On the Alesker-Verbitsky conjecture on hyperK\"ahler manifolds

We solve the quaternionic Monge-Amp\`ere equation on hyperK\"ahler manifolds. In this way we prove the ansatz for the conjecture raised by Alesker and Verbitsky claiming that this equation should be solvable on any hyperK\"ahler with torsion manifold, at least when the canonical bundle is trivial holomorphically. The novelty in our approach is that we do not assume any flatness of the underlying hypercomplex structure which was the case in all the approaches for the higher order a priori estimates so far. The resulting Calabi-Yau type theorem for HKT metrics is discussed.


Introduction
The quaternionic Monge-Ampère equation on compact HKT manifolds was introduced by Alesker and Verbitsky in [AV10]. On a general hyperhermitian manifold (M, I, J, K, g) of quaternionic dimension n it takes the form (Ω + ∂∂ J φ) n = e f Ω n , Ω + ∂∂ J φ > 0. (1.1) The goal of this paper is to prove the higher order estimates for the quaternionic Monge-Ampère equation (1.1) on compact hyperKähler manifolds. The highlight of our result is that we do not assume any flatness or additional integrability of the underlying hypercomplex structure as was always the case for the higher order estimates known so far. The hyperKähler condition, as will be seen during the derivation of the estimates, may in turn be interpreted as a curvature condition which we hope can be removed. As a result we solve this equation on any compact hyperKähler manifold, cf. Remark 6.1.
Our main result is stated as follows: is satisfied.
Equation (1.1) was motivated by the prior research in the local case, cf. [A03], and the attempt to prove the analog of the Calabi conjecture in quaternionic geometry, cf. [AV10,V09], as we explain in the next section. It was conjectured in [AV10] that equation (1.1) can always be solved at least in the case when the canonical bundle of the HKT manifold is trivial holomorphically. Given the recent progress in solving Calabi-Yau type equations for non Kähler metrics, cf. [GL10,TW10b,SzTW17,TW17,Sz18,TW19], this is expected to hold even without the latter assumption, cf. [AS17,Sr19]. To sum up Theorem 1.1 constitutes the ansatz for proving the following conjecture.
(1.6) It may be checked, cf. [AV06] or [Sr18] for an elementary calculation, that in this case where Θ is the canonical trivialization of K I (H n ) and the determinant has to be understood in a proper way -as the Moore determinant, cf. [M22], of the hyperhermitian matrix. The Dirichlet problem for the operator (1.7) was first considered by Alesker in [A03] where continuous solutions were found for continuous right hand sides. After that, the problem was solved in the smooth category by Zhu [Z17]. Later, thanks to the form of (1.7), the pluripotential approach was taken up resulting in providing continuous solutions even for right hand sides in L p spaces. For p ≥ 4 it is due to Wan, cf. [W20], and for p > 2 due to the second named author, cf. [Sr18]. In the latter case the exponent was proven to be optimal. This equation is also covered by the very general approach taken up in the last two decades by Harvey and Lawson. They provide viscosity solutions for the Dirichlet problem even for domains in quaternionic manifolds, cf. [HL09,HL11,HL20]. Coming back to the advances towards proving Conjecture 1.2. Generally speaking the difficulties with obtaining a priori estimates for the equation (1.1) are caused, as we explain in depth in the next section, by the fact that a generic hypercomplex structure locally is not the pull back of the flat structure from H n . The lack of quaternionic coordinates forces one to work in the general case, at best, with holomorphic coordinates for the reference complex structure I. But this in turn results in equation (1.1) depending not only on the coefficients of the metric tensor but also of the endomorphism field J since, in the holomorphic coordinates for I, (1.8) Proving estimates in such coordinates is similar to solving the complex Monge-Ampère equation by performing the calculations in generic real coordinates. Another drawback is that equation (1.1), instead of being of the form (1.7), is an equation on the Pfaffian of the coefficients of the two form Ω + ∂∂ J φ in the complex coordinates Pf Ω φ ij i,j = e f · Pf Ω ij i,j . (1.9) As will be seen in the calculations, the more times the equation (1.9) is differentiated the more the lack of quaternionic derivatives becomes an issue. In the paper [AV10], the C 0 estimates for the solutions to the equation (1.1) were derived by the classical Moser iteration method in the setting exactly as in Conjecture 1.2. There the existence of the holomorphic trivialization is crucial for the argument to work. Later it was shown that the C 0 estimate holds for (1.1) even without the assumption on the holomorphic triviality of K I (M ), cf. [AS17,Sr19], but the methods are much more involved. As for the higher order estimates, as we mentioned, they have been derived only under the assumption that the hypercomplex structure is integrable in the strong sense, i.e. is locally flat. In addition, either the initial metric should be very special or the manifold has to admit in addition a hyperKähler metric compatible with the flat hypercomplex structure. Precisely, in [A13] it was shown that in the case the manifold is a torus or its quotient endowed with the flat hyperKähler metric (this implies in particular the flatness of the hypercomplex structure), the Laplacian bound on the solution of (1.1) holds. Alesker proved also that the analogue of the Evans-Krylov theorem, cf. [E82], holds under the assumption of flatness of the hypercomplex structure. In [GV21] the authors show the Laplacian bound, unlike in Alesker's result depending on the gradient bound, for equation (1.1) on certain eight dimensional nilmanifolds endowed with the flat hypercomplex structures and torus action invariant initial HKT metrics. Shortly before this preprint was written down the preprint [BGV21] appeared on arXiv. The authors take up there the parabolic approach for the equation (1.1) but the assumption under which they are able to prove the convergence of the flow to the solution are exactly as in Alesker's paper [A13].
In the current note we carry out the computations for all the higher order estimates in geodesic coordinates for the Obata connection. This seems to be the main technical input which allows us to overcome the issues coming from the dependence of the equation (1.1) on the second complex structure J. This is still not enough though to deal with the dependence on the metric. Because of that the hyperKähler assumption appears.
Our strategy in this paper is as follows. First of all, motivated by an influential idea of B locki from [B09] (see also [Gu]), we show, cf. Theorem 3.1, the gradient estimate in the setting of Theorem 1.1. Here, the hyperKähler condition plays similar role as non negativity of the holomorphic bisectional curvature of the background Kähler metric in the case of the complex Monge-Ampère equation. This is somewhat surprising since the hyperKähler metric is not necessarily of such a curvature. In the general hypercomplex case the gradient estimate does not seem to follow from arguments as in [B09]. The troubles in this case are caused partially by the failure of the Leibniz rule for the operators (1.5) and (1.6). Let us mention that (direct) gradient estimate is not known for general complex Hessian equations. We use Theorem 3.1 in Section 5 to derive, with the aid of Theorem 4.1, the full C 2 estimate. In this case the proof is standard and relies on an idea of B locki from [B11]. The crucial result is Theorem 4.1 proven in Section 4 which gives the bound on the Laplacian, or equivalently the quaternionic Hessian ∂∂ J φ, for the solutions of (1.1). Let us remark that for general HKT metric a major issue is how to handle third order terms. Roughly speaking the positive term appearing, being formed by the squares of sums of certain third order derivatives, compensates only half of the negative term. At the moment we do not know how to deal with this difficulty and the classical methods of [Y78,B09,GL10] does not seem, cf. Remark 4.2, to work. The hyperKähler assumption allows us to get rid of some terms coming from differentiating the metric coefficients and in turn, by considering a more general perturbation than the classical one in the Pogorelov approach for the Laplacian bound, cf. [Y78,B09,GL10], we are able to ignore the negative term.
As we explain in details in the next section, Theorem 1.1 allows one to draw some conclusions concerning the HKT geometry of the underlying hyperKähler manifold. First of all having a hypercomplex manifold (M, I, J, K) any hyperhermitian metric g provides a canonically associated smooth section of the canonical bundle K I (M ) via the map g −→ Ω n . (1.10) Sections obtained in (1.10) satisfy the properties, defined rigorously in the next section, which we call positivity and J-realness.
Proposition 1.3. Let (M, I, J, K, g) be a compact, connected hyperKähler manifold. Given any positive and J-real trivialization Θ, i.e. of the form e f Ω n for some smooth function f , of the canonical bundle K I (M ) there is an HKT metricĝ such that the associated HKT formΩ satisfieŝ Ω n = Θ. (1.11) What is more the metricĝ may be chosen so that the associated HKT form is of the form a Ω + ∂∂ J φ for a smooth function φ and a positive constant a.
From Proposition 1.3 it is easy to obtain the so called Calabi-Yau type theorem for HKT metrics, yet only on hyperKähler manifolds.
Remark 1.4. Let (M, I, J, K, g) be a compact, connected hyperKähler manifold. Given any positively oriented volume form σ on (M, I) there is an HKT metric g φ , such that the associated HKT form is a Ω + ∂∂ J φ , for a smooth function φ and a positive constant a, such that (1.12) A classical calculation allows us to obtain the following as a result of Remark 1.4.
Proposition 1.5. Let (M, I, J, K, g) be a compact, connected hyperKähler manifold. Given any representative ρ ∈ c BC 1 (M, I) of the first Bott-Chern class of (M, I), see [T15], there is an HKT metric g φ , whose associated HKT form is Ω + ∂∂ J φ, such that Ricc(∇ Ch I,g φ ) = ρ. (1.13) In (1.13) the symbol Ricc(∇ Ch I,g φ ) denotes the well known Chern-Ricci form associated to the Chern connection of the hermitian structure (I, g φ ) on M .
As the non constant conformal deformation of an HKT metric is never an HKT metric the last three results stated above are non trivial, even under the hyperKähler assumption. Provided one can remove the extra assumption on the initial metric the Calabi-Yau theorem for yet another class of metrics would be settled. Proving the Calabi-Yau type theorem for different class of hermitian (non Kähler) metrics was a subject of an intense study in the last decade. In the classical case of Kähler metrics it is known due to Yau [Y78]. For the class of Gauduchon metrics it was settled only recently in [SzTW17], building on [Sz18,TW19], confirming in turn an old conjecture of Gauduchon from '80s. Actually the same result was proven there for strongly Gauduchon metrics as well. In [TW17] a Calabi-Yau type theorem, Corollary 1.3 in there, for balanced metrics was proven, yet like in our case, under an extra assumption that the manifold admits a Kähler metric. In this case the result follows of course from the Calabi-Yau theorem for Kähler metrics as well. The assumption of admitting Kähler metric was later relaxed to admitting merely an Astheno-Kähler one, cf. [SzTW17], in which case the classical Calabi-Yau theorem can not be applied. It is still an open problem whether the Calabi-Yau type theorem for the class of balanced metric holds in general, cf. [TW19,SzTW17].
Let us finish this introduction by the remark in the spirit of the mentioned Harvey and Lawson theory. Like in the local case, in the global situation the equation (1.1) is a companion of the real and complex Monge-Ampère equations which received great attention in the last century and, in certain forms, are related to the fundamental differential geometric problems.
The complex Monge-Ampère equation on complex n dimensional hermitian manifold (M, I, g) taking the form (ω + i∂∂φ) n = e f ω n (1.14) was solved, as we mentioned, on Kähler manifolds by Yau [Y78] and on general hermitian manifold by Tosatti and Weinkove [TW10a,TW10b], cf. also [GL10]. It was proven that it can be solved even on almost complex manifolds, cf. [ChTW19].
As for the real Monge-Ampère equation on a Riemannian manifold (M, g) of dimension n, its rough version does not carry a substantial geometric meaning. It is nevertheless interesting from an analytic point of view. In this mentioned rough form it can be written as det g + (∇ LC ) 2 φ = e f det g. (1.15) In (1.15) the meaning of taking the determinant is that the symmetric bilinear forms g+(∇ LC ) 2 φ and g are treated, via the metric, as the endomorphisms of the tangent bundle and we take the determinant of this endomorphisms. These equations were treated for example in Li [L90], where they were solved under the non negative curvature assumption on g. This assumption was later removed by Urbas in [U02]. Let us just mention that in case of this equation it is hard to obtain an easy normalization of f 's for which the equation can be solved. The very special modification of this equation was treated earlier by Cheng and Yau in [ChY82]. They considered affine manifolds, i.e. smooth real manifolds endowed with a flat torsion free connection, admitting a Riemannian metric, which they called affine Kähler metric, being locally given as the Hessian of a potential function in affine coordinates. The equation they considered is obtained by taking the affine Kähler metric in (1.15) and by exchanging the Levi-Civita connection there for the affine connection.
Acknowledgments: The authors would like to use the opportunity of professor S lawomir Ko lodziej's 60th birthday to express their gratitude to him for his constant support and advice. The first named author is supported by the National Science Center of Poland grant no. 2017/26/E/ST1/00955. The second named author is supported by the National Science Center of Poland grant no. 2019/35/N/ST1/01372.

Preliminaries
In this section we introduce the notation and collect basic facts concerning hyperhermitian manifolds. We also prove technical or computational in nature results needed for the a priori estimates of Sections 3-5 in order to make the presentation there more straightforward.

Hypercomplex geometry
We denote by where i 2 = j 2 = k 2 = −1 and ijk = −1 the field of quaternions with addition and multiplication being defined in the standard way. We consider H n as a right H vector space. Let us recall what have become the standard definition.
Definition 2.1. For a manifold M of the real dimension 4n endowed with a triple of complex structures I, J, K satisfying the quaternion relation A hypercomplex manifold admits many complex structures in particular the ones given (the so called twistor sphere).
Remark 2.2. We warn the reader that for us, in the whole text, endomorphisms act from the right on the tangent space. This convention is compatible with the one usually taken up in papers on hypercomplex geometry.
In that case each tangent space T x M , for x ∈ M , becomes a right H-vector space where multiplication by i, j and k is given by I x , J x and K x respectively.
Clearly the structure group of the hypercomplex manifold is reduced to Gl n (H) and vice versa each such reduction induces the almost complex structures I, J and K as in the definition above. The condition of the almost complex structures being integrable is though not equivalent to the induced Gl n (H) structure being integrable in the strong sense of differential geometry, i.e locally I, J and K are not pull backs of the standard hypercomplex structure induced by i, j and k in H n . In case the latter condition is satisfied such structures were studied in [S75]. The latter is also equivalent to the existence of an atlas whose transition functions are affine maps with the endomorphism parts belonging to Gl n (H). On the bright side the integrability of the almost complex structures I, J and K implies the 0-integrability of the induced Gl n (H) structures (the reverse implication holds due to the Newlander-Nirenberg theorem), i.e. the existence of the Gl n (H) compatible torsion free connection. This is a non obvious result of Obata. Strong integrability of the Gl n (H) structure is equivalent to the Obata connection being in addition flat.
The coordinate expression for this connection can be found in [Ob56]. An invariant global formula can be found for example in Gauduchon's paper [G97b]. The existence of this connection will be very important for the technical results stated below and for the computations involved in deriving the a priori estimates for the equation (1.1).
Remark 2.4. For a more detailed discussion on quaternionic geometry one can refer to the two excellent papers [AM96] and [S86].
When considering a hypercomplex manifold (M, I, J, K) we obtain that acts from the right on T M and from the left on T * M or more generally from the left on differential forms. The convention we use for the latter action is against the commonly used. Namely, given any field of endomorphisms L on T M , acting according to our convention from the right, we define its left action on the space of complex valued smooth differential forms by Remark 2.5. From now on, whenever it happens that on a hypercomplex manifold (M, I, J, K) we do not specify with respect to which complex structure the Hodge bidegree is taken, it is taken with respect to I.
Let (M, I, J, K) be a hypercomplex manifold. Let us remind that we have the Dolbeault operators ∂ := ∂ I and ∂ := ∂ I associated to the complex structure I on M . We are going to introduce the quaternionic analogue of the ∂ operator, or rather d c I := I −1 • d • I. In this we follow Verbitsky, cf. [V02], who defined the differential operator ∂ J by Since the operator J acts on the complex forms by exchanging the bidegree components the operator ∂ J acts on this forms by We also introduce the operator ∂ J defined formally again by twisting but as the operator J is real it is equal to (∂ J ) as well. It was observed by Verbitsky, [V02, V07a], that the bicomplex called by him the quaternionic Dolbeault bicomplex, not only resembles the Dolbeault bicomplex but it is also isomorphic to the so called Salamon complex, cf. [S86], introduced by Salamon in the broader context of quaternionic manifolds (see [V07a] for details).
For the further reference we would like to introduce the notion of J-realness. This is done as follows. The composition of the operator (2.2) with the bar operator is an involution on Λ p,q I (M ) if p + q is even. The bundle of fixed points for this endomorphism in Λ

Hyperhermitian metrics
Definition 2.6. A Riemannian metric g on a hypercomplex manifold (M, I, J, K) is called hyperhermitian if it is hermitian with respect to I, J and K. For a hyperhermitian manifold (M, I, J, K, g) and any L ∈ S M we denote the associated hermitian form by We define also the associated hyperhermitian form It is elementary to check that Ω ∈ Λ 2,0 I,R (M ). Let us elaborate on the role of Ω in encoding the hyperhermitian metric g, cf. [H90] Chapter 2, Lemma 2.72. Suppose (V, g) is a right H-vector space with a hyperhermitian inner product g. Such inner products correspond bijectively to the hyperhermitian sesquilinear forms (2.6) and Ω as above we obtain H =h + jΩ. (2.8) What is more Ω = h(·j, ·). (2.9) Consequentlyh, and in turn also H, is completely determined by Ω satisfying for any v, w ∈ V Ω(vj, wj) = Ω(v, w), or after taking the complexification of V , Ω(·j, ·j) = Ω (2.10) and or after taking the complexification, for any z ∈ V 1,0 i . The conditions (2.10) and (2.11) give the meaning of the inequality in (1.1).
Remark 2.7. In calculations it will be customary to assume that at the point of interest the hyperhermitian structure (T x M, I x , J x , K x , g x ) is isomorphic to the standard model below, cf.
(2.13) and (2.14). This can trivially be seen to be possible by taking the orthonormal basis of the form e 0 , e 0 I, e 0 J, e 0 K, ..., e n−1 , e n−1 I, e n−1 J, e n−1 K for (T x M, I x , J x , K x , g x ). As was discussed in [Sr19] in the presence of a second hyperhermitian metric the first one may still be assumed to be standard while the second one being diagonal.
The right multiplications by i, j, and k act on H n defining the almost complex structures I, J and K respectively. Let us introduce, next to the real coordinates (1.4), the holomorphic coordinates, for the complex structure I, by decomposing (2.12) for i ∈ {0, ..., n−1}. As an easy calculation shows the action of J in this holomorphic coordinates is for j ∈ {0, ..., 2n − 1}. Take a standard inner product on H n , in coordinates from (1.4), We easily get the following expressions for the quantities associated with this hyperhermitian structure (H n , I, J, K, g) (2.14) Remark 2.8. One should note that the real coordinates introduced in (1.4) are not given by taking the real and imaginary part decomposition of the complex coordinates (2.12). The relation between these coordinates is Before introducing certain classes of hyperhermitian metric let us recall that in the paper [G97a] Gauduchon has distinguished in the affine space of all the hermitian connections, i.e. those satisfying ∇I = ∇g = 0 for a hermitian manifold (M, I, g), the affine line of the canonical connections. Among them two classical ones will be important for our presentation.
Proposition 2.9. [G97a] Let (M, I, g) be a hermitian manifold. There exists a unique hermitian connection, denoted by ∇ Ch I,g , which will be called the Chern connection characterized by the fact that ∇ Ch There exists as well a unique hermitian connection, denoted by ∇ B I,g , which will be called the Bismut connection characterized by the fact that its torsion tensor, after lowering the upper index by g, is a three form.
Coming back to the hyperhermitian metric on hypercomplex manifolds, when considering the above connections associated to the hermitian structure (M, L, g) for any L ∈ S M we add a subscript L, eg. ∇ Ch L,g . Let us recall the following definition: Definition 2.10. A hyperhermitian metric g on (M, I, J, K) is called hyperKähler, HK for short, if any of the equivalent conditions are satisfied If moreover M is simply connected the conditions above are equivalent to and I, J, K being induced by this holonomy group.
This class of metrics is standard to consider from the point of view of Berger's Riemannian holonomy theorem. Indeed it implies that Sp(n), and Sp(n) · Sp(1), corresponding respectively to the hyperKähler and quaternionic Kähler metrics, are the only infinite families occurring, cf. [Be87], which correspond to the hypercomplex, respectively quaternionic, geometry. There are only two known deformation classes in each dimension and two isolated examples, due to O'Grady, of hyperKähler manifolds. Partially because of that one may be tempted to look for a natural generalizations of those. A possible attempt is as follows.
Definition 2.11. A hyperhermitian manifold (M, I, J, K, g) is called HKT, which stands for hyperKähler with torsion, if any of the equivalent conditions is satisfied From the second condition it follows that HKT structures are natural differential geometric generalizations of HK structures where the torsion just vanishes. These metrics emerged originally from mathematical physics. More exactly connections with special holonomy and skew torsion occur naturally while studying the target space of sigma models in quantum theory. In the presence of the so called Wess-Zumino term and supersymmetry the HKT metrics appear, cf. [HP96].
An established mathematical treatment of basic properties of HKT manifolds is [GP00]. One should note though that despite the name HKT manifolds do not admit Kähler metrics in general. In fact a result of Verbitsky [V05] shows they can be Kähler only in the case when the manifold already admits HK metric.

Quaternionic Monge-Ampère equation
The quaternionic Monge-Ampère equation (1.1) on HKT manifolds proposed by Alesker and Verbitsky, cf. [AV10], naturally solves the prescribed trivialization problem (1.10)-(1.11). The authors suggested to look for an HKT metric whose associated HKT form is for some smooth real function φ and for which Ω n φ is the section we want to obtain because the ∂∂ J φ perturbation preserves the HKT condition. If such a φ exists then this new HKT metric g φ can be obtained from Ω φ by applying the reasoning from the section above. We denote the associated hermitian forms by adding a subscript g φ , eg. ω I,g φ .
As we noted the canonical bundle K I (M ) of a given HKT manifold is trivial topologically, with Ω n providing a smooth trivialization. Unlike in the Kähler case where, up to the finite covering, topological triviality gives the holomorphic one, here the canonical bundle is not holomorphically trivial in many cases, Hopf surfaces being one example. Arguably, cf. [V09], HKT metrics for which Ω n is holomorphic trivialization constitute a hypercomplex analogue of a Calabi-Yau manifold. These metrics have in particular the property of being balanced with respect to any complex structure from S M . They perfectly fit into the recently active stream of research on generalizations of Calabi-Yau spaces, the so called torsion Calabi-Yau manifolds, cf. [T15,Pi19].
Let us now turn to other consequences of Theorem 1.1 advocated in the introduction. In doing so let us also keep in mind that actually an expectation is that Conjecture 1.2 should hold even after dropping the assumption on the holomorphic triviality of the canonical bundle K I (M ).
For a hyperhermitian manifold (M, I, J, K, g) and the constant c n depending only on the dimension, as can be seen from (2.14), Ω n ∧ Ω n = c n ω 2n I .
Consequently we see that the quaternionic Monge-Ampère equation (1.1) is solvable for any f if and only if is satisfied with suitable φ and C for any F . This shows that Remark 1.4 follows from Theorem 1.1. The above means that any representative of c BC 1 (M, I) can be obtained as the Chern-Ricci curvature of an HKT metric g φ as the standard calculation shows. This was noted already in [Ma11] for what Madsen calls the projected Chern form, cf. Section 7.1.3 in [Ma11]. Indeed, as is well known prescribing the Chern-Ricci curvature is equivalent to solving the Monge-Ampère equation This in turn means, by going from the chart expression to the global one, that ω n I,g φ = e F +b ω n I justifying that Proposition 1.5 follows from Remark 1.4.

Technical results
From now on we assume that the components of all tensors are taken with respect to a holomorphic coordinates z i for I. Another important convention we use is that whenever an unknown, eg. i, appears as an index its range is in {0, ..., 2n − 1}. When an expressions 2i or 2i + 1 involving an unknown appears as an index the range for i is in {0, ..., n − 1}. We often omit the summation symbols when it is clear that the summation takes place even when the Einstein summation convention does not apply directly. First of all let us note the following properties of the operators ∂, ∂ J , ∂, ∂ J introduced in this section. This result is elementary and based only on the integrability and anti commutativity of I and J but we do not know a reference containing the proof.
Lemma 2.12. For a hypercomplex manifold (M, I, J, K) the following holds Proof. One can simply use the facts that This follows from the integrability of J and K. Then by rewriting d as ∂ + ∂ we obtain . Comparing both sides in (2.17) and (2.18) and taking into an account the Hodge bidegrees with respect to I gives the claim. More precisely (2.17) gives while (2.18) gives (2.20) It turns out that ∂∂ J + ∂ J ∂ = 0, as it is the component of bidegree (2, 0) of the left hand side of (2.19), as it is of bidegree (0, 2) of (2.19), as it is of bidegree (1, 1) of (2.19), as it is of bidegree (1, 1) of (2.20). Adding and subtracting (2.21) and (2.22) we obtain The only two remaining identities ∂∂ + ∂∂ = 0, The first conclusion we may draw from this is, what we will use constantly during the computations of the sections to follow, that certain identities involving derivatives of the components of J vanish locally and not only at a fixed point.
Remark 2.13. For any holomorphic coordinates for I we have In coordinates this reads This in turn provides J ī k,l = J ī l,k and by conjugation Jī k,l = Jī l,k for all i, j and k.
In order to have even better control of the derivatives of J we have to stick to the point. Let ∇ Ob be the Obata connection for (M, I, J, K). Since it is the complex, in particular for I, torsion free connection we have, in any holomorphic chart for I, (2.26) The condition

This in turn allows us to obtain the expressions
(2.27) Since the Obata connection is torsion free and I-complex, for any chosen p ∈ M we can choose I-holomorphic, geodesic coordinates which, from (2.27), gives at p the equalities J l k,i = J l k,i = Jl k,ī = J l k,ī = 0. (2.28) Finally, let us find the expression for the ∂∂ J φ perturbation at the point, in local coordinates.
Lemma 2.14. At the point on (M, I, J, K), in any coordinates satisfying (2.13), for any φ we have: (2.29) The last one, after rearrangement, gives (2.30) In the coordinates in which ∂∂ J φ is diagonal, formula (2.30) reads for any i, j.
Using the above discussion we provide, for later reference, the expression for the Chern laplacian involving the quaternionic Hessian. ∂∂ J φ ∧ Ω n−1 Ω n = ∆ Ch I,g φ.
Proof. It is well known that the Chern laplacian can be expressed as Let us choose any holomorphic coordinates such that at the point x ∈ M the hyperhermitian structure is standard, in the sense that (2.13) and (2.14) are satisfied. In those coordinates we see, since On the other hand, because of (2.30) and Ω = dz 2i ∧ dz 2i+1 , we see that also as required.
We recall the basic facts concerning the Pfaffian. The following proposition is probably well known but we do not know the reference. The proof reduces to defining Pfaffian as below and checking the claimed equality (2.33) of polynomials on sufficiently many skew-symmetric matrices.
Proposition 2.16. There exists a polynomial which we denote by Pf , with real coefficients, of degree n on the space of skew-symmetric complex matrices of size 2n, i.e. those satisfying A T = −A, such that det = Pf 2 (2.33) as polynomials on this space.
The polynomial Pf from Proposition 2.16, is defined only up to the sign. We make the following choice. where e i is the canonical basis of C 2n .
With this definition, the mentioned convention for tensors components and notation (1.8), it follows immediately that writing the equation (1.1) in holomorphic coordinates gives the equation (1.9). Note that the original equation (1.1) could have been rewritten in that way only because the associated hyperhermitian forms are of the Hodge type (2, 0).
In deriving the estimates we will differentiate equation (1.9) and in order to do that we would like to know the formula for the derivative of the Pfaffian. As we will be concerned only with the matrices with positive Pfaffian this can be derived from the more familiar formula for determinant derivatives coupled with Proposition 2.16. The result is as follows.
Lemma 2.18. Suppose A = A ij i,j is a complex skew-symmetric 2n × 2n matrix with positive Pfaffian depending on variables t and s. Its derivatives are given by (2.36) Remark 2.19. From now on we assume that at the point of interest the holomorphic coordinates are chosen so that the hyperhermitian structure (I, J, K, g) is standard, i.e. (2.13) and (2.14) hold, the form Ω φ := Ω+∂∂ J φ is diagonal, or equivalently the metric g φ is and that (2.28) holds. Such an arrangement is possible because after choosing the geodesic coordinates we may make a linear change of those gaining the simultaneous diagonalization of both metrics. The vanishing of the Christoffel symbols is preserved under linear change of coordinates. From time to time we may refer to such a coordinates as canonical.

C 1 estimate
The goal of this section is to prove the following C 1 a priori estimate: Proof. Let us define β by It is easy to see, for example by rewriting this in canonical coordinates, that β = 1 4 |dφ| 2 g .
Since, after taking the logarithm, the linearization of the equation (1.1) is, up to the constant, the Chern Laplacian with respect to the hermitian structure (I, g φ ) on M , we note the useful form-type formula for this, cf. Proposition 2.15, in our setting Following B locki, cf. [B09], we consider the quantity for a function γ : R → R to be specified below.
All the computations from now on will be curried out at a maximum point of α. As the operators ∂ and ∂ J are of pure first order we note that (3.7) Taking the bar of (3.2) results in Next, taking ∂ J of both sides, because of the hyperKähler assumption, we get and by taking ∂, from the same reason as above, we end up with From the equation (1.1) Ω + ∂∂ J φ n = e f Ω n by taking ∂ and applying Lemma 2.12 we obtain

From this we obtain
We now turn to evaluating the required quantities of second order present in the expression from (3.9). They are equal to (3.10) At a maximum point of α we have (3.14) Applying (3.10) -(3.13) in (3.9) and rewriting in coordinates gives us that those quantities are equal to (3.18) We have thus obtained from (3.14) (3.19) Note that we may assume β > 1, otherwise we are finished. Under this assumption the first two terms in (3.19) are bounded from below by a quantity not depending on φ. The next two terms are positive and after fixing γ the penultimate one is bounded from below as well. All of this allows us to rewrite the inequality (3.19) as (3.20) Now we can take, as in for example [B09], Under this choice and the C 0 bound we have (3.22) From (3.22) we obtain for any fixed j which coupled with the equation (1.1) written in canonical coordinates as (3.24) Having the bound on Ω φ 2i2i+1 's from (3.24) we obtain the bound for β, from (3.22), at a maximum point of α. This results in a uniform bound for β.
Remark 3.2. The argument above corresponds to a gradient bound for the complex Monge-Ampère equation with a background metric of non negative holomorphic bisectional curvature. Let us briefly discuss the problems in the argument above for general HKT metrics. Due to non vanishing of dΩ extra terms controlled by appear. Hence γ would have to satisfy both (γ ′ ) 2 + γ ′′ ≥ 0 and γ ′ > C which is possible only if the oscillation of φ is small compared to C. Furthermore the idea from [B09] to exploit the terms containing squares of the pure second order derivatives does not seem to work. This is partially explained by the fact that, even in the flat case, though the gradient can be written as β = φ q i φq i , its q j 'th derivative is not given by the Leibniz rule as this fails for the Cauchy-Riemann-Fueter operators (1.5) and (1.6). What one obtains instead of φ q i q j φq i + φ q i φq i q j are φ q i φq i q j and the conjugations of φ q i q j φq i . Lack of orthogonality between these conjugates results in insufficient positivity to beat -the main negative term.

Bound on ∂∂ J φ
In this section we bound partially the Hessian of φ. More specifically we prove the following a priori estimate: Theorem 4.1. Let (M, I, J, K, g) be a compact, connected hyperKähler manifold. There exists a constant C depending on f , sup M |φ| and the hyperhermitian structure (I, J, K, g) such that for any solution φ of the equation (1.1) the estimate holds.
Proof. Let us define this time ηΩ n = Ω φ ∧ Ω n−1 (4.2) and consider the quantity where the function γ is as in the previous section, cf. (3.21). We note that in order to obtain (4.1) it is sufficient to bound η from above, at a maximal point of α, as it is a positive quantity due to Ω φ > 0. This truly implies (4.1) as in the canonical coordinates η = 1 n φ iī + n is the constant plus the sum of the coefficients of ∂∂ J φ which in light of (4.3) are bounded from below.
We note that at a maximum point of α Here we have used the fact that at the extremal point Let us focus for the moment on the term ∂∂ J η appearing in (4.4). Differentiating twice the conjugation of the relation (4.2) (recall that we work under the assumption dΩ = 0) we obtain An easy calculation in the canonical coordinates shows that (4.6) Here we have used the relation (2.23) and its conjugation and the fact that the first derivatives of the components of J vanish at the point, cf. (2.28). Formulas (4.5) and (4.6) allows us to conclude that Now we find another expression for the last quantity in (4.7). Recall that the equation (1.1) can be written in the form (1.9) After taking the logarithm this reads log P f (Ω φ ij ) = f + log P f (Ω ij ). (4.8) Differentiating (4.8) once provides, due to (2.35), because, due to the hyperKähler assumption, (4.10) In particular the first barred derivatives vanish locally and not only at the fixed point. Differentiating (4.9) once more, due to the formula (2.36), yields (4.11) Summing, over p, the formulas (4.11) give us (recall that Ω φ ij i,j is block diagonal) (4.12) Using the hyperKähler assumption and (2.23) as well as (2.28) we obtain the formula and similarly (4.14) From (4.13) and (4.14) we obtain that, for any k, l and p, This gives the expression in terms of derivatives of φ for the fourth order component obtained in (4.12) . (4.16) From (4.16) we see that the quantity from (4.12) satisfies Finally observe that from (4.6), much like in (4.13) and (4.14), it is easy to see that for any i, l.
Having this we return to the estimation of Ω φ ij 's. At a maximum point of α we have (4.19) We may assume η > 1, otherwise we are done. By (4.7), (4.18) and (4.17) the first term on the right hand side of (4.19) is bounded from below. The same holds for the third term of (4.19) as γ is chosen as in (3.21). This choice of γ ensures also the positivity of the coefficients of the two remaining terms on the right hand side of (4.19). This means we can rewrite the inequality (4.19) as (4.20) for positive constants C 1 , C 2 . This allows us, as in the previous section, to obtain the bounds on Ω φ 2i2i+1 's at a maximum point of α. This in turn gives us a bound on η, which is a multiple of the sum of Ω φ 2i2i+1 's, at a maximum point of α which in turn yields the uniform bound on η itself.
Remark 4.2. Let us note that for the general HKT metric the presence of terms coming from differentiating Ω in (4.4) significantly complicates the computations. Instead, one has to bound the term ∂η ∧ ∂ J η η 2 in (4.4). Unfortunately as one can easily see the quantity is only bounded by twice the quantity (4.7) as can bee seen from what we have obtained in (4.16).

Full C 2 estimate
This section fully exploits the fact that under the assumptions of Theorem 1.1 ∇ := ∇ Ob = ∇ LC = ∇ Ch I,g . (5.1) As we shall see this coupled with the previous a priori bounds suffices to bound the full Hessian of φ. The C 2 a priori estimate reads as follows: Theorem 5.1. Let (M, I, J, K, g) be a compact, connected hyperKähler manifold. There exists a constant C depending on f , sup M |φ| and the hyperhermitian structure (I, J, K, g) such that for any solution φ of the equation (1.1) the estimate holds.
Proof. We wish to estimate the quantity θ being, this time, defined for any x ∈ M as as was done originally in [B11] in the case of the complex Monge-Ampère equation on Kähler manifolds. This is sufficient for the bound on the full Hessian (5.2) because θ is the maximum eigenvalue of the Hessian at the point x. More precisely, the sum of all the eigenvalues, being the Laplacian, is bounded from below by as Ω φ > 0. Once the sum is under control from below and λ max is bounded from above we obtain the lower bound for the smallest eigenvalue, λ min , and consequently we get both sided bounds for all the entries of the matrix of ∇ 2 φ. Consider the quantity α this time given by Since we obtained the gradient bound in Theorem 3.1 it is enough to estimate α at a maximum point p ∈ M .
In this section it will be customary to introduce also the real coordinates for i = 0, ..., 2n − 1, different from the one introduced in (1.4) as can be seen from Remark 2.8. As the quantity (5.5) is in general non smooth, due to (5.3) not being smooth, we extend a fixed vector realizing the supremum in the definition of θ(p), to a constant coefficient local vector field This X is fixed for the rest of the proof. Consider instead of (5.5) the quantitỹ which means that also the quantity (5.9) attains a maximum at p. We may assume that θ(p) ≥ 0 (5.13) since otherwise we are done.
We have the following expression in the introduced coordinates (5.6) where Γ k ji are Christoffel symbols in the real frame ∂ t i . From (5.14) we find that where D denotes the flat connection in coordinates (5.6). Our goal is to exploit the estimate 0 ≥ 1 2n ∆ Ch where we have used the fact that at the point p ∂|X| 2 g = ∂ J |X| 2 g = 0. As we already noticed in the previous section, in the canonical coordinates, we have at p (5.17) Differentiating the expression (5.15) forθ we obtaiñ In the estimation (5.18) we have used the facts that Γ k ij vanish at the point p (recall (2.27) and (2.28)), Γ k ij 's derivatives depend only on the derivatives of the initial metric g, the gradient of φ is bounded and |φ t i t j | < C(1 +θ). (5.19) Since we know from Section 4, (4.1), that we can estimate the quantity (5.17) by applying (5.18) and (5.20) (5.21) In order to deal with the last remaining terms involving derivatives of φ in (5.21) let us differentiate the equation (4.8) twice in the direction of X obtaining Rewriting the quantity in (5.22) explicitly gives (5.23) From the formulas we obtained in (4.15) we have the expression for (5.23) in terms of the derivatives of φ as follows 1 2 which is seen to be non negative. We also note that which gives (recall (2.28)),  We note that |φ ij | 2 + |φ ij | 2 ≥ Cθ 2 . This is because after rewriting this term in coordinates as in (5.17) and applying (5.20) we observe it depends only onθ and second derivatives of the metric g alone (as the coefficients of X are constant and we are computing in normal coordinates). The estimations (5.28), (5.32) and (5.33) applied in (5.16) deliver 0 ≥ 1 2n ∆ Ch I,g φα ≥ Cθ 2 − C ′θ − C ′′ .
(5.34) Inequality (5.34) provides the desired estimate onθ in terms of C, C ′ and C ′′ . The desired bound on θ follows from that.
6 Proof of Theorem 1.1 As we advocated in the introduction having Theorem 3.1, Theorem 5.1 and using Corollary 5.7 in [AV10] (or Theorem 1.1.13 of [AS17] or Theorem A in [Sr19]), one obtains the C 2,α a priori estimate for the solutions of (1.1) for some α ∈ (0, 1). The rest of the proof are completely standard. For the convenience of the reader we sketch them and we refer to [A13] Section 5 for a the detailed discussion.
Turning to the proof of Theorem 1.1. For the given f satisfying (1.2) we set up the continuity path (Ω + ∂∂ J φ) n = te f + (1 − t) Ω n (6.1) for t ∈ [0, 1]. In order to prove Theorem 1.1 it is enough to show that the set S of t ∈ [0, 1], for any fixed k ≥ 1 and α ∈ (0, 1), such that there exists φ ∈ C k+2,α solving (6.1) is both open and closed. Openness in our setting is completely standard. One has to prove that for a fixed φ the operator has an open image. In (6.2) the Banach manifolds, with the induced Hölder norms, are given by The fact that (6.2) has an open image was proven in Proposition 5.1 of [A13] and relies on the theory of linear elliptic operators. As for the closedness it is enough to know that once t i ∈ S are such that t i → t then t ∈ S as well. For any t i let us take φ t i solving (6.1) normalized by sup M φ t i = 0. (6.4) Once we know that the sequence φ t i is bounded in C k+3,α the Kondrakov theorem yields a subsequence converging in a C k+3 norm (and hence in C k+2,α norm) to the solution φ of (6.1). All we need then is to have a priori estimates for the solutions of (1.1) normalized by (6.4) up to the order C k+3,α . From Theorem 3.1, Theorem 5.1 and Corollary 5.7 in [AV10] we have the C 2 estimate. Applying now the Evans-Krylov theorem, cf. [E82], to the operator log (Ω + ∂∂ J φ) n Ω n (6.5) defined on those functions for which the hyperhermitian matrix associated to Ω+∂∂ J φ is positive we obtain a C 2,α estimate for some fixed α. From this the standard procedure of bootstrapping provides the bounds of any higher order as in Section 17.5 of [GT01].
Remark 6.1. For simplicity of presentation (and calculations for obtaining a priori estimates) we stated Theorem 1.1 in the setting when the initial metric g is already HK. Actually, our method works equally well just under the assumption that the hypercomplex manifold (M, I, J, K) admits some compatible hyperKähler metric g ′ and the initial hyperhermitian metric g is arbitrary (in particular HKT). In that setting no new, essential, complications arise while performing a priori estimates, provided we still define the test quantities using g ′ . This is because all the new terms are estimable from below by where C is a constant depending on the curvature of the initial metric g, σ ∈ { 1 2 , 1} and δ is the test quantity we are trying to estimate at the moment. Thus, we can always assume δ makes the above term arbitrarily small in comparison with for γ which is chosen in the sections above, since otherwise we already obtain a bound on the test quantity.