Fully nonlinear elliptic equations with gradient terms on compact almost Hermitian manifolds

In this paper, we establish second order estimates for a general class of fully nonlinear equations with linear gradient terms on compact almost Hermitian manifolds. As an application, we first prove the existence of solutions for the Monge-Amp\`ere equation with linear gradient terms for $(n-1)$-plurisubharmonic functions, originated from Gaudochon conjecture, in the almost Hermitian setting. Second, we solve the Monge-Amp\`ere equation and Hessian equations with linear gradient terms. Third, we give the $C^{\infty}$ a priori estimates for the deformed Hermitian-Yang-Mills equation with supercritical phase. At last, we prove the existence of deformed Hermitian-Yang-Mills equation and complex Hessian quotient equations under supersolutions.


Introduction
Let (M, χ, J) be a compact almost Hermitian manifold of real dimension 2n, and ω is a fixed real (1, 1)-form on (M, J). For an arbitrary smooth function u, we write where Z(∂u) denotes a smooth (1,1)-form depending on ∂u linearly which will be specified later, and let µ(u) = (µ 1 (u), . . . , µ n (u)) be the eigenvalues of ω u with respect to χ. For the sake of notational convenience, we sometimes denote µ i (u) by µ i when no confusion will arise. In the current paper, we consider the following fully nonlinear elliptic equations of the form where h ∈ C ∞ (M ) and f is a smooth symmetric function in R n .
The equation (1.1) covers many important elliptic equations in (almost) complex geometry. A typical example of (1.1) is the following equation: Here η is an almost Hermitian metric, ∆ χ denotes the canonical Laplacian operator of χ and W = W (∂u) is a Hermitian tensor that linearly depends on ∂u.
The fully nonlinear elliptic equations with gradient terms on Hermitian manifolds have been researched extensively, we refer the reader to [14,21,22,42,44,45] and references therein. On the framework of almost Hermitian manifolds, to our knowledge most of researches toward equation (1.1) are independent of ∂u. Inspired by these works, we shall consider the equation (1.1) on compact almost Hermitian manifolds.
We have the following estimate: Theorem 1.1. Let (M, χ, J) be a compact almost Hermitian manifold of real dimension 2n. Suppose that u (resp. u) is a smooth solution (resp. C-subsolution) of (1.1). Then we have where C is a constant depending on u, h, Z, ω, f , Γ and (M, χ, J).
As an application, to begin, we solve the equation ( For the complex Monge-Ampère equation, Yau [43] solved it on a Kähler manifold and confirmed the famous Calabi's conjecture (see [4]). In the non-Kähler setting, we refer the reader to [5,9,20,23,38,39,47]. The classical complex Hessian equations also have been studied extensively, see [7,12,24,25,35,46]. Similar to Theorem 1.2, we can solve the complex Monge-Ampère equation and complex Hessian equations with gradient terms. Theorem 1.3. Let (M, χ, J) be a compact almost Hermitian manifold of real dimension 2n and ω be a smooth k-positive real (1, 1)-form. For any integer 1 ≤ k ≤ n, there exists a unique pair (u, c) ∈ C ∞ (M ) × R such that For the deformed Hermitian-Yang-Mills (dHYM) equation we say (1.6) is hypercritical (resp. supercritical) if h ∈ (0, π 2 ) (resp. h ∈ (0, π)). Jacob-Yau [30] showed the existence of solution for dimension 2, and for general dimensions when (M, χ) has non-negative orthogonal bisectional curvature in the hypercritical phase setting. Pingali [31,32] obtained a solution when n = 3. In general dimensions, the equation (1.6) was solved by Collins-Jacob-Yau [10] under the existence of C-subsolutions. The equation (1.6) was also studied by Leung [27,28] to seek vector bundles over a symplectic manifold. Recently, Zhang and the authors [26] provided a priori estimates on compact almost Hermitian manifolds for the hypercritical case. It was researched by Lin [29] in the supercritical phase on compact Hermitian manifolds.
As a corollary, using Theorem 1.1, we are also able to derive a priori estimates for (1.6) in the supercritical case. Corollary 1.4. Let (M, χ, J) be a compact almost Hermitian manifold of real dimension 2n. Suppose that u (resp. u) is the solution (resp. C-subsolution) of equation (1.6) with h ∈ 0, π − δ is a smooth function for a constant δ ∈ (0, π 2 ). Then for each α ∈ (0, 1), we have where C is a constant depending on α, u, h, ω, δ and (M, χ, J).
We now discuss the proof of Theorem 1.1. The zero order estimate can be proved by adapting the arguments of [35,Proposition 11] and [9, Proposition 3.1], which are based on the method of B locki [2,3]. For the second order estimate, following the idea of [6,8,9,35] and by some delicate calculations, the real Hessian ∇ 2 u can be controlled by the first gradient quadratically as follows: The paper is organized as follows. In §2, we will introduce some notations, and recall the definition and an important property of C-subsolution. We also verify that the dHYM equation satisfying the structural conditions. The zero order estimate will be established in §3.1. In §3.2, we shall prove the estimate (1.7). To see this, we apply the maximum principle to the quantity involving the largest eigenvalue λ 1 of real Hessian ∇ 2 u with respect to χ of form  [9, p.1954], we can define (p, q)-forms and operators ∂, ∂ by using the almost complex structure J. Let A 1,1 (M ) denote the set of smooth real (1,1)-forms on (M, J). For any u ∈ C ∞ (M ), we see that In the sequel, we set where Z(∂u) is a real (1, 1)-form defined by Z ij = Z p ij u p + Z p ij up. For any point x 0 ∈ M , let (e 1 , · · · , e n ) be a local unitary (1, 0)-frame with respect to χ near x 0 , and {θ i } n i=1 be its dual coframe. Then in the local chart we have as well asg After making a unitary transformation, we may assume thatg ij (x 0 ) = δ ijg ii (x 0 ). We denoteg ii (x 0 ) by µ i . It is useful to order µ i such that At x 0 , we have the expressions of G ij and G ik,jl (see e.g. [1,18,34]) where the quotient is interpreted as a limit if µ i = µ j . Using (2.1), we obtain (see e.g. [13,34]) G 11 ≤ G 22 ≤ · · · ≤ G nn . On the other hand, the linearized operator of equation (1.1) is By Definition 2.1, for each C-subsolution u, there are constants δ, R > 0 depending only on u, (M, χ, J), f and Γ such that where B R (0) denotes the Euclidean ball with radius R and center 0.
Similar to [19,35], we have the following proposition: for some δ, R > 0. Then there exists a constant θ > 0 depending on δ and the set in (2.5) such that for each µ ′ ∈ ∂Γ σ and |µ ′ | > R, we have either Proof. The proof can be found in [35,Proposition 5], we include it here for convenience to reader. Set It follows from (2.5) that A δ is compact. For each v ∈ A δ , we define Note that f i > 0 for all i. We conclude that which implies that C v is strictly larger than Γ n . Now we define the dual cone of C v by x k > ǫ for all k.
As A δ compact, we can find a uniform constant ǫ such that (2.6) holds for all v ∈ A δ . Let µ ′ ∈ ∂Γ σ , |µ ′ | > R and T µ ′ be the tangent plane to ∂Γ σ at µ ′ . Now we split the proof into two cases: This completes the proof of proposition.
Using previous proposition, we have the following result originated from [6,19,35]. It will play an important role in the proof of Theorem 3.2.
(1) There exists a constant τ depending on f , Γ and σ such that (2) For δ, R > 0, there exists θ > 0 depending only on f , Γ, h, δ, R such that the following holds. If B is a Hermitian matrix satisfying then we have either By assumption (iii) and concavity, there exists a large constant N such that . For (2), we divide into two possibilities: • |µ(A)| ≥ R. We note that the proof of [35, Proposition 6] only needs assumption (i) and (ii). Then the conclusion follows.

A priori estimates
3.1. Zero order estimate.
Proof. Without loss of generality, we may assume that u = 0. Thanks to [35, (44)], we have tr χ ω u > 0 and hence where ∆ χ denotes the canonical Laplacian operator of χ. Following a similar argument of [9, Proposition 2.3], then there exists a uniform constant C such that . Now it suffices to establish the lower bound of the infimum I = inf M u. We can adopt the arguments in [6]. We remark that the only difference here is the presence of the term Z(∂u) in the definition of H(u). However, this term is linear in ∂u, which can be controlled (by ε) on the contact set P in [6].
3.2. Second order estimate. In this subsection, we give the proof of Theorem 1.1. Our first goal is the following theorem: where ∇ denotes the Levi-Civita connection with respect to χ.
Without loss of generality, we assume u = 0 and sup M u = −1. Let λ 1 ≥ λ 2 ≥ · · · ≥ λ 2n be the eigenvalues of ∇ 2 u with respect to χ. For notational convenience, we write | · | = | · | χ . Let us define On an open set Ω = {λ 1 > 0} ⊂ M , we consider for a large constant A to be chosen later, where By a directly calculation we see that We may assume Ω = ∅, otherwise we are done. Since Q(z) → −∞ as z approaches to the boundary of Ω, we further assume Q achieves its maximum at a point x 0 ∈ Ω. It is easy to show that (see [ Near x 0 , there exists a local unitary frame {e i } n i=1 with respect to χ such that (3.7) χ ij = δ ij ,g ij = δ ijg ii ,g 11 ≥g 22 ≥ · · · ≥g nn at x 0 .
Applying the maximum principle at x 0 , we see that (3.9) In the sequel, we shall make the following conventions: (i) all the calculations are done at x 0 , (ii) we will use the Einstein summation, (iii) we usually use C to denote a constant depending on u C 0 , h, ω, Γ, (M, χ, J), and C A to denote a constant further depending on A, (iv) we always assume without loss of generality, that λ 1 ≥ CK for some C, or λ 1 ≥ C A K for some C A , (v) we use subscripts i andj to denote the partial derivatives e i andē j .

Lower bound for L(Q).
(3.10) We remark that the fourth term is the bad term that we need to control. Since F is both concave and elliptic, then the first, second and third term are nonnegative, which play an imporant role in our proof of Theorem 3.2. To prove Proposition 3.3, we shall estimate the lower bounds of L(λ 1 ), L(|ρ| 2 ) and L(|∂u| 2 ), respectively.
First, we give the lower bound of L(λ 1 ).
Proof. The following formulas are well-known (see e.g. [9,34,35]): Then we compute (3.11) Here and hereafter O(λ 1 ) means the terms those can be controlled by Cλ 1 . Similarly, we also obtain Then the claim follows. Applying W to the equation (1.1),

Proof of Claim 2. It is clear that
It follows that Combining this with (3.12), By direct calculation, we see that Substituting this with Claim 1 into (3.13), we obtain (3.14) To deal with the first term, we apply V 1 V 1 to the equation (1.1) and obtain Then Claim 2 follows from (3.14) and (3.15).
Lemma 3.5. For each ε ∈ (0, 1 3 ], at x 0 , we have Proof. We remark that the linear gradient terms in L can be absorbed by N 2 F . Thus the proof is similar to [6]. Finally, we give the lower bound of L(|∂u| 2 ).
Lemma 3.6. At x 0 , we have Proof. By a direct calculation, we deduce where Applying e j to the equation (1.1), where O( √ K) means the terms those can be controlled by C √ K. Similarly, By the Cauchy-Schwarz inequality, This proves the lemma.
We will use the above computations to prove Proposition 3.3.

Proof of Proposition 3.3.
Combining (3.9) and Lemmas 3.4-3.6, we obtain It suffices to deal with the third and last term. For the third term, using (3.5) and the fact N ≤ C A λ 1 , For the last term, using (3.5) again we infer that Combining the above inequalities, we conclude Proposition 3.3.

Proof of Theorem 3.2. First, we define the index set
If J = ∅, then Theorem 3.2 follows. So we assume J = ∅ and let j 0 be the maximal element of J. If j 0 < n, we denote According to the index sets J and S, the proof of Theorem 3.2 can be divided into three cases: Case 2. j 0 < n and S = ∅.
For Case 1 and Case 2, the proof in [6] is still valid in our setting, we shall omit it here. Now we only need to establish Case 3.
Observe that S = ∅. Let i 0 be the minimal element of S and define I = {i 0 + 1, · · · , n}.
Let us decompose the term into three terms based on I.

Calculations of B 3 .
We now devote to prove the following proposition.
Furthermore, n ∈ I implies e n e n u ≥ −C A K and g nn = g nn + e n e n u + [e n , e n ] (0,1) u + Z nn ≥ e n e n u − CK ≥ −C A K.
Now we give the proof of Proposition 3.8.
Proof of Proposition 3.8. By the definition of W 1 in (3.22), we see that Using this together with Cauchy-Schwarz inequality and Lemma 3.9, (3.24) For the second term in RHS of (3.24). Observing that |V 1 e i e q u| ≤ C α,β |e i (u αβ )|+ Cλ 1 , we deduce Under the assumption λ 1 ≥ CA ε , we obtain Now we deal with the first term in RHS of (3.24). For a constant γ > 0 to be chosen later, we see that (3.26) Using the Cauchy-Schwarz inequality again, for the first term, 27) and for the second term, Recalling the definition of the index set I, when q / ∈ I and i ∈ I, Combining this with Lemma 3.9, In addition, from (2.2) and the concavity of f , we get It follows from (3.28) and (3.29) that (3.30) Since ε = e Au(x 0 )
For a), by Lemma 3.9, we deduce q / ∈Ig where we used (3.22) in the last inequality. Combining this with the assumption of Case B, we see that Using Lemma 3.9 again and (3.34), Since ε = e Au(x 0 )
Consequently, the Proposition 3.8 follows from ( where θ is the constant given in Proposition 2.2. There are two possibilities: • −L(u) ≥ θG. In this setting, (3.36) yields that Using the fact A = 10C θ , we deduce This is impossible.
• G 11 ≥ θG. Using the Cauchy-Schwarz inequality, Plugging it into (3.36), This yields λ 1 ≤ C A K and the proof is completely.  3). Assume u is a C-subsolution and u is a smooth solution of (1.1). Then for each k = 0, 1, 2, · · · , we have where C k is a constant depending on k, u, h, Z, ω, f , Γ and (M, χ, J). In this section, we prove Corollary 1.4. First, we give the C 1 estimates of the dHYM equation (2.10).

Proof. Let us define
Here D > 0 are certain constants to be picked up later. 1 Consider the test function Q = e H(η) |∂u| 2 .
Otherwise we are done. Then near x 0 , we can choose a proper local frame {e i } n i=1 such that χ ij = δ ij and the matrix g ij is diagonal at x 0 . It follows from maximum principle that G iī Re e i (η)ē i e j (u)ē j (u) + e i (η)ē iēj (u)e j (u) . For the last term of (4.1). Note that ε ∈ (0, 1 2 ] implies 1 ≤ (1 − ε)(1 + 2ε). Using the definition of Lie bracket again, we see 2 i,j G iī Re{e i (η)ē i e j (u)ē j (u)} =2 i,j G iī Re η i uj e jēi (u) − [e j ,ē i ] 0,1 (u) − [e j ,ē i ] 1,0 (u)  1 From now on, the C below denotes the constants those may change from line to line, and it doesn't depend on D that we yet to choose.
Combining the Theorem 1.1, we establish the second order estimates. Therefore, the equation (1.6) is uniform elliptic. Based on Evans-Krylov theory, we obtain the higher order estimates. This completes the proof of Corollary 1.4.