On uniform estimate of complex elliptic equations on closed Hermitian manifolds

In this paper, we study Hessian equations and complex quotient equations on closed Hermitian manifolds. We directly derive the uniform estimate for the admissible solution. As an application, we solve general Hessian equations on closed K\"ahler manifolds.


Introduction
Let (M n , ω) be a compact Hermitian manifold of complex dimension n ≥ 2 and χ a smooth real (1, 1) form on M. Write ω and χ respectively as We denote χ u := χ + √ −1∂∂u. In the seminal paper [17], Yau  Since then, remarkable progress and geometric applications have been made. The Hermitian case was later solved by Tosatti and Weinkove [16]. In the setting of moment maps, Donaldson [4] proposed an equation on closed Kähler manifolds Surprisingly, in the study of Mabuchi enery, Chen [2] found the same equation in the critical case. In addition, Hou, Ma and Wu [9] considered the Hessian equation on closed Kähler manifolds, for 2 ≤ α ≤ n − 1 and proved a second order estimate. Following this estimate, Dinew and Kolodziej [3] applied a blow-up argument to obtain the gradient estimate and consequently solved the Hessian equation. These equations attract much attention in complex geometry and analysis, and are still fast-developing subjects nowadays. Indeed, all these equations can be included in the same class. For a smooth positive real function f on M, we are concerned with the following complex equations, (1) The complex α-Hessian equation. For 2 ≤ α ≤ n and χ ∈ Γ α ω , χ α u ∧ ω n−α = f ω n , with χ u ∈ Γ α ω . (1.4) (2) The complex (α, β)-quotient equation. For 1 ≤ β < α ≤ n and χ ∈ Γ α ω , χ α u ∧ ω n−α = f χ β u ∧ ω n−β , with χ u ∈ Γ α ω . (1.5) Following [10], [5] and [8], we define the cone class If [χ] ∈ C α,β (f ), we say that χ satisfies the cone condition for equation (1.5) with respect to f . We say that χ ′ is in the cone subject to f .
Here Γ α ω is the set of all the real (1, 1) forms whose eigenvalue set with respect to ω belong to α-positive cone Γ α ⊂ R n . In fact, complex α-Hessian equation can be treated as a particular case of complex quotient equations when β = 0. However, complex Hessian equation has a natural strong cone condition, i.e. C α,0 (f ) = {[χ] : Γ α ω ∩ [χ] = ∅}, which is independent on f . This makes the approaches for Hessian equations easier than those for quotient equations.
In the work of the author [12], we provide a direct uniform estimate for the two types of equations on closed Kähler manifolds. In this paper, we further develop the technique in [12] and apply a key lemma by Zhang [18] to extend the estimate to general Hermitian cases. The lemma in [18] helps us to improve our result in an early version of this paper. Our main result is as follows.
Theorem 1.1. Let (M, g) be a closed Hermitian manifold of complex dimension n ≥ 2 and χ a smooth real (1, 1) form. Let u be a smooth solution to either (1.4) or (1.5) with the corresponding cone condition. Then there is a uniform estimate for u depending only on (M, ω), χ and f .
It is worth a mention that the estimate of the quotient equation depends on the gradient of f at most, while that in [12] needs the data of derivatives up to second order. Zhang also independently solved the complex Hessian equations when χ = ω with a direct uniform estimate. Very recently, Zhang have informed the author that using Gåding's inequality, his method and result can be extended to more general cases. Moreover, Székelyhidi [13] was able to solve the complex Hessian equations with a uniform estimate based on the approach of Blocki [1].
Generally, the uniform estimate is the most difficult one on closed manifolds. When we are able to prove it, it is natural to consider the solvability. As an application of the uniform estimate, we study complex Hessian equations on closed Kähler manifolds. Theorem 1.2. Let (M, g) be a closed Kähler manifold of complex dimension n ≥ 2 and χ a smooth real (1, 1) form in Γ α ω . Then there exists a unique smooth real function u and a unique real number b such that When the equation is Monge-Ampère type, Guan and the author [8] treated the Dirichlet problem and derived the gradient estimate directly. For general cases, a blow-up argument is still necessary so far.
Acknowledgements The author would like to thank S lawomir Dinew, Jixiang Fu and Bo Guan for comments and helpful suggestions. The author also expresses his gratitude to Mulin Li for helpful discussions.

The uniform estimate
It is sufficient to prove the inequality for p large enough. Without loss of generality, we may assume p ≫ 1 throughout this section. The first inequality (2.1) is naturally satisfied, and thus we only need to verify the second inequality (2.2). We refer the readers to [15], [16] and [17] for more details.
For general Hermitian manifolds, we need the following lemma by Zhang [18].
The following lemma was essentially proven by Sun [12]. Proof. Using intergration by parts and Gårding's inequality, (2.7) We shall use Lemma 2.1 and Lemma 2.2 to deal with some troublesome terms due to torsion.
We apply integration by parts to the second term, For a fixed t, we express X = χ tu . For any point p ∈ M, we can find a coordinate chart around p such that g ij = δ ij and X ij is diagonal at p. In this paper, we call such local coordinates normal coordinate charts around p. Then we pointwise control the first term in the right of (2.9) (2.11) We pointwise control the first term in (2.10), (2.12) By Schwarz inequality and Lemma 2.1, we have (2.13) We pointwise control the second term in (2.10), (2.14) By Schwarz inequality and Lemma 2.1, we have Proof. Without loss of generality, it is reasonable to assume that p ≫ 1. We consider the following inequality, It is easy to see that I ≤ C 1 M e −pu ω n for some uniform constant C 1 . We rewrite the integral, and then calculate it directly, By integration by parts for the third term, We can pointwise control four terms in the right hand side of (2.19). By Schwarz's inequality, for any ǫ > 0 there is a uniform constant C such that and where ǫ 1 is to be determined later. By Lemma 2.2, we know that if ǫ 1 ≤ λ 2 , (2.24) If α = 2, we calculate the last term in (2.24), and thus if p is large enough.
If α ≥ 3, we control the last term in (2.24). By Lemma 2.3, (2.27) To control the last term in (2.27), we compute (2.28) Consequently, for sufficiently large p, and hence from (2.27), As in [12], we can show that for some c 0 > 0, (2.31) Lemma 2.5. Let u be a smooth admissible solution to quotient equation (1.5). Then there are uniform constants C, p 0 such that for all p ≥ p 0 , inequality (2.2) holds true.
Proof. Without loss of generality, we can assume Also, by the monotony of S α /S β , we have We consider the integral It is not hard to see for some uniform constant C 1 > 0.

Hessian equations
In this section, we follow the work of Hou, Ma and Wu [9] to obtain the second order estimate quadratically dependent on the gradient. Then the blow-up argument of Dinew and Kolodziej [3] suffices to prove the gradient estimate. Higher order estimates follow from Evans-Krylov theory and Schauder estimate, which is done by Tosatti, Wang, Weinkove and Yang [14]. Therefore, we can apply the conitnuity method shown in [15] and [11] to prove the existence of solution. where C depends on ||F 1 α || C 2 (M ) , α and geometric data.
First of all, we need to clarify the derivatives. Assume that U is a C 2 Hermitian structure. Let W is Lipschitz coninuous but might not be differentiable. In the calculation, we need to understand the formal differentiation of W , which should independent on the choice of local coordinate charts. In other words, the formal derivatives of W are covariant derivatives. Suppose that W (x) is achieved at some unit ξ ∈ T U(ξ + tζ,ξ + tζ) g(ξ + tζ,ξ + tζ) ≤ U(ξ,ξ).
As an immediate result, that is, Considering the eigenvalues of U with respect to g, we can sort them in order λ 1 ≥ λ 2 ≥ · · · ≥ λ n . Suppose that W (p) is attained at some unit η ∈ T (1,0) p M. There are two cases λ 1 > λ 2 and λ 1 = λ 2 .
Now we begin to prove the second order estimates.
Proof of Lemma 3.1. We can rewrite the equation (1.4) as below, Expressing X = χ u and differentiating (3.14) twice under a normal coordinate chart, we obtain and C 0 is to be determined later. Note that we have and Assume that H achieves a maximum at some point p in the unit direction of η.