{'fraction_non_alphanumeric': 0.06670033953475321, 'fraction_numerical': 0.02685411229901507, 'mean_word_length': 3.489543965734442, 'pattern_counts': {'":': 0, '<': 10, '<?xml version=': 0, '>': 23, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 26, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
The Monge-Amp\`ere equation for non-integrable almost complex structures

We show existence and uniqueness of solutions to the Monge-Ampere equation on compact almost complex manifolds with non-integrable almost complex structure.


Introduction
Yau's Theorem for compact Kähler manifolds [56] states that one can prescribe the volume form of a Kähler metric within a given Kähler class. This result, proved forty years ago, occupies a central place in the theory of Kähler manifolds, with wide-ranging applications in geometry and mathematical physics.
More precisely, Yau's Theorem is as follows. Let (M 2n , ω, J) be a compact Kähler manifold of complex dimension n, where ω denotes the Kähler form and J the complex structure. Then given a smooth volume form e F ω n on M there exists a unique smooth function ϕ satisfying There has been great interest in extending Yau's Theorem to non-Kähler settings. Recall that a Kähler metric is a positive definite real (1,1) form ω, namely a Hermitian metric, which satisfies dω = 0.
Within this circle of ideas remain some open conjectures, such as a version of Yau's Theorem for balanced metrics (satisfying d(ω n−1 ) = 0) [17,50,51]. This is related to the some questions about the Strominger system of mathematical physics, see e.g. [19,31,34] and the references therein.
A different type of extension of Yau's Theorem is to the case when J is a non-integrable almost complex structure, and this is subject of the current paper. Around twenty years ago, Gromov posed the following problem to P. Delanöe [12]: let (M 2n , ω) be a compact symplectic manifold, J an almost complex structure compatible with ω and F a smooth function on M with M e F ω n = M ω n . Can one find a smooth function ϕ on M such that ω + d(Jdϕ) is a symplectic form taming J and satisfying (1.2) (ω + d(Jdϕ)) n = e F ω n ? However, Delanöe [12] showed that when n = 2 the answer to this question is negative, and this was later extended to all dimensions by Wang-Zhu [53].
The key ingredient of their results is the construction of a smooth function ϕ 0 such that ω + d(Jdϕ 0 ) is on the boundary of the set of taming symplectic forms (so its (1, 1) part is semipositive definite but not strictly positive), and yet (ω+d(Jdϕ 0 )) n > 0. This is possible because in this case the (2, 0)+(0, 2) part of d(Jdϕ 0 ) contributes a strictly positive amount. This indicates that the problem with Gromov's suggestion is that the 2form d(Jdϕ) is in general not of type (1, 1) with respect to J, due to the fact that J may not be integrable (in fact, d(Jdf ) is of type (1,1) for all functions f if and only if J is integrable). Its (1, 1) part (up to an unimportant factor of 2) will be denoted by √ −1∂∂ϕ = 1 2 (d(Jdϕ)) (1,1) , which agrees with the standard notation when J is integrable (see also section 2 for more explanations). This quantity was apparently first explicitly considered in [25]. We show that an analogue of Gromov's problem does hold after replacing d(Jdϕ) with √ −1∂∂ϕ. In fact we do not even require the manifold to be symplectic. We obtain the following result for almost complex manifolds equipped with an almost Hermitian metric ω.
Our main theorem is the following: Namely, this says that the main result of [46] holds even when J is a non-integrable almost complex structure. The equation (1.3) was first considered by Harvey-Lawson [27] and Pliś [36] in the setting of the Dirichlet problem for J-pseudoconvex domains (see Remark 5 below). We note here that there has been a renewed interest recently in the theory of Jplurisubharmonic functions on almost complex manifolds, i.e. functions ϕ for which √ −1∂∂ϕ > 0 (see e.g. [11,25,27,28,33,36,37]), and our main theorem fits well into this picture.
The key ingredients for proving Theorem 1.1 are the following new a priori estimates: We make now a few remarks about our results: 1. We can also interpret the Monge-Ampère equation (1.3) in terms of Ricci curvature forms. More precisely, we can associate to an almost Hermitian metric ω a canonical connection (see e.g. [52]), whose curvature form can be expressed as a skew-Hermitian matrix of 2-forms {Ω i j } (1 i, j n) in any local unitary frame, and defining we obtain a globally defined closed real 2-form, which is cohomologous to the first Chern class 2πc 1 (M, J) in H 2 (M, R). If ϕ is a smooth function withω := ω + √ −1∂∂ϕ > 0 then (1.3) holds for some constant b if and only if as follows easily from [52, (3.16)]. Note however that in general there are representatives of the first Chern class 2πc 1 (M, J) in H 2 (M, R) which cannot be written in the form Ric(ω) − 1 2 d(JdF ) for any ω and F , even when J is integrable (cf. [45,Corollary 2]). Note also that in general Ric(ω) is different from the Riemannian Ricci curvature of the Levi-Civita connection of the Riemannian metric induced by ω. For example, when (M, J) is the Kodaira-Thurston 4-manifold with a certain explicit almost complex structure, then one can construct an explicit almost Hermitian metric ω with Ric(ω) = 0 (see e.g. [48,Theorem 4.1]), and yet M does not admit any Ricci-flat Riemannian metrics. Indeed, four-dimensional Einstein manifolds satisfy χ(M ) 0 with equality only when the metric is flat [2, 6.32]. The Kodaira-Thurston manifold is a T 2 -bundle over T 2 and therefore has χ(M ) = 0, so if M admitted an Einstein metric, M would be finitely covered by a torus, which is not the case.
2. Suppose now that (M 2n , ω, J) is a compact symplectic manifold with J an almost complex structure tamed by ω. In this case ω (1,1) is an almost Hermitian metric. Given a smooth function F we obtain a pair (ϕ, b) solving , then a simple calculation (cf. [12,Proposition 5]) shows thatω is a symplectic form taming J and 3. A different extension of Yau's Theorem to the symplectic setting was proposed by Donaldson [15]. He considered a symplectic 4-manifold (M, ω, J) with J a tamed almost complex structure, and conjectured that, given a smooth function F , and given a symplectic formω cohomologous to ω, compatible with J and solvingω 2 = e F ω 2 , thenω should have uniform C ∞ a priori estimates. Donaldson showed that this conjecture and various extensions of it would have consequences for symplectic topology. This problem is fundamentally different from our setup, since here the differenceω − ω is not given by a simple operator applied to a function ϕ. Donaldson's conjecture remains open in general, but it is known to hold in some special cases [55,52,47,48,16,6].
4. One can also consider the parabolic Monge-Ampère equation where we require that ω + √ −1∂∂ϕ t > 0. Analogously to the result proved by Cao [7] in the Kähler case and Gill [23] in the Hermitian case, we expect that the techniques developed in this paper can be used to show that a smooth solution ϕ t exists for all t 0, and after suitable normalization converges smoothly to the solution of (1.3).
Similarly, we can consider the equation where now we require that ω − tRic(ω) (1,1) + √ −1∂∂ϕ t > 0. This is in fact equivalent to the evolution equation for almost Hermitian forms ∂ω t ∂t = −Ric(ω t ) (1,1) , ω 0 = ω, which is the Kähler-Ricci flow if J is integrable and ω 0 is Kähler, and the Chern-Ricci flow [23,49] if J is integrable. Again, using the techniques developed in this paper one should be able to characterize the maximal existence time of this flow, exactly as in [42,49].
5. As in the Hermitian case treated in [50], the constant b that appears in Theorem 1.1 is in general a bit mysterious, and cannot be obtained from ω and F via a simple formula (unless J is integrable and ∂∂ω = 0 = ∂∂(ω 2 ), in which case b = log M ω n M e F ω n ). One should really think of the pair (ϕ, b) as the unique solution of the PDE (1.3), and in general one cannot expect solutions of PDEs to have simple expressions. What we do know nevertheless is that |b| sup M |F |, which follows from a simple maximum principle argument.
Outline of the proof. We now discuss the proof of our main results, and relate this to the work of Pliś [36] who, as noted above, investigated this problem on J-pseudoconvex domains (Harvey-Lawson [27] obtained weak solutions in the viscosity sense, and in a more general setting). We prove the a priori estimates of Theorem 1.2 in four steps: the zero order estimate for ϕ, the first order estimate, the real Hessian bound and then higher order estimates.
For the zero order estimate we adapt an approach of Székelyhidi [40], which in turn uses ideas of B locki [3,4]. The idea is to work locally near the infimum of ϕ and use a modification of the Alexandrov-Bakelman-Pucci maximum principle. Székelyhidi's argument holds for a large class of equations, but assuming an integrable complex structure. We show that the extra terms arising from the non-integrability can be controlled. This argument is contained in Section 3. The next step is to bound the first derivatives of ϕ, and we carry this out in Section 4. We use a maximum principle argument, adapted from Pliś [36]. In particular we compute using a unitary frame e 1 , . . . , e n , which turns out to be quite convenient for this problem. The difference with [36] is that, as would be expected in the compact case, we need to replace Pliś's auxiliary global plurisubharmonic function with a certain choice of "barrier function" involving ϕ itself.
The heart of the paper is Section 5, where we prove a bound on the real Hessian of ϕ. Here, the argument diverges substantially from the study of other similar nonlinear PDEs on complex manifolds where the approach is to bound first the complex Hessian of ϕ (see e.g. [5,24] for instances where a real Hessian bound is obtained for the complex Monge-Ampère equation, with J integrable, after having first obtained a complex Hessian bound). There are difficulties in carrying out the analogous computation for the complex Hessian here, because of some linear third order terms which, roughly speaking, involve three "barred" or "unbarred" partial derivatives of ϕ and cannot be controlled by the usual squares of third order terms. Instead, we use an important idea of Pliś [36], which is to directly bound the real Hessian of ϕ. However, as acknowledged by Pliś in a private communication, there is an error in this argument in [36] (in obtaining the first displayed equation of page 981 from the previous lines). We take a different approach, which corrects [36], and turns out to be substantially more intricate. We apply the maximum principle to a quantity involving the largest eigenvalue λ 1 (cf. [54,40,41]) of the real Hessian of ϕ. This gives us some good third order terms which are sufficient, after a series of rather technical lemmas, to push the argument through. One source of complication is the need to rule out the case when the largest eigenvector of the real Hessian of ϕ is in a direction where the complex Hessian is very small (see Lemma 5.6 below).
Once we have the real Hessian of ϕ bounded, it is then straightforward to obtain the higher order estimates by applying directly an Evans-Krylov type result from [44] and then some standard bootstrapping arguments. In Section 6 we describe how to obtain the main result Theorem 1.1 using a continuity argument similar to that of [45].
Before we start with the proofs of the main results, we describe first some basic results, and notation, on almost complex manifolds. Center for Mathematical Sciences and Applications at Harvard University. We would like to thank these institutions for their hospitality and support.

Basic results and notation
Let M 2n be a compact manifold of real dimension 2n (without boundary) equipped with an almost complex structure J, namely an endomorphism of the tangent bundle satisfying J 2 = −Id. The complexified tangent space T C M can be decomposed as a direct sum of the two eigenspaces T (1,0) M and T (0,1) M of J, corresponding to eigenvalues √ −1 and − √ −1 respectively. Similarly, extending J to 1-forms α by (Jα)(X) = −α(JX), we obtain a decomposition of T C M * into the √ −1 and − √ −1 eigenspaces, spanned by the (0, 1) and (1, 0) forms respectively. Thus, with the obvious definitions, any complex differential form of degree k can be expressed uniquely as a linear combination of (p, q) forms with p + q = k.
An almost Hermitian metric g on (M 2n , J) is a Riemannian metric satisfying g(JX, JY ) = g(X, Y ), for all X, Y ∈ T M. Such metrics always exist in abundance on almost complex manifolds. The metric g defines a positive definite real (1, 1) form ω, given by Conversely, such an ω defines an almost Hermitian metric g by g(X, Y ) = ω(X, JY ).
As we have already done in the introduction, we abuse terminology by referring to the positive definite (1, 1) form ω as an almost Hermitian metric.
The exterior derivative acting on (p, q) forms splits as where T changes the bidegree by (2, −1) and T by (−1, 2) (these are essentially given by the Nijenhuis tensor) and so we can define It is immediate to see that this is indeed a real (1, 1) form. A simple calculation shows that √ −1∂∂ϕ = 1 2 (d(Jdϕ)) (1,1) .
We also have (see e.g. [27, (2.5)]) that for any two (1, 0) vector fields V, W, We extend the action of J to p-forms by (Jα)(X 1 , . . . , X p ) := (−1) p α(JX 1 , . . . , JX p ), which satisfies J 2 = (−1) p . If * is the Hodge star operator of the Riemannian metric g defined by ω then the actions of * and J on p-forms commute (see e.g. [22, Lemma 1.10.1]). We will say that ω is Gauduchon if where d * is the adjoint of d with respect to g. A short calculation shows that (2.2) is equivalent to so we see that when J is integrable this condition reduces to the well-known one.
Theorem 2.1 (Gauduchon [20]). Let (M 2n , ω, J), n 2, be a compact almost Hermitian manifold. Then there exists a smooth function u, unique up to addition of a constant, such that the conformal almost Hermitian metric e u ω is Gauduchon.
For convenience, we will give a brief sketch of the proof of this result in the proof of Theorem 2.2.
Let now (M 2n , ω, J) be almost Hermitian, and f a smooth function on M . We define the canonical Laplacian of f by If ∆ denotes the usual Laplace-Beltrami operator of g then we have the relation where τ is the "torsion vector field" of (ω, J) (the dual of its Lee form), see e.g. [43,Lemma 3.2]. For the "openness" part of the continuity method, we will need the following result: [20]). Let (M 2n , ω, J), n 2, be a compact almost Hermitian manifold. Fix a nonnegative integer k, and 0 < α < 1.
if and only if where u is as in Theorem 2.1. In this case the solution f is unique up to addition of a constant.
Proof. Although this result is not explicitly stated there, the proof is contained in [20]. For the convenience of the reader we give a brief sketch, referring to [20] for details. Let ∆ * be the formal L 2 adjoint of ∆ C , which is given by where f ∈ C ∞ (X, R). This is an elliptic second order real differential operator, whose index, like the index of ∆ C , is zero. Since the kernel of ∆ C consists just of constants, it follows that dim R ker ∆ * = 1. The strong maximum principle implies that the only smooth real function in the image of ∆ C which has constant sign is the zero function, and this implies that every function in the kernel of ∆ * has constant sign. Thus we can choose a smooth function f ≡ 0 with ∆ * f = 0 and f 0, and then another application of the strong maximum principle shows that in fact f > 0. Therefore we can write f = e (n−1)u , for some u ∈ C ∞ (X, R), and the fact that ∆ * f = 0 exactly says that e u ω is Gauduchon.
The theorem now follows from the Fredholm alternative, since (2.7) means that h is orthogonal to ker ∆ * , which is then equivalent to h being in the image of ∆ C .
For later use, we have the following result. Proposition 2.3. Let (M 2n , ω, J) be a compact almost Hermitian manifold, with ω its associated real (1, 1) form. Then there is a constant C > 0 depending only on (M, ω, J) such that every smooth function ϕ on M which satisfies Proof. Applying Theorem 2.1 we obtain a smooth function u such that the almost Hermitian metric ω ′ = e u ω is Gauduchon. Let ∆ ′C be the canonical Laplacian of ω ′ , which is an elliptic second order differential operator with the kernel consisting of just constants. Standard linear PDE theory (see e.g.
for a constant C > 0, and for all smooth functions ϕ and all x ∈ M . On the other hand we have since ω ′ is Gauduchon. For the "integration by parts" step above, we have used the elementary pointwise equality which holds for any (2n − p)-form α and p-form β. Therefore, we are free to add a large uniform constant to G(x, y) to make it nonnegative, while preserving the same Green formula. If ϕ satisfies (2.8) then we have and so we immediately deduce that M (−ϕ)ω ′n C, from which (2.9) follows.
We end this section with a remark about the use of the maximum principle. Consider a function f ∈ C 2 (M ). Then and the reverse inequality holds at a local minimum. Indeed, the only difference from the integrable case is a first order term which vanishes at x 0 .

Zero order estimate
Here we follow the argument given in [40,Proposition 10] when J is integrable, which is a modification and improvement of an earlier argument by B locki [3,4].
where ε > 0 will be chosen later. Then We define a set P by Note that 0 ∈ P , and that D 2 v(x) 0 as well as |Dϕ(x)| 5 2 ε for all x ∈ P . Now at any x ∈ B the symmetric bilinear form and E(v) is an error matrix which depends linearly on Dv(x) (see e.g. [44, p.443]). Therefore, using the fact that det(A + B) det A + det B for symmetric nonnegative definite matrices A, B, we have for all x ∈ P , (or, one can argue as in [3]). Moreover, and using that |Dϕ(x)| 5ε/2, together with (3.2), we obtain for a uniform constant C. Therefore at any x ∈ P we have provided ε is sufficiently small (this fixes the value of ε), but the Monge-Ampère equation (1.4) then gives us From this, using again (3.1) and |Dv(x)| < ε/2, we conclude that for all Applying the modified Alexandrov-Bakelman-Pucci maximum principle in [40,Proposition 11] we obtain for a constant C 2n which depends only on the dimension of M , and so where |P | denotes the Lebesgue measure of P . But for any x ∈ P we also have and we may assume that I + ε 2 < 0, so using Proposition 2.3, and so which gives us the desired uniform lower bound for I.
We remark that it is also possible to prove the zero order estimate in our case by the method of Moser iteration [56], much like in [46] (where J is integrable), but the calculations are longer. The Moser iteration method was also used by Delanöe [13] for the equation suggested by Gromov (as discussed in the introduction), but in that case the argument is more similar to that of the Kähler case.

First order estimate
In this section, we prove a first order a priori estimate for ϕ, which uses the zero order estimate of Section 3. This part of the argument is similar to [36,Lemma 3.3], except that here we replace Pliś's auxiliary plurisubharmonic function by a barrier function involving the solution ϕ. Proof. We will prove this estimate by applying the maximum principle to the quantity Q = e f (ϕ) |∂ϕ| 2 g , for a function f = f (ϕ) to be determined later. We will show that at the maximum point of Q, |∂ϕ| g is uniformly bounded from above.
First we discuss coordinates. Let {e 1 , . . . , e n } be a local frame for T (1,0) M and let {θ 1 , . . . , θ n } be a dual coframe. We write g ij = g(e i , e j ). The (1,1) form ω is given by where here and henceforth, as should be clear from the context, we are summing over repeated indices (on occasion, for clarity, we will include the summation). We defineω and writeg ij for the associated metric, defined byω = and hence We now assume for the rest of this section that our local frame {e 1 , . . . , e n } is unitary with respect to g, so that g ij = δ ij . Therefore in these coordinates, |∂ϕ| 2 g = k ϕ k ϕ k , where we are writing ϕ k = e k (ϕ) and ϕ k = e k (ϕ) . Fix a point x 0 at which Q achieves its maximum on M . Then, after making a unitary transformation, we may and do assume thatg ij is diagonal at x 0 .
Define a second order elliptic operator Compute at x 0 , using (2.11), (4.2) We now compute each of these three terms in turn. First where we have used (4.1) for the last line. Next,    Using these last two inequalities in (4.4), and combining with Young's inequality, we obtain for ε ∈ (0, 1/2] (to be determined later), (4.8) We now deal with the third term on the right hand side of (4.2), 2Re(g ii e i (e f )e i (|∂ϕ| 2 g )) = 2Re kg ii e f f ′ ϕ i ϕ k e i e k (ϕ) + 2Re kg ii e f f ′ ϕ i ϕ k e i e k (ϕ) . For the first of these terms, we have ii . (4.10) For the second term of (4.9), 2Re kg using the fact that for any real numbers a and b, and ε ∈ (0, 1/2], Combining (4.9), (4.10), (4.11), kg ii e f |e i e k (ϕ)| 2 . (4.12) We now put together (4.2), (4.3), (4.8), (4.12) to obtain, at x 0 , for a uniform constant C 0 . We now choose the function f = f (ϕ) and constant ε > 0 as follows. Define for a large constant A to be determined shortly. Note that since sup M ϕ = 0, the constant ε is small. Compute, at x 0 , In particular, note that f ′ is negative. Choosing A = 12C 0 we obtain lower bounds, at x 0 , for a uniform constant C > 0. In (4.13), after dividing by e f , and increasing C if necessary, we obtain Dividing by |∂ϕ| 2 g (which we may assume is larger than 1, without loss of generality), we obtain uniform upper bounds for ig ii and |∂ϕ| 2 g at x 0 . Combining the upper bound of ig ii with the equation detg = e F , we obtain a uniform upper bound forg ii for each i. Then as required. This completes the proof of the proposition.
Before we end this section, we state the following lemma which we will need later. It is an immediate consequence of the previous proposition, and the estimate (4.8), (but taking now ε = 1/2).

Second order estimate
In this section we prove: where ∇ 2 ϕ denotes the real Hessian of ϕ with respect to the metric g (using its Levi-Civita connection).
To be precise, the dependence of the constant C on F is as follows: C depends only on upper bounds for sup M F, sup M |∂F | g , and a lower bound for ∇ 2 F w.r.t. g. It does not depend on inf M F .
Proof. We first make the preliminary observation that everywhere on M , where λ 1 (∇ 2 ϕ) is the largest eigenvalue of the real Hessian ∇ 2 ϕ (with respect to the metric g). Indeed, if we write λ 1 (∇ 2 ϕ) λ 2 (∇ 2 ϕ) . . . λ 2n (∇ 2 ϕ) for all the eigenvalues, then On the other hand (2.5) gives using Proposition 4.1. This inequality implies that λ 1 −C, and so and it also implies that |λ 2n | Cλ 1 + C, and (5.1) follows. Therefore it suffices to bound λ 1 (∇ 2 ϕ) from above. To achieve this we apply the maximum principle to the quantity Q = log λ 1 (∇ 2 ϕ) + h(|∂ϕ| 2 g ) + e −Aϕ , on the set {x ∈ M | λ 1 (∇ 2 ϕ(x)) > 0}, which we may assume is nonempty, without loss of generality. Here h is given by and A > 1 is a constant to be determined (which will be uniform, in the sense that it will depend only on the background data). Observe that h(|∂ϕ| 2 g ) is uniformly bounded, and we have where we evaluate h and its derivatives at |∂ϕ| 2 g . Note that Q is a continuous function on its domain, and goes to −∞ on its boundary (if this is nonempty), and hence achieves a maximum at a point x 0 ∈ M with λ 1 (∇ 2 ϕ(x 0 )) > 0. However, Q is not smooth in general, since the eigenspace associated to λ 1 may have dimension strictly larger than 1. We deal with this using a perturbation argument, as in [40,41].
First we discuss the choice of coordinate system in a neighborhood of x 0 . Let V 1 be a unit vector (with respect to g) corresponding to the largest eigenvalue λ 1 of ∇ 2 ϕ, so that at x 0 , Since g is almost Hermitian, it follows easily that we can find a coordinate system x 1 , x 2 , . . . , x 2n centered at x 0 , such that in the frame ∂ 1 , . . . , ∂ 2n the almost complex structure J is standard at x 0 (i.e. J∂ 1 = ∂ 2 , etc.) and the vectors ∂ 1 , . . . , ∂ 2n are g-orthonormal at x 0 . Furthermore, after making a quadratic change of coordinates, we assume that the first derivatives of g vanish at x 0 : (5.5) ∂ γ g αβ | x 0 = 0, for all α, β, γ = 1, . . . , 2n.
If at x 0 we let (5.6) then these form a frame of (1, 0) vectors at x 0 , and we have that g ij := g(e i , e j ) = δ ij (i.e. the frame is g-unitary). By performing a further linear change of coordinates at x 0 , which commutes with J, we may assume that at x 0 we have g ij = δ ij and (g ij ) is diagonal with This does not affect condition (5.5), or the fact that J is standard at x 0 , and since g is almost Hermitian we see that the (new) real vectors ∂ 1 , . . . , ∂ 2n are still g-orthonormal at x 0 . We extend e 1 , . . . , e n smoothly to a g-unitary frame of (1, 0) vectors in a neighborhood of x 0 . The coordinate system and the local unitary frame are now fixed. Extend V 1 to an orthonormal basis V 1 , . . . , V 2n of eigenvectors of ∇ 2 ϕ (with respect to g) at x 0 , corresponding to eigenvalues λ 1 (∇ 2 ϕ) λ 2 (∇ 2 ϕ) . . . λ 2n (∇ 2 ϕ). Write {V α β } 2n α=1 for the components of the vector V β at x 0 , with respect to the coordinates x 1 , . . . , x 2n described above. We extend V 1 , V 2 , . . . , V 2n to be vector fields in a neighborhood of x 0 by taking the components to be constant. Note that we do not assert that the V i are eigenvectors for ∇ 2 ϕ outside x 0 .
We apply a perturbation argument. We define near x 0 a smooth section B = (B αβ ) of T * M ⊗T * M by setting its value in our coordinates x 1 , . . . , x 2n at x 0 to be B αβ = δ αβ − V α 1 V β 1 , and extending it to be constant in these coordinates nearby x 0 . At x 0 , and B(Y, Y ) = 1 for any unit vector Y g-orthogonal to V 1 . It is convenient to work with endomorphisms of T M instead of symmetric sections of T * M ⊗ T * M . We define a local endomorphism Φ = (Φ α β ) of T M by Now instead of λ 1 (∇ 2 ϕ), we consider λ 1 (Φ), the largest eigenvalue of the endomorphism Φ. Note that B αβ is nonnegative definite and hence λ 1 (Φ) λ 1 (∇ 2 ϕ) in a neighborhood of x 0 while, by definition of B and V 1 , λ 1 (Φ) = λ 1 (∇ 2 ϕ) at x 0 . Moreover, at x 0 , the eigenspace of Φ corresponding to the largest eigenvalue now has dimension 1, and hence λ 1 = λ 1 (Φ) is smooth in a neighborhood of x 0 .
The first and second derivatives of λ 1 at x 0 are given by the following lemma.
Proof. Consider the constant orthogonal matrix V = (V α β ), where V α β were defined above. It has the property that V T ΦV is diagonal at x 0 . Define Θ = (Θ α β ) := V T ΦV , which has the same eigenvalues as Φ. Since Θ is diagonal at x 0 , with λ 1 distinct from all the other eigenvalues, we have the following well-known formulas (see e.g. [39]): Hence, by the chain rule, we compute at x 0 , Similarly, as required.
As in the previous section, we denote by L the operator L =g ij (e i e j − [e i , e j ] (0,1) ).
We first prove a lower bound for L(λ 1 ). In what follows, we assume without loss of generality that λ 1 >> 1 at x 0 . We also note that from (1.4) and the arithmetic-geometric mean inequality, for a uniform c > 0.
where we are writing Proof. Using (5.5), (5.8) and the fact thatg is diagonal at x 0 , Next, we claim that Given this, the lemma follows, since in our coordinates, our equation (1.4) is log detg = log det g + F.
We now give the proof of (5.12). We have since, for any α, β, First we deal with the second term of (5.14). We claim that ii .
Next we note that for a uniform constant C, and so combining this with (5.14), (5.15), In what follows, we write E for a term (which may change from line to line) which can be bounded by Cλ 1 ig ii . We havẽ Recalling (4.1), and combining this with (5.16), we obtain (5.12).
We use the above lemma to obtain a lower bound for L(Q) at x 0 : Lemma 5.4. For any ε with 0 < ε 1/2, we have at x 0 , We deal with the bad third order terms on the second line. At x 0 we can write for some complex numbers ν iα which are uniformly bounded, hence and so where we used (5.2). Thus the lemma follows from (5.18).
We need to deal with the "bad" negative term in (5.17). We split up into different cases. The constant ε > 0 will be different in each case.
In this case we just choose ε = 1 2 . We use the fact that, at x 0 , the first derivative ofQ vanishes. Hence, using the elementary inequality |a + b| 2 4|a| 2 + 4 3 |b| 2 , for a, b ∈ C, (5.20) Hence from (5.17), discarding some positive terms, we obtain at x 0 , But from (5.4), we have h ′′ = 2(h ′ ) 2 . In case (a) note that all the numbers g ii , 1 i n, are comparable to each other up to a uniform constant (which depends on A). Using again (5.9), as well as Propositions 3.1 and 4.1, we have for a uniform constant C A (depending on the uniform constant A), and since all theg ii are comparable to each other this gives But at x 0 we can define the complex covariant derivatives ϕ e i e k and ϕ e i e k in the obvious way, which satisfy ϕ e i e k − e i e k (ϕ) = −(∇ e i e k )(ϕ), ϕ e i e k − e i e k (ϕ) = −(∇ e i e k )(ϕ), and these differences are uniformly bounded thanks to Proposition 4.1. Therefore and recalling (5.6), we see that and so λ 1 (x 0 ) is uniformly bounded. Thanks to Propositions 3.1 and 4.1, this shows that Q is bounded from above at x 0 , and hence everywhere.
On the other hand, in case (b), we obtain for a uniform C 0 . Then as long as we see that at x 0 we have ig ii C, and so also (5.26) tr gg (trgg) n−1ω n ω n C.
Therefore at x 0 we have thatg is uniformly equivalent to g, and (5.24) then shows that (5.21) holds again, and so Q is bounded from above at x 0 . This completes Case 1.

Case 2. Neither (a) nor (b) of Case 1 hold.
This is the difficult case, and it will take up the rest of this section. Let We have n ∈ I (since A > 1), and 1 ∈ I (otherwise we are in part (a) of Case 1). So for example when n = 2 we have I = {1}. We will use four lemmas to complete the proof. The first deals with the relatively harmless part of the bad term (5.19).
Lemma 5.5. Assuming that A is larger than 6n sup M |∂ϕ| 2 g , we have Proof. Using the fact that the first derivative ofQ vanishes at x 0 , as in (5.20), we compute where in the last line we used the assumption A 6n sup M |∂ϕ| 2 g .
The next lemma shows that, roughly speaking, and modulo terms of order O(λ −1 1 ), the largest eigenvector V 1 lies in the directions where the Hermitian metric (g ii ) at x 0 is not too small. More precisely, we define in our coordinate patch a (1, 0) vector field bỹ We write at x 0 , (5.29)ẽ 1 = n q=1 ν q e q , n q=1 |ν q | 2 = 1, for complex numbers ν 1 , . . . , ν n , where the second equation follows from the fact thatẽ 1 is g-unit at x 0 . Then we have: Lemma 5.6. For a uniform constant C A depending on A, we have where the ν q are given by (5.29).
Proof. To prove the claim, we see from Hence by the definition of I, For convenience, write I = {1, 2, . . . , j} for some j with 1 j < n. Then, arguing exactly as in (5.21), (5.22), (5.23), the estimate (5.30) implies that where we write C A for a uniform constant depending on A (recall that A is yet to be determined, but will be chosen uniformly).
The lemma then follows easily from the definition of the e i .
The goal for our last two lemmas is to obtain a lower bound for the first three terms on the right hand side of (5.17), excluding those bounded by Lemma 5.5. The first of these two lemmas is a technical intermediate step.
We make use of real numbers µ 2 , µ 3 , . . . , µ 2n defined by where we recall that at x 0 the vector JV 1 is g-unit and g-orthogonal to V 1 .
Proof. We first claim that where E denotes a term satisfying |E| Cλ 1 for a uniform C. Defining ϕ V 1ẽ1 in the obvious way, we have at x 0 , since the error terms arising from switching the order of operators give terms involving only two derivatives of ϕ.
With ν 1 , . . . , ν n ∈ C as in (5.29), we have at x 0 , Recalling (5.32), Hence, for a uniform C, using Lemma 5.6. For γ > 0, we bound (5.39) But we also have, by the Cauchy-Schwarz inequality, recalling that we have α>1 µ 2 α = 1. So the conclusion is that Also, using (4.1), we see that at x 0 we havẽ for any q, using that λ 1 may be assumed to be large. Therefore, if λ 1 for the constant C A of (5.38), and if also λ 1 C ′ , then we have Therefore we obtain the inequalitỹ We use this to prove the final lemma: for a uniform constant C > 0, assuming without loss of generality that at Proof. It suffices to show that, at x 0 , We now prove (5.42). At x 0 we have 43) using that at x 0 we have, recalling (5.7) and (5.32), The proof splits into two cases.
Case (i). Assume that at x 0 we have Then Lemma 5.7 gives and we can now choose so that the second and fourth term on the right hand side cancel each other, while the first term dominates the last one, and this establishes (5.42).
Case (ii). Assume on the other hand that at x 0 we have Then (5.43) gives and so In general, (5.43) implies that as long as 2λ 1 C/ε 2 . We now choose γ = 1 ε 2 , where ε still has to be chosen. This gives if ε 1 6 , and from Lemma 5.7 we get But thanks to (5.46) we have and so provided λ 1 C ε 3 at x 0 . We now complete the proof of Proposition 5.1. Combining Lemmas 5.4, 5.5 and 5.8, and recalling that h ′′ = 2(h ′ ) 2 , we have ii (|e i e k (ϕ)| 2 + |e i e k (ϕ)| 2 ) + (A 2 e −Aϕ − 6εA 2 e −2Aϕ )g ii |e i (ϕ)| 2 − Ane −Aϕ , as long as ε 1 6 , where C 1 is uniform constant. We now make our choices of A and ε. First choose A = 6C 1 + 1, and, at the expense of increasing C 1 , we may assume that A max(C 0 + 1, 6n sup Note that now that the values of ε and A have been fixed, we may indeed assume without loss of generality that at x 0 we have λ 1 C ε 3 and λ 1 C A ε . We conclude that at x 0 we have (5.47) ig ii + h ′ 2 kg ii (|e i e k (ϕ)| 2 + |e i e k (ϕ)| 2 ) C.

Completion of the proofs of the Main Theorems
In this section, we complete the proofs of the main results stated in the introduction.
Proof of Theorem 1.2. Thanks to Propositions 3.1 and 5.1, we can obtain a priori C 2,α estimates from the main result of [44], for some 0 < α < 1 (and in fact for all 0 < α < 1 by [10]). Higher order estimates follow after differentiating the equation and applying the usual bootstrapping method.
Our main result now follows easily: Proof of Theorem 1.1. Now we have Theorem 1.2, we can follow closely the arguments of [45]. Indeed, we will see that the (1, 1) form √ −1∂∂ϕ = 1 2 (d(Jdϕ)) (1,1) has the right properties for this part of the proof to go through essentially unchanged from the integrable case. We include the brief arguments for the sake of completeness. Consider the family of equations (6.1) (ω + √ −1∂∂ϕ t ) n = e tF +bt ω n , ω + √ −1∂∂ϕ t > 0, for t ∈ [0, 1], where b t ∈ R. For a fixed α ∈ (0, 1), we define T to be the set of t ∈ [0, 1] such that (6.1) admits a solution (ϕ t , b t ) ∈ C 3,α (M ) × R. Clearly 0 ∈ T . We will show that T is open and closed, since this will show that 1 ∈ T , and then adding a constant to ϕ 1 we will obtain a solution of (1.3).
To prove that T is closed we will apply Theorem 1.2. Suppose ϕ t ∈ C 3,α (M ) solves (6.1). Then, as above, we can differentiate the equation and bootstrap to obtain that ϕ t is smooth. Applying the maximum principle (2.11) to ϕ t in (6.1) we obtain the bound |b t | sup M |F |. We may add a t-dependent constant to ϕ t solving (6.1) so that sup M ϕ t = 0. Now we apply Theorem 1.2 to ϕ t to obtain C 3,α (in fact, C ∞ ) estimates which are independent of t. This shows that T is closed. Hence we have proved the existence of a C 3,α , and hence smooth, solution of (1.3).