Remarks on a result of Chen-Cheng

Zhiqin Lu; Reza Seyyedali

doi:10.1515/coma-2024-0004

Article Open Access

Remarks on a result of Chen-Cheng

Zhiqin Lu and Reza Seyyedali

Published/Copyright: July 18, 2024

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From the journal Complex Manifolds Volume 11 Issue 1

Abstract

In their seminal work, Chen and Cheng proved a priori estimates for the constant scalar curvature metrics on compact Kähler manifolds. They also prove C 3 , α -estimate for the potential of the Kähler metrics under boundedness assumption on the scalar curvature and the entropy. The goal of this article is to replace the uniform boundedness of the scalar curvature to the L p -boundedness of the scalar curvature.

Keywords: Calabi flow; Chen-Cheng estimate; scalar curvature

MSC 2010: 53B35

1 Introduction

A fundamental theorem in the realm of complex analysis is the uniformization theorem. One of the implications of the uniformization theorem is that every compact Riemann surface admits a metric with consistent Gaussian curvature. This principle can be extended in numerous ways to manifolds of higher dimensions. Within complex geometry, the aspiration is to discover canonical metrics on a Kähler manifold, those that align with the complex structure and exhibit curvature with specified characteristics. Kähler-Einstein metrics, constant scalar curvature Kähler metrics, and extremal metrics are prime examples of such metrics.

The existence of Kähler-Einstein metrics on compact complex manifolds was proved by Yau for manifolds with a trivial canonical class [13,14]. In the case of negative first Chern classes, both Aubin and Yau independently affirmed the existence of Kähler-Einstein metrics [1,13,14]. However, the scenario is most challenging for Fano manifolds, where the first Chern class is positive, and there exist known barriers to the realization of Kähler-Einstein metrics. As conjectured by Yau, these barriers should all correlate with the stability of the manifolds.

The challenge concerning Fano manifolds was eventually overcome by Chen et al. [4–6] and Tian [8] a few years back. Regarding cscK metrics, the Yau-Tian-Donaldson conjecture proposes that the presence of such metrics corresponds to a form of stability. The cscK metrics scenario is notably more intricate than that of Kähler-Einstein metrics, primarily because the constant scalar curvature equation is a fourth-order fully nonlinear elliptic partial differential equation (PDE), while our understanding of fourth-order nonlinear PDEs is still limited. In contrast, the Kähler-Einstein equation is a second-order fully nonlinear elliptic PDE, a field that has been extensively explored over the years.

Progress in the constant scalar curvature equation had been stagnant until the recent breakthrough of Chen and Cheng [2,3], who established a priori estimates for cscK equations, providing significant insights that the Kähler potential and all its derivatives of a cscK metric can be controlled in terms of the relative entropy.

Let M be a Kähler manifold of dimension n and ω be its Kähler form. For any Kähler potential φ , define ω φ = ω + − 1 ∂ ∂ ¯ φ . We consider the equations

(1) ω φ n = ( ω + − 1 ∂ ∂ ¯ φ ) n = e F ω n , sup M φ = 0 , Δ ω φ F = − R + tr ω φ η ,

where R is the scalar curvature of the metric ω φ , and η is a fixed smooth ( 1 , 1 ) -form. The prototype of η is the Ricci curvature Ric ( ω ) of ω .

In their papers [2,3], Chen and Cheng proved the following:

Theorem 1.1

[2, 3] For any p ≥ 1 , there exists a constant C that depends on n , p , ω , η , ‖ R ‖ ∞ , and ∫ M e F 1 + F 2 ω n such that ‖ F ‖ W 2 , p , ‖ φ ‖ W 4 , p ≤ C . In particular, F and φ are uniformly bounded in C 1 , α and C 3 , α , respectively, for any α ∈ ( 0 , 1 ) .

With some modifications to the argument in [2], we slightly generalize the proceeding theorem. Namely, we replace the uniform bound on the scalar curvature with the L p -bound for some p > 0 .

Let Φ ( t ) = 1 + t 2 . Define A F and A R , p by

A F n = ∫ M e F Φ ( F ) ω n , A R , p n = ∫ M e F Φ ( R ) p ω n

for p > 0 . A F gives an upper bound for the entropy

∫ M F e F ω n ≤ A F ,

and A R , p gives an upper bound for the L p -norm of R with respect to ω φ

∫ M ∣ R ∣ p ω φ n 1 ∕ p ≤ A R , p n ∕ p .

The main results of this article are the following theorems.

Theorem 1.2

For any p > n , there exists a constant C that depends on n , p , ω , A F , and A R , p such that ‖ F ‖ ∞ ≤ C and ‖ φ ‖ ∞ ≤ C .

Theorem 1.3

Let n = dim M . Then, there exist p n > 2 n that depends only on n such that ‖ F ‖ W 2 , p n ≤ C and ‖ φ ‖ W 4 , p n ≤ C for a constant C depending on n , ω , η , A F , and A R , p n .

Moreover, for any p ≥ p n , there exists a constant C p that depend on n , ω , η , A F , and A R , p such that ‖ F ‖ W 2 , p ≤ C p and ‖ φ ‖ W 4 , p ≤ C p .

Note that in Theorem 1.3, W 2 , p and W 4 , p are optimal regularity for φ and F , respectively, because of (1) and the fact that R is L p for some p > 0 .

Theorem 1.3 gives a priori C 3 , α and C 1 , α estimate for φ and F , respectively for some α = α ( p , n ) ∈ ( 0 , 1 ) by Sobolev embedding theorem.

This article is organized as follows. In Section 2, we prove Theorem 1.2. Our argument does not use the Alexandrov maximum principle and the cut-off function as in Chen and Cheng [2,3]. Instead, we use Kołodziej’s theorem to prove the boundedness of the auxiliary functions. We then prove the result using the classical maximum principle.

In Section 3, we prove that there is an L p -estimate of n + Δ φ . The C 2 estimate is obtained in Section 4 using Moser iteration. The arguments in Sections 3 and 4 are essentially the same as those in [3].

Throughout this article, we shall use ∫ M f to denote ∫ M f ω n , where ω is the background metric of the manifold. We use ‖ f ‖ p to denote the L p -norm of function f with respect to the background metric ω .

2 Proof of Theorem 1.2

The section’s main goal is to prove a uniform estimate for φ and F . This section’s constant C depends on n = dim M , ω , and η , which may differ line by line.

Lemma 2.1

Let h : M → R be a positive function and φ and ν be Kähler potentials such that

( ω + − 1 ∂ ∂ ¯ φ ) n = e F ω n ,

( ω + − 1 ∂ ∂ ¯ ν ) n = e F h n ω n .

Then, Δ φ ν ≥ n h − tr ω φ ( ω ) . Here, ω φ = ω + − 1 ∂ ∂ ¯ φ and Δ φ is the Laplacian with respect to the metric ω φ .

Proof

This follows by applying the AM-GM inequality to tr ω φ ( ω + − 1 ∂ ∂ ¯ ν ) .□

Let α = α ( M , ω ) be the α -invariant of ( M , ω ) . By definition, for any smooth function φ : M → R such that ω + − 1 ∂ ∂ ¯ φ > 0 , we have

∫ M e − 1 2 α ( φ − sup M φ ) ω n ≤ C

for some C > 0 independent to φ .

Theorem 2.1

For any p > n , there exists δ 0 = δ 0 depending on n , p , ω , η , A F , ‖ R ‖ p such that for any δ < δ 0 , we have

∫ M e ( 1 + δ ) F ≤ C ,

where C = C ( n , p , δ 0 , ω , η , A F , ∣ ∣ R ∣ ∣ p ) .

Proof

For a fixed p > n , we define functions ψ and ρ as the solutions of the following:

(2) ( ω + − 1 ∂ ∂ ¯ ψ ) n = A F − n e F Φ ( F ) ω n = A F − n Φ ( F ) ω φ n , sup M ψ = 0 ;

(3) ( ω + − 1 ∂ ∂ ¯ ρ ) n = A R , p − n e F Φ ( R ) p ω n = A R , p − n Φ ( R ) p ω φ n , sup M ρ = 0 .

Let 0 < ε ≤ 1 and u = F + ε ψ + ε ρ − λ φ = v − λ φ , where v = F + ε ψ + ε ρ and λ = ∣ η ∣ ω + 2 . Let δ > 0 . Then, by Lemma 2.1, we have

(4) e − δ u Δ φ ( e δ u ) ≥ δ Δ φ u ≥ δ ( − R + tr ω φ η ) + ε δ ( n A F − 1 Φ ( F ) 1 n − tr ω φ ω ) + ε δ ( n A R , p − 1 Φ ( R ) p n − tr ω φ ω ) − n δ λ + δ λ tr ω φ ω = δ ( − R + ε n A F − 1 Φ ( F ) 1 n + ε n A R , p − 1 Φ ( R ) p n − λ n ) + δ ( tr ω φ η − 2 ε tr ω φ ω + λ tr ω φ ω ) ≥ δ ( − R + ε n A F − 1 Φ ( F ) 1 n + ε n A R , p − 1 Φ ( R ) p n − λ n ) .

The last inequality holds since ε ≤ 1 .

Let

δ 0 = λ − 1 min ( α , 1 ) ,

where α = α ( M , [ ω ] ) is the α -invariant of M . We choose 0 < δ < 1 2 δ 0 . Fixing δ , we choose ε > 0 small so that

2 ( 1 + δ ) ⋅ ε < min ( α , 1 ) .

Let

Φ ˆ ( F ) = ε n A F − 1 Φ ( F ) 1 ∕ n .

Then

ε A R , p − 1 Φ ( R ) p n − R ≥ − C ( ε , p , A R , p ) ,

since A R , p is bounded and p > n . Therefore, (4) implies that

(5) Δ φ e δ u ≥ δ e δ u ( Φ ˆ ( F ) − C )

for some constant C > 0 . As a result, we have

∫ M e δ u ( Φ ˆ ( F ) − C ) ω φ n ≤ 0 .

We let

E 1 = { x ∣ Φ ˆ ( F ) − C ≥ 1 } ; E 2 = { x ∣ Φ ˆ ( F ) − C < 1 } .

On E 2 , F is bounded, say F ≤ C . Thus, we have

∫ E 1 e δ u + F ≤ ∫ E 1 e δ u ( Φ ˆ ( F ) − C ) ω φ n ≤ − ∫ E 2 e δ u ( Φ ˆ ( F ) − C ) ω φ n .

Since Φ ˆ ( F ) is nonnegative, and on E 2 , we have u ≤ C − λ φ , we have

∫ E 1 e δ u + F ≤ C ∫ E 2 e − δ λ φ ≤ C ∫ M e − δ λ φ ≤ C ,

since δ λ is less than half of the α -invariant. By definition of u , we have

∫ E 1 e ( 1 + δ ) F + ε δ ( ψ + ρ ) ≤ ∫ E 1 e δ u + F ≤ C .

Since

ω + − 1 ∂ ∂ ¯ ψ + ρ 2 > 0 ,

using the Hölder inequality, we have

∫ E 1 e ( 1 + δ ∕ 2 ) F = ∫ E 1 e ( 1 + δ ∕ 2 ) F + ( 1 + δ ∕ 2 ) 1 + δ ε δ ( ψ + ρ ) ⋅ e − ( 1 + δ ∕ 2 ) 1 + δ ε δ ( ψ + ρ ) ≤ ∫ E 1 e ( 1 + δ ) F + ε δ ( ψ + ρ ) 1 + δ ∕ 2 1 + δ ⋅ ∫ E 1 e − 1 + δ ∕ 2 δ ∕ 2 ε δ ( ψ + ρ ) δ ∕ 2 1 + δ ≤ C ,

since 1 + δ ∕ 2 δ ∕ 2 ε δ is less than half of the α -invariant. Combining the above with the fact that F is bounded on E 2 , we have

□ ∫ M e ( 1 + δ ∕ 2 ) F ≤ C .

The following proof of Theorem 1.2 is slightly different from that of Chen and Cheng [2].

Proof of Theorem 1.2

As in (2) and (3), we define functions ψ and ρ as the solutions of the following:

(6) ( ω + − 1 ∂ ∂ ¯ ψ ) n = A F − n e F Φ ( F ) ω n = A F − n Φ ( F ) ω φ n , sup M ψ = 0 ;

(7) ( ω + − 1 ∂ ∂ ¯ ρ ) n = A R , p ′ − n e F Φ ( R ) p ′ ω n = A R , p ′ − n Φ ( R ) p ′ ω φ n , sup M ρ = 0 ,

where p ′ = ( p + n ) ∕ 2 .

We shall use the result of Kołodziej [7] to prove that the functions φ , ψ , ρ are uniformly bounded.

That φ is bounded directly follows from Theorem 2.1 and Kołodziej’s theorem.

Since x 1 + δ e − x ≤ C for any real number x > 0 , for δ < δ 0 ∕ 2 , we have

∫ M Φ ( F ) 1 + δ e ( 1 + δ ) F ≤ C ( n , p , δ 0 , ω , η , A F , ∣ ∣ R ∣ ∣ p ) .

Hence, Kołodziej’s theorem implies that ψ is uniformly bounded.

Finally, we prove that ρ is uniformly bounded. Let 0 < σ < δ < δ 0 ∕ 2 and a = 1 + σ . We have

(8) ∫ M Φ ( R ) a p ′ e a F = ∫ M Φ ( R ) a p ′ e σ F ω φ n ≤ ∫ M Φ ( R ) a p ′ δ δ − σ ω φ n δ − σ δ ∫ M e δ F ω φ n σ δ ≤ C ∫ M Φ ( R ) a p ′ δ δ − σ ω φ n δ − σ δ .

The first inequality follows from the Hölder inequality and the last inequality follows from Theorem 2.1. Now choose σ sufficiently small such that a p ′ δ δ − σ < p . Therefore, the Hölder inequality implies that

∫ M Φ ( R ) a p ′ e a F ≤ C ∫ M Φ ( R ) p ω φ n .

This, together with Kołodziej’s theorem implies that ‖ ρ ‖ ∞ ≤ C = C ( n , ω , η A F , A R , n + 1 ) .

Let u = F + ψ + ρ − λ φ , where λ = ∣ η ∣ ω + 2 . Then, we have

Δ φ u ≥ − R + n A F − 1 Φ ( F ) 1 n + n A R − 1 Φ ( R ) p + n 2 n − C ≥ n A F − 1 Φ ( F ) 1 n − C ,

since ( p + n ) ∕ 2 n > 1 . Let x 0 be a maximum point of u . Then, by the above,

F ( x 0 ) ≤ C .

As a result, for any x ∈ M , we have

u ( x ) ≤ u ( x 0 ) = F ( x 0 ) + ψ ( x 0 ) + ρ ( x 0 ) − λ φ ( x 0 ) ≤ C .

This implies that F ( x ) ≤ C .

Now, let u ′ = − F + ψ + ρ − λ φ . Then, by a similarly computation, we have

Δ φ u ′ ≥ ε n A F − 1 Φ ( F ) 1 n − C .

The same argument would imply that F ≥ − C . This completes the proof of the theorem.□

3 W 2 , p estimate

In this section, we prove that for any p > 0 , n + Δ φ , where φ is the solution of (1), is in L p ( M ) .

This section’s constants C and C i depend on n = dim M , p > 0 , ω , and η , which may differ line by line.

Theorem 3.1

Let γ = n n − 1 and p > 0 be a positive number. Then

∫ M ( n + Δ φ ) p ≤ C .

where C depends on n , p , ω , η , ‖ φ ‖ ∞ , ‖ F ‖ ∞ , and ‖ R ‖ ( n − 1 ) p γ .

To prove Theorem 3.1, we first prove the following gradient estimate.

Proposition 3.1

For any p ≥ 1 , there exist constants c 1 and c 2 depending on n , p , ω , η , ‖ φ ‖ ∞ , ‖ F ‖ ∞ , and ‖ R ‖ ( n − 1 ) p such that

‖ ∇ φ ‖ 2 p ≤ c 1 + c 2 ‖ R ‖ ( n − 1 ) p ( n − 1 ) ∕ 2 .

Proof

Let

u = e − ( F + λ φ ) + 1 2 φ 2 ( ∣ ∇ φ ∣ 2 + K ) ,

where K is an absolute constant (e.g., we can take K = 10 ). Then, we have

Δ φ u ≥ C u n n − 1 − ( c + ∣ R ∣ ) u

by [1, p. 918, equation (2.31)], where C , c are positive constants depending on n , p , ω , η , ‖ φ ‖ ∞ , ‖ F ‖ ∞ . Let p > 0 and let γ be defined in Theorem 3.1. Then, we have

1 p + 1 Δ φ u p + 1 = u p Δ φ u + p u p − 1 ∣ ∇ φ u ∣ 2 ≥ u p Δ φ u ≥ C u p + γ − ( c + ∣ R ∣ ) u p + 1 .

Using Young’s inequality ∣ R ∣ u p + 1 ≤ ∣ R ∣ ( p + γ ) ( n − 1 ) + u p + γ , we have

1 p + 1 Δ φ u p + 1 ≥ C u p + γ − C 1 − C 2 ∣ R ∣ ( n − 1 ) ( p + γ ) .

Integrating the above inequality to the volume form ω φ n , we have

C ∫ M u p + γ ω φ n ≤ C 1 + C 2 ∫ M ∣ R ∣ ( n − 1 ) ( p + γ ) .

Since F is bounded, ω φ n and ω n are equivalent. Thus, we have

‖ u ‖ L p + γ ≤ c 1 + c 2 ‖ R ‖ ( n − 1 ) ( p + γ ) n − 1 .

Thus, the proposition is valid for p > γ . But, then from the Hölder inequality, it is valid for any p > 0 .□

Proof of Theorem 3.1

Let α > 2 be a constant depending on p only, and to be determined later. Let λ be a constant depending on M . Let

u = e − α ( F + λ φ ) ( n + Δ φ ) .

By Yau’s estimate, we have

Δ φ u ≥ e − ( α + 1 n − 1 ) F − α λ φ λ α 2 − C ( n + Δ φ ) 1 + 1 n − 1 − λ α n e − α ( F + λ φ ) ( n + Δ φ ) + α e − α ( F + λ φ ) R ( n + Δ φ ) + e − α ( F + λ φ ) ( Δ F − R ω ) ,

where R ω is the scalar curvature of the metric ω . By choosing λ big enough such that λ α 2 − C ≥ λ α 4 , we have

Δ φ u ≥ C 1 u γ − C 2 ∣ R ∣ u + e − α ( F + λ φ ) Δ F − C 3 .

We then have

(9) 1 p + 1 Δ φ u p + 1 = u p Δ φ u + p u p − 1 ∣ ∇ φ u ∣ 2 ≥ p u p − 1 ∣ ∇ φ u ∣ 2 + C 1 u p + γ − C 2 ∣ R ∣ ( p + γ ) ( n − 1 ) + e − α ( F + λ φ ) Δ F u p − C 3 ,

where we used Young’s inequality (with possibly different C 1 > 0 , C 2 , and C 3 ). Integrating the above to the volume form ω φ n and using the fact that F is bounded and R is in L ( p + γ ) ( n − 1 ) , we have

(10) C 1 ∫ M u p + γ + p ∫ M u p − 1 ∣ ∇ φ u ∣ 2 ≤ C 3 − ∫ M e − α ( F + λ φ ) Δ F u p ω φ n .

Let F ˆ = ( 1 − α ) F − α λ φ . Using integration by parts, we have

(11) − ∫ M e − α ( F + λ φ ) Δ F u p ω φ n = − ∫ M Δ F u p e F ˆ = − ( α − 1 ) ∫ u p ∣ ∇ F ∣ 2 e F ˆ − λ α ∫ u p ∇ F ∇ φ e F ˆ + p ∫ u p − 1 ∇ F ∇ u e F ˆ .

By the AM-GM inequality, we have

(12) − α − 1 2 ∫ u p ∣ ∇ F ∣ 2 e F ˆ − λ α ∫ u p ∇ F ∇ φ e F ˆ ≤ λ 2 α 2 2 ( α − 1 ) ∫ u p ∣ ∇ φ ∣ 2 e F ˆ .

Since F ˆ is bounded, we have

λ 2 α 2 2 ( α − 1 ) ∫ u p ∣ ∇ φ ∣ 2 e F ˆ ≤ C 4 ∫ u p ∣ ∇ φ ∣ 2 ,

where C 4 is a constant that depends on λ , α , and ‖ F ˆ ‖ L ∞ . Using Young’s Inequality, we obtain

∫ u p ∣ ∇ φ ∣ 2 ≤ 1 2 ∫ u p + γ + C 5 ∫ ∣ ∇ φ ∣ 2 ( p + γ ) ∕ γ

for a constant depending only on n . By Proposition 3.1, we have

∫ M ∣ ∇ φ ∣ 2 ( p + γ ) ∕ γ ≤ ( c 1 + c 2 ‖ R ‖ ( n − 1 ) 2 ( p + γ ) ∕ n ( n − 1 ) ∕ 2 ) .

Since ( n − 1 ) 2 ( p + γ ) ∕ n ≤ ( p + γ ) ( n − 1 ) , from (12), we conclude that

(13) − α − 1 2 ∫ u p ∣ ∇ F ∣ 2 e F ˆ − λ α ∫ u p ∇ F ∇ φ e F ˆ ≤ C 6 ,

where C 6 depends on λ , α , and ‖ F ˆ ‖ L ∞ .

On the other hand, we have

(14) − α − 1 2 ∫ u p ∣ ∇ F ∣ 2 e F ˆ + p ∫ u p − 1 ∣ ∇ F ∣ ∣ ∇ u ∣ e F ˆ ≤ p 2 2 ( α − 1 ) ∫ u p − 2 ∣ ∇ u ∣ 2 e F ˆ .

By the Cauchy-Schwarz inequality, we have

∣ ∇ u ∣ 2 = ∑ i 1 + φ i i ¯ ⋅ ∣ u i ∣ 1 + φ i i ¯ 2 ≤ ( n + Δ φ ) ⋅ ∣ ∇ φ u ∣ 2 .

Thus

∣ ∇ u ∣ 2 ≤ e α ( F + λ φ ) u ∣ ∇ φ u ∣ 2 ≤ C 7 u ∣ ∇ φ u ∣ 2 e − F ˆ

for C 7 = ‖ e F ‖ L ∞ . Hence, we have

p 2 2 ( α − 1 ) ∫ u p − 2 ∣ ∇ u ∣ 2 e F ˆ ≤ C 7 p 2 2 ( α − 1 ) ∫ u p − 1 ∣ ∇ φ u ∣ 2 .

We choose α large enough so that

C 7 p 2 2 ( α − 1 ) ≤ p 2 .

Then, we have

(15) p 2 2 ( α − 1 ) ∫ u p − 2 ∣ ∇ u ∣ 2 e F ˆ ≤ p 2 ∫ u p − 1 ∣ ∇ φ u ∣ 2 .

Thus, from (11), using (13) and (15), we have

− ∫ M e − α ( F + λ φ ) Δ F u p ω φ n ≤ C 6 + p 2 ∫ u p − 1 ∣ ∇ φ u ∣ 2 .

Combining with (10), we have

C 1 ∫ M u p + γ + p ∫ M u p − 1 ∣ ∇ φ u ∣ 2 ≤ C 3 + C 6 + p 2 ∫ u p − 1 ∣ ∇ φ u ∣ 2 .

Thus,

∫ M ( n + Δ φ ) p ≤ C

is valid for any p > γ . By the monotonicity of the L p -norm, the above inequality is valid for any p > 0 .□

4 C 2 -estimate

In this section, we shall give the C 2 and high-order estimates. This section’s constants C and C i depend on n , ω , and η , which may differ line by line. But contrary to the previous section, these constants are independent of p > 0 .

Theorem 4.1

For each n, there exist positive numbers p n , q n (depending only on n ) and C such that ‖ n + Δ φ ‖ ∞ ≤ C . Here, C depends on n , ω , η , ‖ φ ‖ ∞ , ‖ F ‖ ∞ , ‖ R ‖ p n , and ‖ n + Δ φ ‖ q n .

We start with a Sobolev-type of inequality proved in [2].

Lemma 4.1

Let n be the complex dimension of M. Then, for any ε ∈ ( 0 , 1 n + 1 ) , there exists a constant C that depends on ω and ε such that

‖ u ‖ β 2 ≤ C ‖ n + Δ φ ‖ 1 − ε ε 2 ∫ M ∣ ∇ φ u ∣ φ 2 + ‖ u ‖ 1 2 ,

where β = 2 1 + 1 − ( n + 1 ) ε n − 1 + ε = 2 n ( 1 − ε ) n − 1 + ε .

Proof

The proof is given in [2]. For the reader’s convenience, we include the argument here. We have the following Sobolev inequality:

∫ M ∣ u ∣ 2 n ∕ ( 2 n − 1 ) ≤ C ∫ M ∣ ∇ u ∣ + ∫ M ∣ u ∣ 2 n 2 n − 1 .

Replacing u by u 2 n − 1 2 n β in the above inequality, and by interpolation, we obtain

(16) ∫ M ∣ u ∣ β ≤ C ∫ M ∣ ∇ u ∣ 2 α + ∫ M ∣ u ∣ 2 α β 2 α ,

where α = 1 − ε .

By the Cauchy-Schwarz inequality, we have

∣ ∇ u ∣ 2 = ∑ i 1 + φ i i ¯ ⋅ ∣ u i ∣ 1 + φ i i ¯ 2 ≤ ( n + Δ φ ) ⋅ ∣ ∇ φ u ∣ 2 .

Thus, using (16), we have

□ ∫ M ∣ u ∣ β 2 α β ≤ C ∫ M ∣ ∇ u ∣ 2 α + ∫ M u 2 α ≤ C ∫ M ∣ ∇ φ u ∣ 2 α ( n + Δ φ ) α + ∫ M ∣ u ∣ 2 α ≤ C ∫ M ∣ ∇ φ u ∣ 2 α ∫ M ( n + Δ φ ) α 1 − α 1 − α + C ∫ M ∣ u ∣ 2 α .

Proof of Theorem 4.1

We let

u = e F ∕ 2 ∣ ∇ φ F ∣ φ 2 + ( n + Δ φ ) + 1 .

Then, by [1, equation (4.13)], we have

(17) Δ φ u ≥ − C ( n + Δ φ ) n − 1 u + 2 e F ∕ 2 ⟨ ∇ φ F , ∇ φ Δ φ F ⟩ − C ∣ R ∣ u − C .

Multiplying (17) by u 2 p and integrating by parts and using the fact that F is bounded, we have

(18) 2 p ∫ M u 2 p − 1 ∣ ∇ φ u ∣ 2 ω φ n ≤ C ∫ M ( n + Δ φ ) n − 1 u 2 p + 1 + C ∫ M ∣ R ∣ u 2 p + 1 + C ∫ M u 2 p − 2 ∫ M e F ∕ 2 ⟨ ∇ φ F , ∇ φ Δ φ F ⟩ u 2 p ω φ n .

In the above last term, we use the same idea as in the proof of Theorem 3.1 to obtain

− ∫ M e F ∕ 2 ⟨ ∇ φ F , ∇ φ Δ φ F ⟩ u 2 p ω φ n = ∫ M e F ∕ 2 ( Δ φ F ) 2 u 2 p ω φ n + 1 2 ∫ e F ∕ 2 ( Δ φ F ) ∣ ∇ φ F ∣ 2 u 2 p ω φ n + 2 p ∫ M e F ∕ 2 ( Δ φ F ) ⟨ ∇ φ F , ∇ φ u ⟩ u 2 p − 1 ω φ n .

Using the Cauchy-Schwarz inequality, for any ε 0 > 0 , we have

(19) ∫ M e F ∕ 2 ( Δ φ F ) ⟨ ∇ φ F , ∇ φ u ⟩ u 2 p − 1 ω φ n ≤ C ε 0 − 1 ∫ M ( Δ φ F ) 2 u 2 p ω φ n + ε 0 ∫ M ∣ ⟨ ∇ φ F , ∇ φ u ⟩ ∣ 2 u 2 p − 2 ω φ n ≤ C ε 0 − 1 ∫ M ( Δ φ F ) 2 u 2 p + C ε 0 ∫ M ∣ ∇ φ u ∣ 2 u 2 p − 1 .

As a result, we have

− ∫ M e F ∕ 2 ⟨ ∇ φ F , ∇ φ Δ φ F ⟩ u 2 p ω φ n ≤ C ε 0 ∫ M ∣ ∇ φ u ∣ 2 u 2 p − 1 + C ( ε 0 − 1 + 1 ) ∫ M ( Δ φ F ) 2 u 2 p + C 2 p ∫ M ∣ Δ φ F ∣ u 2 p + 1 .

By choosing ε 0 small enough, from (18), we have

(20) p ∫ M ∣ ∇ φ u ∣ 2 u 2 p − 1 ≤ C 1 ∫ M ( n + Δ φ ) n − 1 u 2 p + 1 + C 2 ∫ M ∣ R ∣ u 2 p + 1 + C 3 ∫ M ( Δ φ F ) 2 u 2 p + C 4 p ∫ M ∣ Δ φ F ∣ u 2 p + 1 .

Using equation (1), we have

∣ Δ φ F ∣ ≤ ∣ R ∣ + ∣ Tr ω φ η ∣ ≤ ∣ R ∣ + C ( n + Δ φ ) n − 1 .

Therefore, from (20), we obtain

∫ M ∣ ∇ φ u ∣ 2 u 2 p − 1 ≤ C 1 ∫ M ( n + Δ φ ) 2 n − 2 u 2 p + 1 + C 2 ∫ M ( 1 + ∣ R ∣ 2 ) u 2 p + 1 .

Hence,

p − 2 ∫ M ∇ φ u p + 1 2 2 ≤ C ∫ M u 2 p − 1 ∣ ∇ φ u ∣ 2 ≤ C ∫ M ( ( n + Δ φ ) 2 n − 2 + 1 + ∣ R ∣ 2 ) u 2 p + 1 .

Now, we fix an ε ∈ ( 0 , 1 n + 1 ) . Let β = 2 ( 1 + δ ) , where

δ = 1 − ( n + 1 ) ε n − 1 + ε

as in Lemma 4.1. Then, we have

u p + 1 2 β 2 ≤ C ‖ n + Δ φ ‖ 1 − ε ε ∫ M ∇ φ u p + 1 2 φ 2 + C u p + 1 2 1 2 ≤ C p 2 ‖ n + Δ φ ‖ 1 − ε ε ∫ M ( ( n + Δ φ ) 2 n − 2 + 1 + ∣ R ∣ 2 ) u 2 p + 1 .

On the other hand, let 2 < θ < β and let θ * = ( 1 − 2 θ − 1 ) − 1 . Then, for any function H , by the Hölder inequality, we have

∫ M H u 2 p + 1 ≤ ‖ H ‖ θ * ⋅ ∫ M u ( 2 p + 1 ) θ 2 2 θ .

In particular, we have

∫ M ∣ R ∣ 2 u 2 p + 1 ≤ ‖ R ‖ 2 θ * 2 ⋅ ∫ M u ( 2 p + 1 ) θ 2 2 θ

and

∫ M ( n + Δ φ ) 2 n − 2 u 2 p + 1 ≤ ‖ n + Δ φ ‖ ( 2 n − 2 ) θ * 2 n − 2 ⋅ ∫ M u ( 2 p + 1 ) θ 2 2 θ .

Assuming ‖ R ‖ 2 θ * ≤ C , ‖ n + Δ φ ‖ ( 2 n − 2 ) θ * + ‖ n + Δ φ ‖ 1 − ε ε ≤ C , we have

u p + 1 2 β 2 ≤ C p 2 u p + 1 2 θ 2 .

This implies that for any p ≥ 1 2 , we have

‖ u ‖ ( p + 1 2 ) β ≤ ( C p 2 ) 2 2 p + 1 ‖ u ‖ ( p + 1 2 ) θ .

Applying Moser’s iteration, one obtains

‖ u ‖ ∞ ≤ C ‖ u ‖ θ .

On the other hand

‖ u ‖ ∞ θ ≤ C ‖ u ‖ θ θ = ∫ M ∣ u ∣ θ ≤ C ‖ u ‖ ∞ θ − 1 ‖ u ‖ 1 .

which implies that

‖ u ‖ ∞ ≤ C ‖ u ‖ 1 ≤ C ∫ M ( ∣ ∇ φ F ∣ φ 2 + ( n + Δ φ ) + 1 ) .

Since

∫ M ( n + Δ φ ) ω n = n

and

∫ M ∣ ∇ φ F ∣ φ 2 ω φ n = − ∫ M F Δ φ F ≤ ∫ M ∣ F ∣ ( ∣ R ∣ + C ( n + Δ φ ) n − 1 ) ≤ C ,

we have

□ ‖ u ‖ ∞ ≤ C .

Remark 1

Choosing ε = 1 2 n + 1 , we obtain q n = 4 n 2 − 4 . On the other hand, Theorem 3.1 implies that a bound on ‖ R ‖ ( n − 1 ) 2 ( 4 n 2 − 4 ) n gives a bound on ‖ n + Δ φ ‖ 4 n 2 − 4 . Therefore, we can show that C in the statement of Theorem 4.1 depends on n , ω , ‖ φ ‖ ∞ , ‖ F ‖ ∞ , ‖ R ‖ p n , where p n = 4 ( n − 1 ) 3 ( n + 1 ) n .

One might hope to improve the estimate by lowering p n . However, we have not been able to improve the bound yet.

Now, the proof of Theorem 1.3 is straightforward.

Proof of Theorem 1.3

Suppose that φ satisfies equation (1). Then, Theorems 1.2, 3.1, and 4.1 imply that there exists p n such that

‖ n + Δ φ ‖ ∞ ≤ C = C ( n , ω , η , ‖ R ‖ p n ) .

This implies that eigenvalues of ω φ = ω + − 1 ∂ ∂ ¯ φ are bounded from above by C . On the other hand, by Theorem 1.2, ‖ F ‖ ∞ ≤ C . Therefore, eigenvalues of ω φ = ω + − 1 ∂ ∂ ¯ φ are bounded below by a positive constant that only depends on n , ω , η , ‖ R ‖ p n . Hence, the equation

Δ ω φ F = − R + tr ω φ η

is uniformly elliptic. Therefore, DeGiorgi-Nash-Moser theorem implies that there exists α ∈ ( 0 , 1 ) such that ‖ F ‖ C α ≤ C . This together with the C 2 bound on φ , we obtain that φ is bounded in C 2 , α [11].

Hence, the Carlderon-Zygmond estimate implies that F is bounded in W 2 , p n . Now differentiating the Monge-Ampere equation implies that φ is bounded in W 4 , p n . □

Acknowledgements

The authors would like to express his deep appreciation to Professor Yusuke Sakane for valuable advice. The authors also thank Professor Hiroshi Sawai for comments. The authors appreciates very much the referee’s comments that lead to many improvements in this article.

Funding information: Zhiqin Lu is partially supported by the DMS-19-08513.
Author contribution: The author have accepted responsibility for the entire content of this manuscript and consented to its submission to the journal, reviewed all the results and approved the final version of the manuscript. ZL and RS each contributed 50% of the manuscript.
Conflict of interest: The authors have no competing interests to declare that are relevant to the content of this article.
Data availability statement: All data generated or analysed during this study are included in this published article.

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Received: 2024-03-19

Revised: 2024-05-15

Accepted: 2024-05-28

Published Online: 2024-07-18

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https://doi.org/10.1515/coma-2024-0004

Keywords for this article

Calabi flow; Chen-Cheng estimate; scalar curvature

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