**4. Applications**

On an almost Kähler manifold, using the bidegree decompositions of *d* and *d*<sup>∗</sup>, one may derive from (1) the relation

$$[\Lambda, \partial] = i\vec{\partial}^\* \lrcorner$$

involving Λ, *∂* and the adjoint of ¯ *∂*. For a non-Kähler Hermitian manifold there is an additional term

$$[\Lambda, \partial] = i(\bar{\partial}^\* + t^\*)$$

where *τ*¯ = [<sup>Λ</sup>, [ ¯ *∂*, *L*]] is the zero-order *torsion operator* (see [5,6]). In the case of (0, *q*)-forms this gives

$$
\Lambda \partial \mathfrak{a} = i \mathfrak{d}^\* a + i [\Lambda, \mathfrak{d}^\*] L \mathfrak{a}.
$$

Next we use Theorem 1 to derive this local identity also in the non-integrable case.

**Proposition 1.** *For all α* ∈ A0,*q in an almost Hermitian manifold we have*

$$
\Lambda \partial \mathfrak{a} = i \mathfrak{d}^\* \mathfrak{a} + i [\Lambda, \mathfrak{d}^\*] L \mathfrak{a}.
$$

**Proof.** By bidegree reasons *α* is a primitive form and we have *dα* = *α*0 + *Lα*1 + *<sup>L</sup>*2*α*2 where *αi* are primitive. By expanding each term in the equality of Theorem 1 with respect to the bidegree decomposition *d* = *μ*¯ + ¯ *∂* + *∂* + *μ*, in the case *j* = 0, we obtain:

$$[\Lambda, d]\mathfrak{a} = \Lambda d\mathfrak{a} = \Lambda(\mathfrak{d} + \mu)\mathfrak{a}\_{\prime}$$

$$\star \mathbb{I}^{-1} \, d\mathbb{I} \, \star \mathfrak{a} = i(\mathfrak{d}^\* - \bar{\mu}^\*)\mathfrak{a}\_{\prime}$$

and

$$\mathbb{T}^{-1}\left[d^\*,\Lambda\right]\mathbb{T}L\mathfrak{a} = i\left[\Lambda,\bar{\partial}^\*-\bar{\mu}^\*\right]L\mathfrak{a}.$$

In particular, all terms decompose into sums of pure bidegrees (0, *q* − 1) and (1, *q* − <sup>2</sup>). Note as well that the remaining term

$$f\_{m,0}(2) 
La\_2$$

given in Theorem 1 has pure bidegree (1, *q* − <sup>2</sup>), since *α*2 must have bidegree (0, *q* − <sup>3</sup>). By putting together all terms of bidegree (0, *q* − 1) we obtain the desired identity.

**Remark 2.** *The proof of Proposition 1 gives a second identity relating the operators* Λ*, μ and μ*¯ *and their adjoints, which also contains the term fn*,*q*,<sup>0</sup>(2)*<sup>L</sup>α*2*. For forms in* A0,2*, this extra term vanishes by bidegree reasons, since α*2 = 0*. Then the second identity reads*

$$
\Lambda \mu \alpha = -i \bar{\mu}^\* \alpha - i [\Lambda, \bar{\mu}^\*] L \alpha.
$$

*This corrects the identity*

$$[\Lambda, \mu] = -i\mu^\*$$

*known in the almost Kähler case for arbitrary forms (see [4]).*

The previous proposition can be used to give a uniqueness result for the Dirichlet problem on compact domains with a boundary.

**Corollary 1.** *Let* Ω *be a compact domain in an almost complex manifold* (*<sup>M</sup>*, *J*)*, with smooth boundary, and let g* : Ω → C*, and φ* : *∂*Ω → C *be smooth. Then the Dirichlet problem,*

$$
\partial \mathfrak{A} \mathfrak{u} = \mathfrak{g} \qquad \text{with} \quad \mathfrak{u}|\_{\partial \Omega} = \mathfrak{q},
$$

*has at most one solution u* : Ω → C*.*

*In particular, if* (*<sup>M</sup>*, *J*) *is a compact connected almost complex manifold, and f* : *M* → C *is a smooth map of almost complex manifolds, then f is constant.*

**Proof.** It suffices to show the only solution to the homogenous equation with *g* = 0 is a constant function.

In any coordinate chart *ψ* : *V* → R2*n* containing any maximum point, we pullback *J* to *ψ*(*V*) and consider the *J*-preserving map *u* ◦ *ψ*−<sup>1</sup> : *ψ*(*V*) → C. The components of *d* are natural with respect to this *J*-preserving map and we use a compatible metric on *ψ*(*V*) to define Λ and ¯ *∂*<sup>∗</sup>. Then by Proposition 1 with *q* = 1 we obtain

$$-i\Lambda \partial \bar{\partial} \mu = \bar{\partial}^\* \bar{\partial} \mu + [\Lambda, \bar{\partial}^\*] L \bar{\partial} \mu$$

on *ψ*(*V*). Note ¯ *∂*∗ ¯ *∂* is quadratic, self-adjoint, and positive, and [<sup>Λ</sup>, ¯ *∂*∗]*L* ¯ *∂* is first order since [<sup>Λ</sup>, ¯ *∂*∗] = [*d*, *L*]∗ is zeroth order, because [*d*, *<sup>L</sup>*]*η* = *dω* ∧ *η*. Then the right hand side is zero, so the maximum principle due to E. Hopf applies [11], showing *u* is constant in a neighborhood of the maximum point and therefore, by connectedness, *u* is constant.

The final claim follows taking Ω = *M*, with empty boundary, *g* = 0, and noting the condition that *f* is a map of almost complex manifolds implies ¯ *∂ f* = 0.

**Remark 3.** *In [8], we introduce a Dolbeault cohomology theory that is valid for all almost complex manifolds. The above corollary is key in showing that, for a compact connected almost complex manifold, this cohomology is well-behaved in lowest bidegree, in the sense that H*0,0 Dol(*M*) ∼= C*.*

Finally, we refer the reader to the work of Feehan and Leness [9], where the relation of Proposition 1, for *q* = 1, is used to show that the moduli spaces of unitary anti-self-dual connections over any almost Hermitian 4-manifold is almost Hermitian, whenever the Nijenhuis tensor has sufficiently small *C*0-norm.

**Author Contributions:** Conceptualization, J.C. and S.O.W.; Writing—original draft, J.C. and S.O.W.; Writing—review & editing, J.C. and S.O.W. Both authors have read and agreed to the published version of the manuscript.

**Funding:** J.C. would like to acknowledge partial support from the AEI/FEDER, UE (MTM2016-76453-C2-2-P) and the Serra Húnter Program. S.O.W. acknowledges the support provided by a PSC-CUNY Award, jointly funded by The Professional Staff Congress and The City University of New York.

**Acknowledgments:** The authors would like to thank Paul Feehan for encouraging us to develop some previous notes into the present paper.

**Conflicts of Interest:** The authors declare no conflict of interest.
