Hessian equations of Krylov type on compact Hermitian manifolds

Jundong Zhou; Yawei Chu

doi:10.1515/math-2022-0504

Open Access Published by De Gruyter Open Access October 11, 2022

Hessian equations of Krylov type on compact Hermitian manifolds

Jundong Zhou and Yawei Chu

From the journal Open Mathematics

https://doi.org/10.1515/math-2022-0504

Abstract

In this article, we are concerned with the equations of Krylov type on compact Hermitian manifolds, which are in the form of the linear combinations of the elementary symmetric functions of a Hermitian matrix. Under the assumption of the 𝒞-subsolution, we obtain a priori estimates in Γ k − 1 cone. By using the method of continuity, we prove an existence theorem, which generalizes the relevant results. As an application, we give an alternative way to solve the deformed Hermitian Yang-Mills equation on compact Kähler threefold.

Keywords: Hessian equations; Hermitian manifolds; subsolution condition

MSC 2010: 35J60; 35B45; 53C55

1 Introduction

Let ( M , ω ) be a compact Kähler manifold of complex dimension n . In 1978, Yau [1] proved the famous Calabi-Yau conjecture by solving the following complex Monge-Ampère equation on M

( ω + − 1 ∂ ∂ ¯ u ) n = f ω n ,

with positive function f . There have been many generalizations of Yau’s work. One extension of Yau’s Theorem is to the case of Hermitian manifolds, which is initiated by Cherrier [2] in 1987. The Monge-Ampère equation on compact Hermitian manifolds was solved by Tosatti and Weinkove [3], building on several earlier works. See [2,4,5, 6,7] and the references therein.

The complex Hessian equation can be expressed as follows:

( ω + − 1 ∂ ∂ ¯ u ) k ∧ ω n − k = f ω n , 2 ≤ k ≤ n − 1 .

On compact Kähler manifolds ( M , ω ) , Hou [8] proved the existence of a smooth admissible solution of the complex Hessian equation by assuming the nonnegativity of the holomorphic bisectional curvature. Later, Hou et al. [9] obtained the second-order estimate without any curvature assumption. Using Hou et al.’s estimate, Dinew and Kolodziej [10] applied a blow-up argument to prove the gradient estimate and solved the complex Hessian equation on compact Kähler manifolds. The corresponding problem on Hermitian manifolds was solved by Zhang [11] and Székelyhidi [12] independently.

The complex Hessian quotient equations include the complex Monge-Ampère equation and the complex Hessian equation. Let χ be a real (1, 1) form, the complex Hessian quotient equations can be expressed as follows:

( χ + − 1 ∂ ∂ ¯ u ) k ∧ ω n − k = f ( z ) ( χ + − 1 ∂ ∂ ¯ u ) l ∧ ω n − l , 1 ≤ l < k ≤ n , z ∈ M .

When f ( z ) is constant, one special case is the so-called Donaldson equation [13]:

( χ + − 1 ∂ ∂ ¯ u ) n = c ( χ + − 1 ∂ ∂ ¯ u ) n − 1 ∧ ω k .

After some progresses made in [14,15, 16,17], Song and Weinokove [18] solved the Donaldson equation on closed Kähler manifolds via J -flow. Fang et al. [19] extended the Donaldson equation to

( χ + − 1 ∂ ∂ ¯ u ) n = c k ( χ + − 1 ∂ ∂ ¯ u ) n − k ∧ ω k

and solved this equation on closed Kähler manifolds by assuming a cone condition. When f ( z ) is not constant, analogous results were obtained by Sun [20,21] on compact Hermitian manifolds.

In this article, we are concerned with Hessian equations of Krylov type in the form of the linear combinations of the Hessian, which can be written as follows:

(1.1) ( χ + − 1 ∂ ∂ ¯ u ) k ∧ ω n − k = ∑ l = 0 k − 1 α l ( χ + − 1 ∂ ∂ ¯ u ) l ∧ ω n − l , 2 ≤ k ≤ n .

The Dirichlet problem of (1.1) on ( k − 1 ) -convex domain Ω in R n was first studied by Krylov [22] about 20 years ago. He observed that if α l ( x ) ≥ 0 for 0 ≤ l ≤ k − 1 , the natural admissible cone to make (1.1) elliptic is also the Γ k -cone, which is the same as the k -Hessian equation case, where

Γ k = { λ ∈ R n ∣ σ 1 ( λ ) > 0 , … , σ k ( λ ) > 0 } .

Guan and Zhang [23] solved the equation of Krylov type on the problem of prescribing convex combination of area measures. Pingali [24] proved a priori estimates to the following equation in Kähler case:

( χ + − 1 ∂ ∂ ¯ u ) n = ∑ l = 0 n − 1 C n l α l ( χ + − 1 ∂ ∂ ¯ u ) n − k ∧ ω l ,

where α l ≥ 0 are smooth real functions such that either α l = 0 or α l > 0 , and ∑ l = 0 n − 2 α l > 0 . Recently, Phong and Tô [25] solved Hessian equations of Krylov type on compact Kähler manifolds, where α l are non-negative constants for 0 ≤ l ≤ k − 1 . When α l are non-negative smooth functions for 0 ≤ l ≤ k − 1 , analogous results on compact Kähler manifolds are obtained by Chen [26] and Zhou [27] independently.

Naturally, we want to extend this result to Hermitian manifolds. On the other hand, Zhou [27] believed that the condition on α k − 1 ( x ) > 0 is not necessary. In fact, Guan and Zhang [23] considered equation (1.1) without the sign requirement for the coefficient function α k − 1 ( x ) .

In this article, we mainly concern equation (1.1) on Hermitian manifold without any sign requirement for α k − 1 ( x ) . Let Γ k − 1 g be the set of all the real (1, 1) forms, eigenvalues of which belong to Γ k − 1 . To ensure the ellipticity and non degeneracy of the equation in Γ k − 1 , we require smooth real functions α l to satisfy the conditions: for 0 ≤ l ≤ k − 2 , either α l > 0 or α l ≡ 0 , and ∑ l = 0 k − 2 α l > 0 . Let χ u = χ + − 1 ∂ ∂ ¯ u and χ u ̲ = χ + − 1 ∂ ∂ ¯ u ̲ . To state our main results, we need also the following condition of C -subsolution, which is similar to C -subsolution introduced by Székelyhidi [12].

Definition 1.1

A smooth real function u ̲ is a C -subsolution to (1.1), if χ u ̲ ∈ Γ k − 1 g , and at each point x ∈ M , the set

λ ( χ ˜ ) ∈ Γ k − 1 ∣ χ ˜ k ∧ ω n − k = ∑ l = 0 k − 1 α l ( x ) χ ˜ l ∧ ω n − l and χ ˜ − χ u ̲ ≥ 0

is bounded.

Theorem 1.2

Let ( M , g ) be a compact Hermitian manifold, χ a real (1, 1) form on M . Suppose that u ̲ is a C -subsolution of equation (1.1) and at each point x ∈ M ,

(1.2) χ u ̲ ( x ) k ∧ ω n − k ≤ ∑ l = 0 k − 1 α l ( x ) χ u ̲ ( x ) l ∧ ω n − l .

Then there exists a smooth real function u on M and a unique constant b solving

(1.3) χ u k ∧ ω n − k = ∑ l = 0 k − 2 α l ( x ) χ u l ∧ ω n − l + ( α k − 1 + b ) χ u k − 1 ∧ ω n − k + 1 ,

with sup M ( u − u ̲ ) = 0 and χ u ∈ Γ k − 1 g .

Corollary 1.3

Let ( M , g ) be a compact Kähler manifold, χ a closed ( 1 , 1 ) -form. Suppose that u ̲ is a C -subsolution of equation (1.1) and

(1.4) ∫ M χ k ∧ ω n − k ≤ ∑ l = 0 k − 1 c l ∫ M χ ∧ ω n − l ,

where c l = inf M α l , 0 ≤ l ≤ k − 1 . Then there exists a smooth real function u on M and a unique constant b solving

(1.5) χ u k ∧ ω n − k = ∑ l = 0 k − 2 α l ( x ) χ u l ∧ ω n − l + ( α k − 1 + b ) χ u k − 1 ∧ ω n − k + 1 ,

with sup M ( u − u ̲ ) = 0 and χ u ∈ Γ k − 1 g .

Lately, Pingali [28] proved an existence result of the deformed Hermitian Yang-Mills equation with phase angle θ ˆ ∈ 1 2 π , 3 2 π on compact Kähler threefold. Let Ω be a closed ( 1 , 1 ) form, Ω u = Ω + − 1 ∂ ∂ ¯ u . From [24], the deformed Hermitian Yang-Mills equation on compact Kähler threefold can be written as follows:

(1.6) Ω u 3 = 3 sec 2 ( θ ˆ ) Ω u ∧ ω 2 + 2 tan ( θ ˆ ) sec 2 ( θ ˆ ) ω 3 .

As an application of Corollary 1.3, we give an alternative way to solve the deformed Hermitian Yang-Mills equation on compact Kähler threefold.

Corollary 1.4

Let ( M , g ) be a compact Kähler threefold, constant phase angle θ ˆ ∈ 1 2 π , 3 2 π , and Ω a positive definite closed ( 1 , 1 ) form, satisfying the following conditions:

(1.7) 3 Ω 2 − 3 sec 2 ( θ ˆ ) ω 2 > 0 ,

(1.8) ∫ M Ω 3 = 3 sec 2 ( θ ˆ ) ∫ M Ω ∧ ω 2 + 2 tan ( θ ˆ ) sec 2 ( θ ˆ ) ∫ M ω 3 ,

(1.9) Ω + sec ( θ ˆ ) ω ∈ Γ 2 g .

Then there exists a smooth solution to equation (1.6) with sup M u = 0 and Ω u ∈ Γ 3 g .

The rest of this article is organized as follows. In Section 2, we set up some notations and provide some preliminary results. In Section 3, we give the C 0 estimate by the Alexandroff-Bakelman-Pucci maximum principle. In Section 4, we establish the C 2 estimate for equation (1.1) by the method of Hou et al. [9] and the C -subsolution condition. In Section 5, we give the gradient estimate. In Section 6, we give the proof of Theorem 1.2, Corollaries 1.3, and 1.4 by the method of continuity. Although the method is very standard in the study of elliptic PDEs, it is not easy to carry out this method on a compact Hermitian manifold. Since the essential C -subsolution condition depends on α 0 , … , α k − 1 , we have to find a uniform C -subsolution condition for the solution flow of the continuity method.

2 Preliminaries

In this section, we set up the notation and establish some lemmas. Let σ k ( λ ) denote the k th elementary symmetric function

σ k ( λ ) = ∑ 1 ≤ i 1 < ⋯ < i k ≤ n λ i 1 ⋯ λ i k , for λ = ( λ 1 , … , λ n ) ∈ R n , 1 ≤ k ≤ n .

For completeness, we define σ 0 ( λ ) = 1 and σ − 1 ( λ ) = 0 . Let σ k ( λ ∣ i ) denote the symmetric function with λ i = 0 and σ k ( λ ∣ i j ) the symmetric function with λ i = λ j = 0 . Also denote by σ k ( A ∣ i ) the symmetric function with A deleting the i th row and i th column, and σ k ( A ∣ i j ) the symmetric function with A deleting the i th, j th rows and i th, j th columns, for all 1 ≤ i , j ≤ n . In local coordinates,

X i j ¯ = X ∂ ∂ z i , ∂ ∂ z ¯ i = χ i j ¯ + u i j ¯ , X ̲ i j ¯ = χ i j ¯ + u ̲ i j ¯ .

Define λ ( χ u ) as the eigenvalue set of { X i j ¯ } with respect to { g i j ¯ } . In local coordinates, equation (1.1) can be written in the following form:

(2.1) σ k ( λ ( χ u ) ) = ∑ l = 0 k − 1 β l ( x ) σ l ( λ ( χ u ) ) ,

where

σ l ( λ ( χ u ) ) C n l = χ u l ∧ ω n − l ω n , β l ( x ) = C n k C n l α l ( x ) .

Equivalently, we can rewrite equation (2.1) as follows:

(2.2) σ k ( λ ( χ u ) ) σ k − 1 ( λ ( χ u ) ) − ∑ l = 0 k − 2 β l σ l ( λ ( χ u ) ) σ k − 1 ( λ ( χ u ) ) = β k − 1 ( x ) .

Lemma 2.1

[29,30] For λ ∈ Γ m and m > l ≥ 0 , r > s ≥ 0 , m ≥ r , l ≥ s , we have

σ m ( λ ) / C n m σ l ( λ ) / C n l 1 m − l ≤ σ r ( λ ) / C n r σ s ( λ ) / C n s 1 r − s .

The following lemma is similar to Lemma 2.3 in [27], but we need to discuss it more widely, that is, λ ( χ u ) ∈ Γ k − 1 instead of λ ( χ u ) ∈ Γ k .

Lemma 2.2

If u ∈ C 2 ( M ) is a solution of (2.2), λ ( χ u ) ∈ Γ k − 1 and β l ( x ) > 0 , 0 ≤ l ≤ k − 2 , then

(2.3) σ l ( λ ) σ k − 1 ( λ ) ≤ C ( n , k , inf 0 ≤ l ≤ k − 2 β l , sup ∣ β k − 1 ∣ ) for 0 ≤ l ≤ k − 2 .

Proof

If σ k σ k − 1 ≤ 1 , then we obtain from equation (2.2)

β l ( x ) σ l σ k − 1 ≤ σ k σ k − 1 − β ( x ) ≤ 1 − β ( x ) ≤ C ( sup M ∣ β k − 1 ∣ ) , for 0 ≤ l ≤ k − 2 .

If σ k σ k − 1 > 1 , i.e., σ k − 1 σ k < 1 , we see from Lemma 2.1 that

σ l σ k − 1 ≤ ( C n k ) k − 1 − l C n l ( C n k − 1 ) k − l σ k − 1 σ k k − 1 − l ≤ ( C n k ) k − 1 − l C n l ( C n k − 1 ) k − l ≤ C ( n , k )

for 0 ≤ l ≤ k − 2 , which completes the proof of Lemma 2.2.□

For any point x 0 ∈ M , choose a local frame such that X i j ¯ = δ i j X i i ¯ . For the convenience of notations, we will write equation (2.2) as follows:

(2.4) F ( X ) = F k ( X ) + ∑ l = 0 k − 2 β l F l ( X ) = β k − 1 ( x ) ,

where F k ( X ) = σ k ( λ ( X ) ) σ k − 1 ( λ ( X ) ) and F l ( X ) = − σ l ( λ ( X ) ) σ k − 1 ( λ ( X ) ) . Let

F i j ¯ ≔ ∂ F ∂ X i j ¯ = ∂ F ∂ λ k ∂ λ k ∂ X i j ¯ ,

then at x 0 , we have

F i j ¯ = F i i ¯ δ i j .

Let

ℱ ≔ ∑ i F i i ¯ .

Lemma 2.3

[23] If λ ∈ Γ k − 1 and α l ( x ) > 0 , 0 ≤ l ≤ k − 2 , then the operator F is elliptic and concave in Γ k − 1 .

From Lemma 2.4 in [27] and Lemma 2.2, we have the following lemma.

Lemma 2.4

If u ∈ C 2 ( M ) is a solution of (2.4), λ ( χ u ) ∈ Γ k − 1 , then at x 0 ,

(2.5) n − k + 1 k ≤ ℱ ≤ C ( n , k , inf 0 ≤ l ≤ k − 2 α l , sup ∣ α k − 1 ∣ ) .

Lemma 2.5

Under assumptions of Theorem 1.2, there is a constant θ > 0 such that

(2.6) F i i ¯ ( u i i ¯ − u ̲ i i ¯ ) ≤ − θ ( 1 + ℱ ) ,

(2.7) F 1 1 ¯ ≥ θ .

Proof

Without loss of generality, we may assume that X 1 1 ¯ ≥ ⋯ ≥ X n n ¯ . Thus,

F n n ¯ ≥ ⋯ ≥ F 1 1 ¯ .

Since u ̲ is a C -subsolution, if ε > 0 is small enough, χ u ̲ − ε ω still satisfies Definition 1.1. Since M is compact, there are uniform constants N > 0 and δ > 0 such that

(2.8) F ( X ˜ ) > β k − 1 + δ ,

where

X ˜ = X ̲ − ε g + N 0 ⋯ 0 0 0 ⋯ 0 0 0 ⋯ 0 n × n .

Direct calculation yields

(2.9) F i i ¯ ( u i i ¯ − u ̲ i i ¯ ) = F i i ¯ ( X i i ¯ − X ̲ i i ¯ ) = F i i ¯ ( X i i ¯ − X ˜ i i ¯ ) + F 1 1 ¯ N − ε ℱ .

Since F is concave in Γ k − 1 , from (2.8), we obtain

(2.10) ∑ i = 1 n F i i ¯ ( X i i ¯ − X ˜ i i ¯ ) ≤ F ( X ) − F ( X ˜ ) ≤ − δ .

By substituting (2.10) into (2.9), we obtain

F i i ¯ ( u i i ¯ − u ̲ i i ¯ ) ≤ − δ + F 1 1 ¯ N − ε ℱ .

Set θ = min δ 2 , ε , δ 2 N . If F 1 1 ¯ N ≤ δ 2 , we have (2.6); otherwise, (2.7) must be true.□

3 C 0 estimate

In this section, we obtain the C 0 estimate by using the Alexandroff-Bakelman-Pucci maximum principle and prove the following Proposition 3.1, which is similar to the approach of Székelyhidi [12].

Proposition 3.1

Let α l ( x ) > 0 for 0 ≤ l ≤ k − 2 and χ be a smooth real ( 1 , 1 ) form on ( M , g ) . Assume that u and u ̲ are solution and C -subsolution to (1.1) with λ ( χ u ) ∈ Γ k − 1 , λ ( χ u ̲ ) ∈ Γ k − 1 , respectively. We normalize u such that sup M ( u − u ̲ ) = 0 . There is a constant C depending on the given data, such that

(3.1) sup M ∣ u ∣ < C .

Proof

To simplify notation, we can assume u ̲ = 0 , otherwise we modify the background form χ . Therefore, sup M u = 0 . The following goal is to prove that L = inf M u has a uniform lower bound. Notice that λ ( χ u ) ∈ Γ k − 1 , so λ ( χ u ) ∈ Γ 1 , that is,

Δ u = g i p ¯ u i p ¯ > − g i p ¯ χ i p ¯ ≥ − C ^ .

Let G : M × M → R be the Green’s function of a Gauduchon metric conformal to g . From [1], there is a uniform constant K such that

G ( x , y ) + K ≥ 0 , ∀ ( x , y ) ∈ M × M , and ∫ y ∈ M G ( x , y ) ω n ( y ) = 0 .

Since sup M u = 0 , there is a point x 0 ∈ M such that u ( x 0 ) = 0 . Hence,

u ( x 0 ) = ∫ M u d μ − ∫ y ∈ M G ( x 0 , y ) Δ u ( y ) ω n ( y ) = ∫ M u d μ − ∫ y ∈ M ( G ( x 0 , y ) + K ) Δ u ( y ) ω n ( y ) ≤ ∫ M u d μ + C ^ K ,

which yields

∫ M ∣ u ∣ d μ ≤ C ^ K .

Next, we choose local coordinates at the minimum point of u and L = inf M u = u ( 0 ) . Let B ( 1 ) = { z : ∣ z ∣ < 1 } and v = u + ε ∣ z ∣ 2 for a small ε > 0 . From the Alexandroff-Bakelman-Pucci maximum principle, we obtain

(3.2) c 0 ε 2 n ≤ ∫ Ω det ( D 2 v ) ,

where

Ω = x ∈ B ( 1 ) : ∣ D v ( x ) ∣ < ε 2 , v ( y ) ≥ v ( x ) + D v ( x ) ⋅ ( y − x ) , ∀ y ∈ B ( 1 ) .

Let λ ˜ − λ ( χ u ̲ ) ≥ 0 and

σ k ( λ ˜ ) σ k − 1 ( λ ˜ ) − ∑ l = 0 k − 2 β l σ l ( λ ˜ ) σ k − 1 ( λ ˜ ) = β k − 1 ( x ) .

u ̲ is a C subsolution, which means that ∣ λ ˜ ∣ is bounded. Since M is compact, there is uniform constant η > 0 such that λ ( χ u ̲ ) − η 1 satisfies Definition 1.1. Since Ω is a contact set, we have D 2 v ( x ) ≥ 0 , for x ∈ Ω , which implies u i j ¯ ( x ) + ε δ i j ≥ 0 . Choosing ε such that 0 < ε ≤ η , on Ω , we have

λ ( χ u ) − ( λ ( χ u ̲ ) − η 1 ) ≥ λ ( χ u ) − ( λ ( χ u ̲ ) − ε 1 ) = λ ( u i j ¯ ) + ε 1 ≥ 0 .

Since

σ k ( λ ( χ u ) ) σ k − 1 ( λ ( χ u ) ) − ∑ l = 0 k − 2 β l σ l ( λ ( χ u ) ) σ k − 1 ( λ ( χ u ) ) = β k − 1 ( x ) ,

we obtain that ∣ λ ( χ u ) ∣ is bounded, which yields ∣ u i j ¯ ∣ ≤ C . Then

det ( D 2 v ( x ) ) ≤ 2 2 n det ( v i j ¯ ) 2 ≤ C .

From this and (3.2), we obtain

(3.3) c 0 ε 2 n ≤ ∫ Ω det ( D 2 v ) ≤ C ⋅ vol ( Ω ) .

On the other hand, we have for x ∈ Ω

v ( 0 ) ≥ v ( x ) − D v ( x ) ⋅ x > v ( x ) − ε 2 ,

∣ v ( x ) ∣ > ∣ L + ε 2 ∣ .

Then,

∫ M ∣ v ( x ) ∣ ≥ ∫ Ω ∣ v ( x ) ∣ ≥ ∣ L + ε 2 ∣ ⋅ vol ( Ω ) .

Since ∫ M ∣ v ( x ) ∣ is uniformly bounded, this inequality contradicts (3.3) if L is very large.□

4 C 2 estimate

In this section, we establish the C 2 estimate to equation (1.1). Our calculation is similar to that in [27], but on Hermitian manifolds, equation (1.1) are much more difficult to treat due to the torsion terms.

4.1 Notations and lemma

In local coordinates z = ( z 1 , … , z n ) , the Chern connection ∇ and torsion are given, respectively, by

∇ ∂ ∂ z i ∂ ∂ z j = Γ i j k ∂ ∂ z k , Γ i j k = g k l ¯ ∂ g j l ¯ ∂ z i , T i j k = Γ i j k − Γ j i k ,

while the curvature tensor R i j ¯ k l ¯ by

R i j ¯ k l ¯ = g p l ¯ ∂ Γ i k p ∂ z ¯ j .

For u ∈ C 4 ( M ) , we denote

u i j = ∇ j ∇ i u , u i j ¯ = ∇ j ¯ ∇ i u .

We have (see [4,11,31])

(4.1) u i j ¯ l = u l j ¯ i + T i l p u p j ¯ , u i j ¯ k = u i k j ¯ − g l m ¯ R k j ¯ i m ¯ u l , u i j ¯ k ¯ = u i k ¯ j ¯ + T j k l ¯ u i l ¯ , u i j ¯ k = u k i j ¯ − g l m ¯ R i j ¯ k m ¯ u l + T i k l u l j ¯ ,

and

(4.2) u i j ¯ k l ¯ = u k l ¯ i j ¯ + g p q ¯ ( R k l ¯ i q ¯ u p j ¯ − R i j ¯ k q ¯ u p l ¯ ) + T i k p u p j ¯ l ¯ + T j l q ¯ u i q ¯ k − T i k p T j l q ¯ u p q ¯ .

Let A i j ¯ = g j p ¯ X i p ¯ , λ ( A ) = ( λ 1 , … , λ n ) and λ 1 ≥ ⋯ ≥ λ n . For a fixed point x 0 ∈ M , choose a local coordinates such that A i j ¯ = A i i ¯ δ i j . Since λ 1 , … , λ n need not be distinct at x 0 , we will perturb χ u slightly such that λ 1 , … , λ n become smooth functions near x 0 . Let D be a diagonal matrix such that D 11 = 0 and 0 < D 22 < ⋯ < D n n are small, satisfying D n n < 2 D 22 . Define the matrix A ˜ = A − D . At x 0 , A ˜ has eigenvalues

λ ˜ 1 = λ 1 , λ ˜ i = λ i − D i i , i ≥ 2 .

Lemma 4.1

(4.3) λ ˜ 1 , i i ¯ ≥ X i i ¯ 1 1 ¯ + 2 Re ( X 1 1 ¯ i T 1 i 1 ¯ ) − C 0 λ 1 − C 0 .

Proof

Commuting derivative of λ 1 ˜ gives

λ ˜ 1 , i = ∂ λ ˜ 1 ∂ A ˜ p q ¯ ∂ A ˜ p q ¯ ∂ z i = X 1 1 ¯ i − ( D 11 ) i ,

(4.4) λ ˜ 1 , i i ¯ = ∂ 2 λ ˜ 1 ∂ A ˜ r s ¯ ∂ A ˜ p q ¯ ∂ A ˜ p q ¯ ∂ z i ∂ A ˜ r s ¯ ∂ z ¯ i + ∂ λ ˜ 1 ∂ A ˜ p q ¯ ∂ 2 A ˜ p q ¯ ∂ z ˜ i ∂ z i = ∑ p ≥ 2 ∣ X 1 p ¯ i ∣ 2 + ∣ X p 1 ¯ i ∣ 2 λ 1 − λ ˜ p − 2 ∑ p ≥ 2 Re ( ( D 1 p ) i ¯ X p 1 ¯ i ) + Re ( ( D p 1 ) i ¯ X 1 p ¯ i ) λ 1 − λ ˜ p + ∑ p ≥ 2 ( D 1 p ) i ( D p 1 ) i ¯ + ( D p 1 ) i ( D 1 p ) i ¯ λ 1 − λ ˜ p + X 1 1 ¯ i i ¯ + ( D 11 ) i i ¯ .

λ ( A ) ∈ Γ 1 implies that ∣ λ p ∣ < ( n − 1 ) λ 1 , p ≥ 2 . If the matrix D is sufficiently small, then ∣ λ ˜ p ∣ < ( n − 1 ) λ 1 , p ≥ 2 , which means that

1 n λ 1 ≤ 1 λ 1 − λ ˜ p ≤ 1 D p p .

We are trying to bound λ 1 from mentioned earlier, so we can assume λ 1 > 1 . Hence,

(4.5) ∑ p ≥ 2 ( D 1 p ) i ( D p 1 ) i ¯ + ( D p 1 ) i ( D 1 p ) i ¯ λ 1 − λ ˜ p + ( D 11 ) i i ¯ ≥ − C 0 .

From here on, C 0 will always denote such a constant, which depends on the given data and may vary from line to line. Using

2 Re ( ( D 1 p ) i ¯ X p 1 ¯ i ) ≤ 1 2 ∣ X p 1 ¯ i ∣ 2 + C 0 ,

we have

(4.6) ∑ p ≥ 2 ∣ X 1 p ¯ i ∣ 2 + ∣ X p 1 ¯ i ∣ 2 λ 1 − λ ˜ p − 2 ∑ p ≥ 2 Re ( ( D 1 p ) i ¯ X p 1 ¯ i ) + Re ( ( D p 1 ) i ¯ X 1 p ¯ i ) λ 1 − λ ˜ p ≥ 1 2 n λ 1 ∑ p ≥ 2 ( ∣ X 1 p ¯ i ∣ 2 + ∣ X p 1 ¯ i ∣ 2 ) − C 0 .

From (4.2), we obtain

(4.7) u 1 1 ¯ i i ¯ = u i i ¯ 1 1 ¯ + R i i ¯ 1 p ¯ u p 1 ¯ − R 1 1 ¯ i p ¯ u p i ¯ + T 1 i p u p 1 ¯ i ¯ + T 1 i p ¯ u 1 p ¯ i − T 1 i p T 1 i q ¯ u p q ¯ .

From this, we have

(4.8) X 1 1 ¯ i i ¯ = X i i ¯ 1 1 ¯ + χ 1 1 ¯ i i ¯ − χ i i ¯ 1 1 ¯ + R i i ¯ 1 p ¯ u p 1 ¯ − R 1 1 ¯ i p ¯ u p i ¯ + T 1 i p u p 1 ¯ i ¯ + T 1 i p ¯ u 1 p ¯ i − T 1 i p T 1 i q ¯ u p q ¯ ≥ X i i ¯ 1 1 ¯ + λ 1 R i i ¯ 1 1 ¯ − λ i R 1 1 ¯ i i ¯ + 2 Re ( X 1 p ¯ i T 1 i p ¯ ) − λ p ∣ T 1 i p ∣ 2 − C 0 ≥ X i i ¯ 1 1 ¯ + 2 Re ( X 1 1 ¯ i T 1 i 1 ¯ ) − 1 2 n λ 1 ∑ p ≥ 2 ∣ X 1 p ¯ i ∣ 2 − C 0 λ 1 − C 0 .

Substituting (4.5), (4.6), and (4.8) into (4.4) gives (4.3).□

4.2 C 2 estimate

Proposition 4.2

sup M ∣ ∂ ∂ ¯ u ∣ ≤ C ( sup M ∣ ∇ u ∣ 2 + 1 ) ,

where C is a uniform constant.

Proof

We assume that the C subsolution u ̲ = 0 , since otherwise we modify the background form χ . We normalize u so that sup M u = 0 . Consider the function

(4.9) W = log λ ˜ 1 + φ ( ∣ ∇ u ∣ 2 ) + ψ ( u ) .

Here,

φ ( t ) = − 1 2 log 1 − t 2 K , 0 ≤ t ≤ K − 1 , ψ ( t ) = − E log 1 + t 2 L , − L + 1 ≤ t ≤ 0 ,

where

K = sup M ∣ ∇ u ∣ 2 + 1 , L = sup M ∣ u ∣ + 1 , E = 2 L ( C 1 + 1 ) ,

and C 1 is to be determined later. Direct calculation gives

(4.10) 0 < 1 4 K ≤ φ ′ ≤ 1 2 K , φ ″ = 2 ( φ ′ ) 2 ,

and

(4.11) C 1 + 1 ≤ − ψ ′ ≤ 2 ( C 1 + 1 ) , ψ ″ ≥ 4 ε 1 − ε ( ψ ′ ) 2 , for all ε ≤ 1 4 E + 1 .

Since M is compact, W attains its maximum at some point x 0 ∈ M . From now on, all the calculations will be carried out at the point x 0 and the Einstein summation convention will be used. Calculating covariant derivatives, we obtain

(4.12) 0 = W i = X 1 1 ¯ i λ 1 + φ ′ ( ∣ ∇ u ∣ 2 ) i + ψ ′ u i − ( D 11 ) i λ 1 , 1 ≤ i ≤ n ,

(4.13) 0 ≥ W i i ¯ = λ ˜ 1 , i i ¯ λ 1 − λ ˜ 1 , i λ ˜ 1 , i ¯ λ 1 2 + ψ ′ u i i ¯ + ψ ″ ∣ u i ∣ 2 + φ ′ ( ∣ ∇ u ∣ 2 ) i i ¯ + φ ″ ∣ ( ∣ ∇ u ∣ 2 ) i ∣ 2 .

Multiplying (4.13) by F i i ¯ and summing it over index i yield

(4.14) 0 ≥ F i i ¯ λ ˜ 1 , i i ¯ λ 1 − F i i ¯ ∣ λ ˜ 1 , i ∣ 2 λ 1 2 + ψ ′ F i i ¯ u i i ¯ + ψ ″ F i i ¯ ∣ u i ∣ 2 + φ ′ F i i ¯ ( ∣ ∇ u ∣ 2 ) i i ¯ + φ ″ F i i ¯ ∣ ( ∣ ∇ u ∣ 2 ) i ∣ 2 .

We will control some terms in (4.14). Covariant differentiating equation (2.4) twice in the ∂ ∂ z 1 direction and the ∂ ∂ z ¯ 1 direction, we have

(4.15) F i i ¯ X i i ¯ 1 + ∑ l = 0 k − 2 ( β l ) 1 F l = ( β k − 1 ) 1

and

(4.16) F i j ¯ , p q ¯ X i j ¯ 1 X p q ¯ 1 ¯ + F i i X i i ¯ 1 1 ¯ + 2 Re ∑ l = 0 k − 2 ( β l ) 1 ¯ F l i i ¯ X i i ¯ 1 + ∑ l = 0 k − 2 ( β l ) 1 1 ¯ F l = ( β k − 1 ) 1 1 ¯ .

Direct calculation deduces that

(4.17) F i i ¯ X i i ¯ = F i i ¯ λ i = F k i i ¯ λ i + ∑ l = 0 k − 2 β l F l i i ¯ λ i = β k − 1 − ∑ l = 0 k − 2 ( k − l ) β l F l .

From Lemmas 4.1 and (4.16), we can estimate the first term in (4.14)

(4.18) F i i ¯ λ ˜ 1 , i i ¯ λ 1 ≥ 1 λ 1 F i i ¯ X i i ¯ 1 1 ¯ + 2 λ 1 F i i ¯ Re ( X 1 1 ¯ i T 1 i 1 ¯ ) − C 0 ℱ = − 1 λ 1 F i j ¯ , p q ¯ X i j ¯ 1 X p q ¯ 1 ¯ − 2 λ 1 Re ∑ l = 0 k − 2 ( β l ) 1 ¯ F l i i ¯ X i i ¯ 1 − 1 λ 1 ∑ l = 0 k − 2 ( β l ) 1 1 ¯ F l + ( β k − 1 ) 1 1 ¯ λ 1 + 2 λ 1 F i i ¯ Re ( X 1 1 ¯ i T 1 i 1 ¯ ) − C 0 ℱ .

It is shown by Krylov in [22] that the σ k − 1 σ l 1 k − l − 1 is concave in Γ k − 1 for 0 ≤ l ≤ k − 2 , which means that

( − F l ) − 1 k − l − 1 i i ¯ , j j ¯ X i i ¯ 1 X j j ¯ 1 ¯ ≤ 0 .

Direct computation gives

− F l i i ¯ , j j ¯ X i i ¯ 1 X j j ¯ 1 ≥ k − l k − l − 1 ( − F l ) − 1 ∣ F l i i ¯ X i i ¯ 1 ∣ 2 ,

which yields

− F i i ¯ , j j ¯ X i i ¯ 1 X j j ¯ 1 ¯ λ 1 − 2 λ 1 Re ∑ l = 0 k − 2 ( β l ) 1 ¯ F l i i ¯ X i i ¯ 1 ≥ ∑ l = 0 k − 2 k − l k − l − 1 β l λ 1 ( − F l ) − 1 F l i i ¯ X i i ¯ 1 + k − l − 1 k − l ( β l ) 1 ¯ β l F l 2 + ∑ l = 0 k − 2 k − l − 1 k − l ( β l ) 1 2 β l λ 1 F l

≥ ∑ l = 0 k − 2 k − l − 1 k − l ( β l ) 1 2 β l λ 1 F l ≥ − C 0 ,

where the last inequality is given by Lemma 2.2. Noting that

− F i j ¯ , p q ¯ X i j ¯ 1 X p q ¯ 1 ¯ ≥ − F i i ¯ , j j ¯ X i i ¯ 1 X j j ¯ 1 ¯ − F i 1 ¯ , 1 i ¯ ∣ X i 1 ¯ 1 ∣ 2 ,

we have

(4.19) − 1 λ 1 F i j ¯ , p q ¯ X i j ¯ 1 X p q ¯ 1 ¯ − 2 λ 1 Re ∑ l = 0 k − 2 ( β l ) 1 ¯ F l i i ¯ X i i ¯ 1 ≥ − F i 1 ¯ , 1 i ¯ ∣ X i 1 ¯ 1 ∣ 2 − C 0 .

Substituting (4.19) into (4.18) and by Lemma 2.2,

(4.20) F i i ¯ λ ˜ 1 , i i ¯ λ 1 ≥ − F i 1 ¯ , 1 i ¯ ∣ X i 1 ¯ 1 ∣ 2 λ 1 + 2 λ 1 F i i ¯ Re ( X 1 1 ¯ i T 1 i 1 ¯ ) − C 0 ℱ − C 0 .

Since

(4.21) X 1 1 ¯ i = χ 1 1 ¯ i + u 1 1 ¯ i = ( χ 11 i − χ i 11 + T i 1 p χ p 1 ¯ ) + X i 1 ¯ 1 − T i 1 1 λ 1 ,

we have

(4.22) ∣ X 1 1 ¯ i ∣ 2 ≤ ∣ X i 1 ¯ 1 ∣ 2 − 2 λ 1 Re ( X i 1 ¯ 1 T i 1 1 ¯ ) + C 0 ( λ 1 2 + ∣ X 1 1 ¯ i ∣ ) .

From

λ ˜ 1 , i = X 1 1 ¯ i − ( D 11 ) i ,

we estimate the second term in (4.14)

(4.23) − F i i ¯ ∣ λ ˜ 1 , i ∣ 2 λ 1 2 = − F i i ¯ ∣ X 1 1 ¯ i ∣ 2 λ 1 2 + 2 λ 1 2 F i i ¯