, i.e., ζ i (Z j ) = δ i j for i, j = , . . . , n. Since g is almost Hermitian, its components satsfy g ij = g¯i¯j = and g ij = g¯j i =ḡ¯i j .
Using these local frame {Zr} and coframe {ζ r }, we have We write T R M for the real tangent space of M. Then its complexi ed tangent space is given by T C M = T R M⊗ R C. By extending J C-linearly and g, C-bilinearly to T C M, they are also de ned on T C M and we observe that the complexi ed tangent space T C M can be decomposed as   If the ( , )-part of the torsion of an almost Hermitian connection vanishes everywhere, then the connction is called the second canonical connection or the Chern connection. We will refer the connection as the Chern connection and denote it by ∇. Now let ∇ be the Chern connection on M. We denote the structure coe cients of Lie bracket by Notice that J is integrable if and only if the B¯r ij 's vanish. For any p-form ψ, there holds that for any vector elds X , . . . , X p+ on M (cf. [3]). We directly compute that We can split the exterior di erential operator d : In terms of these components, the condition d = can be written as A direct computation yields for any φ ∈ C ∞ (M, R), and we have

. The torsion and curvature on almost complex manifolds
Since the Chern connection ∇ preserves J, we have where Γ r ij = g rs Z i (g js ) − g rs g jl B¯l is , Γ p ip = Z i (log det g) − Bs is . We can obtain that Γ¯r ij = B¯r ij since the ( , )-part of the torsion of the Chern connection vanishes everywhere.
Note that the mixed derivatives ∇ i Z¯j do not depend on g (cf. [1]). Let {γ i j } be the connection form, which is de ned by γ i j = Γ i sj ζ s + Γ ī sj ζs. The torsion T of the Chern connection ∇ is given by T i = dζ i − ζ p ∧ γ i p , T¯i = dζ¯i − ζp ∧ γ¯ip, which has no ( , )-part and the only non-vanishing components are as follows: We will write the equation above by B = Γ + T , and we also have Ts ij = −Bs ij . These tell us that T splits into We also lower the index of torsion and denote it by Note that T depends only on J and it can be regarded as the Nijenhuis tensor of J, that is, J is integrable if and only if T vanishes. Proof. We compute that from (2.7), Proof. We compute from (2.8), and that Since we have∂ where we have used that∂∂φ = −(∂∂ + AĀ +ĀA)φ = −∂∂φ since Aφ =Āφ = , we obtain that by combining (3.1) and ( In this proof, in order to avoid a notational quagmire, we adopt the following *convention A * A between two quantities A and A with respect to a metric g: (1) Summation over pairs of maching upper and lower indices.
(2) Contraction on upper indices with respect to the metric.
(3) Contraction on lower indices with respect to the dual metrics.
Let {Zr} be a local ( , )-frame with respect to g around a xed point p ∈ M (we call it a local g-unitary frame in the following) and let {ζ r } be a local associated coframe with respect to {Zr}, i.e., ζ i (Z j ) = δ i j for i, j = , . . . , n. Note that unitary frames always exist locally since we can take any frame and apply the Gram-Schmidt process. Then with respect to such a frame, we have g ij = δ ij , Z k (g ij ) = for any i, j, k = , . . . , n, and the Christo el symbols satisfy Fix a local g-unitary frame {Zr} in this proof. We choose a smooth function φ arbitrary. Then we have the following formula:    Remark 3.1. If we assume that ∂∂∂φ = (resp.∂∂∂φ = ), since a real-valued smooth function φ is chosen arbitrary, the coe cient of the second order term vanishes, that is, we have that Bs kj = (resp. B s kj = ), which tells us that the almost complex structure J is then integrable. Note that in the quasi-Kähler case, which includes almost Kähler and nearly Kähler cases, since in these cases we have T k ij = for all i, j and k (cf. [2]), we have that from (3.5),