CUBO, A Mathematical Journal

Vol. 24, no. 02, pp. 239–261, August 2022

DOI: 10.56754/0719-0646.2402.0239

On an a priori L∞ estimate for a class of
Monge-Ampère type equations on compact
almost Hermitian manifolds

Masaya Kawamura
1, B

1Department of General Education,

National Institute of Technology, Kagawa

College 355, Chokushi-cho, Takamatsu,

Kagawa, Japan, 761-8058.

kawamura-m@t.kagawa-nct.ac.jp B

ABSTRACT

We investigate Monge-Ampère type equations on almost

Hermitian manifolds and show an a priori L∞ estimate for

a smooth solution of these equations.

RESUMEN

Investigamos ecuaciones de tipo Monge-Ampère en va-

riedades casi Hermitianas y mostramos una estimación L∞

a priori para una solución suave de estas ecuaciones.

Keywords and Phrases: Monge-Ampère type equation, almost Hermitian manifold, Chern connection.

2020 AMS Mathematics Subject Classification: 32Q60, 53C15, 53C55.

Accepted: 7 April, 2022

Received: 19 July, 2021

c©2022 M. Kawamura. This open access article is licensed under a Creative Commons

Attribution-NonCommercial 4.0 International License.

http://cubo.ufro.cl/
http://dx.doi.org/10.56754/0719-0646.2402.0239
mailto:kawamura-m@t.kagawa-nct.ac.jp
https://orcid.org/0000-0003-1303-4237
mailto:kawamura-m@t.kagawa-nct.ac.jp


240 M. Kawamura CUBO
24, 2 (2022)

1 Introduction

Let (M2n,J,ω) be a compact almost Hermitian manifold of real dimension 2n with n ≥ 2. Let χ
be a smooth real (1,1)-form on M. We define for a function u ∈ C2(M),

χu := χ +
√
−1∂∂̄u

and

[χ] := {χu|u ∈ C2(M)}, [χ]+ := {χ′ ∈ [χ]|χ′ > 0}, H(M,χ) := {u ∈ C2(M)|χu > 0}

and

Cα(ψ) := {[χ]|∃χ′ ∈ [χ]+,nχ′n−1 > (n − α)ψχ′n−α−1 ∧ ωα}.

We consider the following fully nonlinear Monge-Ampère type equations, which are called the

(n,n − α)-quotient equations for 1 ≤ α ≤ n:

χnu = ψχ
n−α
u ∧ ωα with χu > 0, (1.1)

where ψ is a smooth positive function. We will call a function u ∈ C2(M) admissible if it satisfies
that u ∈ H(M,χ). When solutions u are admissible, the equations (1.1) are elliptic. Since the
equation (1.1) is invariant under the addition of constants to u, we may assume that u satisfies

the normalized condition such that

sup
M

u = 0. (1.2)

W. Sun has studied a class of fully nonlinear elliptic equations on closed Hermitian manifolds

and derived some a priori estimates for these equations (cf. [5, 6]). In [5], W. Sun has proven a

uniform a priori C∞ estimates of a smooth solution of the equation (1.1) and shown the existence

of a solution of (1.1) on a closed Hermitian manifold. In [12], J. Zhang has shown that on a

compact almost Hermitian manifold (M2n,J,ω), if there exists an admissible C-subsolution and an
admissible supersolution for the equation (1.1) for χ = ω, there exists a pair of (u,b) with b ∈ R
such that u ∈ H(M,ω), supM u = 0, ωnu = ebψωn−α ∧ ωα for 1 ≤ α ≤ n on M. L. Chen has
studied a Hessian equation with its structure as a combination of elementary symmetric functions

on a closed Kähler manifold and Chen has provided a sufficient and necessary condition for the

solvability of this equation in [1]. Q. Tu and N. Xiang have investigated the Dirichlet problem

for a class of Hessian type equation with its structure as a combination of elementary symmetric

functions on a closed Hermitian manifold with smooth boundary and they have derived a priori

estimates for the complex mixed Hessian equation in [9].

In this paper, we show that we have the a priori L∞ estimate for a smooth solution of the equation

(1.1) on general almost Hermitian manifolds.



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On an a priori L∞ estimate for a class of Monge-Ampère type... 241

Theorem 1.1. Let (M,J,ω) be a compact almost Hermitian manifold of real dimension 2n with

n ≥ 2 and u be a smooth admissible solution to (1.1). Suppose that χ ∈ Cα(ψ). Then there is a
uniform a priori L∞ estimate for u depending only on (M,J,ω), χ, ψ.

This paper is organized as follows: in section 2, we recall some basic definitions and computations

on an almost Hermitian manifold (M,J,ω). In section 3, for an arbitrary chosen smooth function

ϕ on M, we show the result that ∂∂∂̄ϕ and ∂̄∂∂̄ϕ depend only on the first derivative of ϕ and

some geometric quantities of (M,J,ω). In section 4, we give a proof for Theorem 1.1. Notice that

we assume the Einstein convention omitting the symbol of sum over repeated indexes in all this

paper.

2 Preliminaries

2.1 The Nijenhuis tensor of the almost complex structure

Let M be a 2n-dimensional smooth differentiable manifold. An almost complex structure on M

is an endomorphism J of TM, J ∈ Γ(End(TM)), satisfying J2 = −IdT M , where TM is the real
tangent vector bundle of M. The pair (M,J) is called an almost complex manifold. Let (M,J)

be an almost complex manifold. We define a bilinear map on C∞(M) for X,Y ∈ Γ(TM) by

4N(X,Y ) := [JX,JY ] − J[JX,Y ] − J[X,JY ] − [X,Y ], (2.1)

which is the Nijenhuis tensor of J. The Nijenhuis tensor N satisfies N(X,Y ) = −N(Y,X),
N(JX,Y ) = −JN(X,Y ), N(X,JY ) = −JN(X,Y ), N(JX,JY ) = −N(X,Y ). For any (1,0)-
vector fields W and V , N(V,W) = −[V,W ](0,1), N(V,W̄) = N(V̄ ,W) = 0 and N(V̄ ,W̄) =
−[V̄ ,W̄ ](1,0) since we have 4N(V,W) = −2([V,W ] +

√
−1J[V,W ]), 4N(V̄ ,W̄) = −2([V̄ ,W̄] −

√
−1J[V̄ ,W̄ ]). An almost complex structure J is called integrable if N = 0 on M. Giving a

complex structure to a differentiable manifold M is equivalent to giving an integrable almost

complex structure to M (cf. [4]). A Riemannian metric g on M is called J-invariant if J is

compatible with g, i.e., for any X,Y ∈ Γ(TM), g(X,Y ) = g(JX,JY ). In this case, the pair (J,g)
is called an almost Hermitian structure.

The complexified tangent vector bundle is given by T CM = TM ⊗R C for the real tangent vector
bundle TM. By extending J C-linearly and g C-bilinearly to T CM, they are also defined on

T CM and we observe that the complexified tangent vector bundle T CM can be decomposed as

T CM = T 1,0M⊕T 0,1M, where T 1,0M, T 0,1M are the eigenspaces of J corresponding to eigenvalues
√
−1 and −

√
−1, respectively:

T
1,0
M = {X −

√
−1JX

∣∣X ∈ TM}, T 0,1M = {X +
√
−1JX

∣∣X ∈ TM}. (2.2)



242 M. Kawamura CUBO
24, 2 (2022)

Let ΛrM =
⊕

p+q=r Λ
p,qM for 0 ≤ r ≤ 2n denote the decomposition of complex differential

r-forms into (p,q)-forms, where Λp,qM = Λp(Λ1,0M) ⊗ Λq(Λ0,1M),

Λ1,0M = {η +
√
−1Jη

∣∣η ∈ Λ1M}, Λ0,1M = {η −
√
−1Jη

∣∣η ∈ Λ1M} (2.3)

and Λ1M denotes the dual of T CM.

Let {Zr} be a local (1,0)-frame on (M,J) with an almost Hermitian metric g and let {ζr} be a
local associated coframe with respect to {Zr}, i.e., ζi(Zj) = δij for i,j = 1, . . . ,n. Since g is almost
Hermitian, its components satisfy gij = gīj̄ = 0 and gij̄ = gj̄i = ḡīj. Using these local frame {Zr}
and coframe {ζr}, we have

N(Zī,Zj̄) = −[Zī,Zj̄](1,0) =: Nkīj̄Zk, N(Zi,Zj) = −[Zi,Zj]
(0,1) = Nk

īj̄
Zk̄,

and

N =
1

2
Nk

īj̄
Zk̄ ⊗ (ζi ∧ ζj) +

1

2
Nk

īj̄
Zk ⊗ (ζī ∧ ζj̄). (2.4)

Let (M,J,g) be an almost Hermitian manifold with dimR M = 2n. An affine connection D on

T CM is called almost Hermitian connection if Dg = DJ = 0. For the almost Hermitian connection,

we have the following Lemma (cf. [10, 13]).

Lemma 2.1. Let (M,J,g) be an almost Hermitian manifold with dimR M = 2n. Then for any

given vector valued (1,1)-form Θ = (Θi)1≤i≤n, there exists a unique almost Hermitian connection

∇ on (M,J,g) such that the (1,1)-part of the torsion is equal to the given Θ.

If the (1,1)-part of the torsion of an almost Hermitian connection vanishes everywhere, then the

connection is called the second canonical connection or the Chern connection. We will refer the

connection as the Chern connection and denote it by ∇. Now let ∇ be the Chern connection on
M. We denote the structure coefficients of Lie bracket by

[Zi,Zj] = B
r
ijZr + B

r̄
ijZr̄, [Zi,Zj̄] = B

r
ij̄
Zr + B

r̄
ij̄
Zr̄, [Zī,Zj̄] = B

r
īj̄
Zr + B

r̄
īj̄
Zr̄.

We have Bkij = −Bkji since [Zi,Zj] = −[Zj,Zi]. Notice that J is integrable if and only if the Br̄ij’s
vanish.

For any p-form ψ, there holds that

dψ(X1, . . . ,Xp+1) =

p+1∑

i=1

(−1)i+1Xi(ψ(X1, . . . ,X̂i, . . . ,Xp+1))

+
∑

i<j

(−1)i+jψ([Xi,Xj],X1, . . . ,X̂i, . . . ,X̂j, . . . ,Xp+1) (2.5)

for any vector fields X1, . . . ,Xp+1 on M (cf. [13]). We directly compute that

dζs = −1
2
Bsklζ

k ∧ ζl − Bs
kl̄
ζk ∧ ζ l̄ − 1

2
Bs

k̄l̄
ζk̄ ∧ ζ l̄. (2.6)



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On an a priori L∞ estimate for a class of Monge-Ampère type... 243

For any real (1,1)-form η =
√
−1ηij̄ζi ∧ ζj̄, we have

∂η =

√
−1
2

(
Zi(ηjk̄) − Zj(ηik̄) − Bsijηsk̄ − Bs̄ik̄ηjs̄ + B

s̄
jk̄
ηis̄

)
ζ
i ∧ ζj ∧ ζk̄, (2.7)

∂̄η =

√
−1
2

(
Zj̄(ηkī) − Zī(ηkj̄) − Bskīηsj̄ + B

s
kj̄
ηs̄i + B

s̄
īj̄
ηks̄

)
ζk ∧ ζī ∧ ζj̄. (2.8)

We can split the exterior differential operator d : ΛpM ⊗R C → Λp+1M ⊗R C, into four components

d = A + ∂ + ∂̄ + Ā

with

∂ : Λp,qM → Λp+1,qM, ∂̄ : Λp,qM → Λp,q+1M,

A : Λp,qM → Λp+2,q−1M, Ā : Λp,qM → Λp−1,q+2M.

In terms of these components, the condition d2 = 0 can be written as

A2 = 0, ∂A + A∂ = 0, ∂̄Ā + Ā∂̄ = 0, Ā2 = 0,

A∂̄ + ∂2 + ∂̄A = 0, AĀ + ∂∂̄ + ∂̄∂ + ĀA = 0, ∂Ā + ∂̄2 + Ā∂ = 0. (2.9)

A direct computation yields for any ϕ ∈ C∞(M,R),

√
−1∂∂̄ϕ = 1

2
(dJdϕ)(1,1) =

√
−1(ZiZj̄ − [Zi,Zj̄](0,1))(ϕ)ζi ∧ ζj̄, (2.10)

so we write locally

∂i∂j̄ϕ = (ZiZj̄ − [Zi,Zj̄](0,1))ϕ. (2.11)

2.2 The torsion and the curvature on almost complex manifolds

Since the Chern connection ∇ preserves J, we have

∇iZj := ∇ZiZj = ΓrijZr, ∇iZj̄ := ∇ZiZj̄ = Γr̄ij̄Zr̄,

where Γrij = g
rs̄Zi(gjs̄) − grs̄gjl̄Bl̄is̄. We can obtain that Γr̄ij̄ = B

r̄
ij̄

since the (1,1)-part of the

torsion of the Chern connection vanishes everywhere.

Note that the mixed derivatives ∇iZj̄ do not depend on g (cf. [10]). Let {γij} be the connection
form, which is defined by γij = Γ

i
sjζ

s + Γis̄jζ
s̄. The torsion T of the Chern connection ∇ is given

by T i = dζi − ζp ∧ γip, T ī = dζī − ζp̄ ∧ γīp̄, which has no (1,1)-part and the only non-vanishing
components are as follows:

T
s
ij = Γ

s
ij − Γsji − Bsij, T s̄ij = −Bs̄ij.



244 M. Kawamura CUBO
24, 2 (2022)

These tell us that T = (T i) splits into T = T ′ + T ′′, where T ′ ∈ Γ(Λ2,0M ⊗ T 1,0M), T ′′ ∈
Γ(Λ0,2M ⊗ T 1,0M).

We denote by Ω the curvature of the Chern connection ∇. We can regard Ω as a section of
Λ2M ⊗Λ1,1M, Ω ∈ Γ(Λ2M ⊗Λ1,1M) and Ω splits in Ω = H+R+H̄, where R ∈ Γ(Λ1,1M ⊗Λ1,1M),
H ∈ Γ(Λ2,0M ⊗ Λ1,1M). The curvature form can be expressed by Ωij = dγij + γis ∧ γsj .

In terms of Zr’s, we have

R r
ij̄k

= Ωrk(Zi,Zj̄) = Zi(Γ
r
j̄k
) − Zj̄(Γrik) + ΓrisΓsj̄k − Γ

r
j̄s
Γsik − Bsij̄Γ

r
sk + B

s̄
j̄i
Γrs̄k = −R rj̄ik , (2.12)

H rijk = Ωrk(Zi,Zj) = Zi(Γrjk) − Zj(Γrik) + ΓrisΓsjk − ΓrjsΓsik − BsijΓrsk − Bs̄ijΓrs̄k = −H rjik , (2.13)

H r
īj̄k

= Ωrk(Zī,Zj̄) = Zī(Γ
r
j̄k
) − Zj̄(Γrīk) + Γ

r
īs
Γs
j̄k

− Γr
j̄s
Γs
īk
− Bs

īj̄
Γrsk − Bs̄īj̄Γ

r
s̄k = −H rj̄īk . (2.14)

Lemma 2.2 (The first Bianchi identity for the Chern curvature). For any X,Y,Z ∈ T CM,
∑

Ω(X,Y )Z =
∑ (

T(T(X,Y ),Z) + ∇XT(Y,Z)
)
,

where the sum is taken over all cyclic permutations.

This identity induces the following formulae:

R l
ij̄k

= R l
kj̄i

− T r̄ikT lr̄j̄ + ∇j̄T
l
ki = R lkj̄i − B

r̄
ikB

l
r̄j̄

+ ∇j̄T lki, (2.15)

H lijk = T r̄jiT k̄r̄l̄ + ∇l̄T
k̄
ji = −Br̄jiT k̄r̄l̄ + ∇l̄T

k̄
ji, (2.16)

where used that Rijk̄l̄ = Rīj̄kl = Hjl̄ik = Hj̄l̄ik̄ = Hl̄ijk = Hl̄ij̄k̄ = 0.

Let {Zr} be a local unitary (1,0)-frame with respect to g around a fixed point p ∈ M. Note
that unitary frames always exist locally since we can take any frame and apply the Gram-Schmidt

process. Then with respect to a local g-unitary frame, we have gij̄ = δij for any i,j,k = 1, . . . ,n,

and the Christoffel symbols satisfy

Γkij = −Γ
j̄

ik̄
, Γk̄

īj̄
= −Γj

īk
,

since we have

Γkij = g(∇iZj,Zk̄) = Zi(gjk̄) − g(Zj,∇iZk̄) = −Γ
j̄

ik̄
,

Γk̄
īj̄
= g(Zk,∇īZj̄) = Zī(gkj̄) − g(∇īZk,Zj̄) = −Γ

j

īk
.

And also we have

R r
ij̄k

= Zi(Γ
r
j̄k
) − Zj̄(Γrik) + ΓrisΓsj̄k − Γ

r
j̄s
Γsik − Bsij̄Γ

r
sk + B

s̄
j̄i
Γrs̄k

= −Zi(Γk̄j̄r̄) + Zj̄(Γ
k̄
ir̄) + Γ

s̄
ir̄Γ

k̄
j̄s̄

− Γs̄
j̄r̄
Γk̄is̄ + B

s
ij̄
Γk̄sr̄ − Bs̄j̄iΓ

k̄
s̄r̄

= −R k̄
ij̄r̄

, (2.17)



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On an a priori L∞ estimate for a class of Monge-Ampère type... 245

H rijk = Zi(Γrjk) − Zj(Γrik) + ΓrisΓsjk − ΓrjsΓsik − BsijΓrsk − Bs̄ijΓrs̄k
= −Zi(Γk̄jr̄) − Zj(Γk̄ir̄) + Γs̄ir̄Γk̄js̄ − Γs̄jr̄Γk̄is̄ + BsijΓk̄sr̄ + Bs̄ijΓk̄s̄r̄
= −H k̄ijr̄ (2.18)

and

R r
ij̄k

= Zī(Γ
r̄
jk̄
) − Zj(Γr̄īk̄) + Γ

r̄
īs̄
Γs̄
jk̄

− Γr̄js̄Γs̄īk̄ − B
s̄
īj
Γr̄
s̄k̄

+ Bs
jī
Γr̄
sk̄

= Zj(Γ
k
īr
) − Zī(Γkjr) + ΓsīrΓ

k
js − ΓsjrΓkīs − B

s
jī
Γksr + B

s̄
īj
Γks̄r

= R k
jīr

, (2.19)

H r
ijk

= Zī(Γ
r̄
j̄k̄
) − Zj̄(Γr̄īk̄) + Γ

r̄
īs̄
Γs̄
j̄k̄

− Γr̄
j̄s̄
Γs̄
īk̄
− Bs̄

īj̄
Γr̄
s̄k̄

− Bs
īj̄
Γr̄
sk̄

= −Zī(Γkj̄r) + Zj̄(Γ
k
īr
) + Γs

īr
Γk
j̄s

− Γs
j̄r
Γk
īs
− Bs̄

j̄ī
Γks̄r − Bsj̄īΓ

k
sr

= H k
j̄īr

. (2.20)

Hence we obtain Rij̄kr̄ = −Rij̄r̄k, Hijkr̄ = −Hijr̄k and Rij̄kr̄ = Rjīrk̄, Hijkr̄ = Hj̄īrk̄ by using a
local unitary (1,0)-frame with respect to g.

3 Some results for a smooth function on almost Hermitian

manifolds

Let (M,J,g) be an almost Hermitian manifold. Here note that B
q̄

jb̄
, B

q

j̄b
’s do not depend on the

metric g, which depend only on the almost complex structure J since the mixed derivatives ∇jZb̄,
∇j̄Zb do not depend on g. Since we have B

q

bj̄
= −Bq

j̄b
, we have that B

q

bj̄
, B

q̄

b̄j
’s also do not depend

on g (cf. [10]). Also note that Bs̄ri, B
s
r̄ī

do not depend on g, depend only on J. We can choose

a local unitary frame {Zr} around an arbitrary chosen point p0 ∈ M such that gij̄(p0) = δij and
∇Z(p0) = 0 (cf. [11]). Then we have Γkij(p0) = 0 since ∇iZj(p0) = Γkij(p0)Zk = 0, also we obtain
that

[Zi,Zj̄](p0) = ∇iZj(p0) − ∇jZi(p0) − T(Zi,Zj̄)(p0) = 0 for all i,j = 1, . . . ,n. (3.1)

Then we have that 0 = [Zi,Zj̄](p0) = B
k
ij̄
(p0)Zk + B

k̄
ij̄
(p0)Zk̄, which gives that B

k
ij̄
(p0) = 0 for all

i,j,k = 1, . . . ,n and that Bk̄
ij̄
(p0) = 0 for all i,j,k = 1, . . . ,n. By choosing such a local unitary

frame around a point p0, we have that the torsion tensor T
′ satisfies that T kij(p0) = −Bkij(p0) for

all i,j,k = 1, . . . ,n, and for instance from the formula (2.11), we have that ϕij̄(p0) = ∂i∂j̄ϕ(p0) =

ZiZj̄(ϕ)(p0) = Zj̄Zi(ϕ)(p0) = ϕj̄i(p0) for a smooth real-valued function ϕ. We show the following

critical lemma for proving the main result. We choose and fix a local unitary frame {Zr} around
an arbitrary chosen point p0 ∈ M such that gij̄(p0) = δij and ∇Z(p0) = 0. Our computations will
be done at the point p0.



246 M. Kawamura CUBO
24, 2 (2022)

We introduce some results for a smooth function on almost Hermitian manifolds. We write that

ϕs := ∇sϕ = ∂ϕ(Zs) = Zs(ϕ).

Lemma 3.1. One has for a smooth real-valued function ϕ on M,

∂∂∂̄ϕ(Zk,Zj,Zī) = ∂̄(B
s̄
kj)(Zī)∂̄ϕ(Zs̄). (3.2)

Proof. We compute that from (2.7),

∂∂∂̄ϕ(Zk,Zj,Zī) = Zk(ϕjī) − Zj(ϕkī) − Bskjϕs̄i − Bs̄kīϕjs̄ + B
s̄
jī
ϕks̄

= Zk(ZjZī(ϕ) − Bs̄jīϕs̄) − Zj(ZkZī(ϕ) − B
s̄
kī
ϕs̄) − Bskj(ZsZī(ϕ) − Br̄s̄iϕr̄)

= ZkZjZī(ϕ) − ZjZkZī(ϕ) − BskjZsZī(ϕ) − Zk(Bs̄jī)ϕs̄ + Zj(B
s̄
kī
)ϕs̄

= [Zk,Zj]Zī(ϕ) − BskjZsZī(ϕ) − Zk(Bs̄jī)ϕs̄ + Zj(B
r̄
s̄i
)ϕr̄

= Bs̄kjZs̄Zī(ϕ) − Zk(Bs̄jī)ϕs̄ + Zj(B
s̄
kī
)ϕs̄

= Bs̄kj[Zs̄,Zī](ϕ) + B
s̄
kjZīZs̄(ϕ) −

{
Zk(Γ

s̄
jī
) − Zj(Γs̄kī)

}
ϕs̄

= Bs̄kjB
r
s̄ī
ϕr + B

s̄
kjB

r̄
s̄̄i
ϕr̄ + B

s̄
kjZīZs̄(ϕ) − H sk̄j̄i ϕs, (3.3)

where we have used that Γk̄
ij̄
(p0) = B

k̄
ij̄
(p0) = 0, Γ

k
īj
(p0) = B

k
īj
(p0) = 0, Γ

k
ij(p0) = 0 for all

i,j,k = 1, . . . ,n, and that from (2.14),

H s
k̄j̄i

(p0) =
{
Zk̄(Γ

s
j̄i
) − Zj̄(Γsk̄i) + Γ

s
k̄r
Γr
j̄i
− Γs

j̄r
Γr
k̄i
− Br

k̄j̄
Γsri − Br̄k̄j̄Γ

s
r̄i

}
(p0)

= Zk̄(Γ
s
j̄i
)(p0) − Zj̄(Γsk̄i)(p0).

We compute that

Bs̄kjZīZs̄(ϕ) = Zī(B
s̄
kjZs̄(ϕ)) − Zī(Bs̄kj)Zs̄(ϕ)

= Zī(∂
2ϕ(Zk,Zj)) − Zī(Bs̄kj)∂̄ϕ(Zs̄)

= ∂̄∂2ϕ(Zī,Zk,Zj) − Zī(Bs̄kj)∂̄ϕ(Zs̄), (3.4)

where we used that

∂
2
ϕ(Zk,Zj) = ZkZj(ϕ) − ZjZk(ϕ) − BskjZs(ϕ)

= [Zk,Zj](ϕ) − BskjZs(ϕ)

= Bs̄kjZs̄(ϕ), (3.5)

∂̄∂
2
ϕ(Zī,Zk,Zj) = Zī(∂

2
ϕ(Zk,Zj)) − ∂2ϕ([Zī,Zk],Zj) + ∂2ϕ([Zī,Zj],Zk)

= Zī(∂
2ϕ(Zk,Zj)) − Br̄sjBsīkϕr̄ + B

r̄
skB

s
īj
ϕr̄

= Zī(∂
2
ϕ(Zk,Zj)).

By combining (3.3) with (3.4), we obtain

∂∂∂̄ϕ(Zk,Zj,Zī) = ∂̄∂
2ϕ(Zī,Zk,Zj) + B

s̄
kjB

r
s̄ī
∂ϕ(Zr) +

{
Br̄kjB

s̄
r̄ī
− Zī(Bs̄kj) − H sk̄j̄i

}
∂̄ϕ(Zs̄)

= ∂̄∂2ϕ(Zī,Zk,Zj) + B
s̄
kjB

r
s̄ī
∂ϕ(Zr),



CUBO
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On an a priori L∞ estimate for a class of Monge-Ampère type... 247

where we have used that from (2.16) and (2.20),

H s
k̄j̄i

= H ijks
= −Br̄kjT s̄r̄ī + ∇īT

s̄
kj

= Br̄kjB
s̄
r̄ī
− Zī(Bs̄kj). (3.6)

We compute by using (3.5),

∂∂∂̄ϕ(Zk,Zj,Zī) = ∂̄∂
2
ϕ(Zī,Zk,Zj) + B

s̄
kjB

r
s̄̄i
∂ϕ(Zr)

= ∂̄(Bs̄kj∂̄ϕ(Zs̄))(Zī) + T
s̄
kjT

r
s̄̄i
∂ϕ(Zr)

= ∂̄(Bs̄kj)(Zī)∂̄ϕ(Zs̄) + B
s̄
kj∂̄

2ϕ(Zī,Zs̄) + T
s̄
kjT

r
s̄̄i
∂ϕ(Zr)

= ∂̄(Bs̄kj)(Zī)∂̄ϕ(Zs̄) − T s̄kjBrīs̄∂ϕ(Zr) + T
s̄
kjT

r
s̄̄i
∂ϕ(Zr)

= ∂̄(Bs̄kj)(Zī)∂̄ϕ(Zs̄),

where we have used that Br
īs̄
= −T r

īs̄
= T r

s̄̄i
.

Lemma 3.2. One has for a smooth real-valued function ϕ on M,

∂̄∂∂̄ϕ(Zk̄,Zi,Zj̄) = ∂(B
s
k̄j̄
)(Zi)∂ϕ(Zs). (3.7)

Proof. We compute that from (2.8), using Bs
ij̄
(p0) = 0 for all i,j,s = 1, . . . ,n, B

r̄
is̄(p0) = 0 for all

i,r,s = 1, . . . ,n and [Zk̄,Zi](p0) = 0 for all i,k = 1, . . . ,n,

∂̄∂∂̄ϕ(Zk̄,Zi,Zj̄) = Zk̄(ϕij̄) − Zj̄(ϕik̄) − Bsij̄ϕsk̄ + B
s
ik̄
ϕsj̄ + B

s̄
j̄k̄
ϕis̄

= Zk̄(ZiZj̄(ϕ) − Bs̄ij̄ϕs̄) − Zj̄(ZiZk̄(ϕ) − B
s̄
ik̄
ϕs̄) + B

s̄
j̄k̄
ϕis̄

= Zk̄ZiZj̄(ϕ) − Zj̄ZiZk̄(ϕ) + Bs̄j̄k̄(ZiZs̄(ϕ) − B
r̄
is̄ϕr̄)

−Zk̄(Bs̄ij̄)ϕs̄ + Zj̄(B
s̄
ik̄
)ϕs̄

= ZiZk̄Zj̄(ϕ) + [Zk̄,Zi]Zj̄(ϕ) − ZiZj̄Zk̄(ϕ) − [Zj̄,Zi]Zk̄(ϕ)

+Bs̄
j̄k̄
ZiZs̄(ϕ) − Zk̄(Bs̄ij̄)ϕs̄ + Zj̄(B

s̄
ik̄
)ϕs̄

= Zi[Zk̄,Zj̄](ϕ) − Bs̄k̄j̄ZiZs̄(ϕ) − Zk̄(Γ
s̄
ij̄
)ϕs̄ + Zj̄(Γ

s̄
ik̄
)ϕs̄

= Bs
k̄j̄
ZiZs(ϕ) + Zi(B

s
k̄j̄
)ϕs + Zi(B

s̄
k̄j̄
)ϕs̄ − Zk̄(Γs̄ij̄)ϕs̄ + Zj̄(Γ

s̄
ik̄
)ϕs̄

= Bs
k̄j̄
ZiZs(ϕ) − Zi(T sk̄j̄)ϕs −

{
Zi(Γ

s̄
k̄j̄
) − Zi(Γs̄j̄k̄) − Zi(B

s̄
k̄j̄
)
}
ϕs̄

−
{
Zk̄(Γ

s̄
ij̄
) − Zi(Γs̄k̄j̄)

}
ϕs̄ +

{
Zj̄(Γ

s̄
ik̄
) − Zi(Γs̄j̄k̄)

}
ϕs̄

= Bs
k̄j̄
ZiZs(ϕ) − Zi(T sk̄j̄)ϕs − Zi(T

s̄
k̄j̄
)ϕs̄ − R skīj ϕs̄ + R

s
jīk

ϕs̄, (3.8)

where we have used that Bs̄
k̄j̄

= −Bs̄
j̄k̄

and that

ZiZk̄Zj̄(ϕ) − ZiZj̄Zk̄(ϕ) = Zi[Zk̄,Zj̄](ϕ)

= Zi(B
s
k̄j̄
Zs + B

s̄
k̄j̄
Zs̄)(ϕ)

= Zi(B
s
k̄j̄
)ϕs + B

s
k̄j̄
ZiZs(ϕ) + Zi(B

s̄
k̄j̄
)ϕs̄ + B

s̄
k̄j̄
ZiZs̄(ϕ),



248 M. Kawamura CUBO
24, 2 (2022)

and from (2.12),

R s
kīj

(p0) =
{
Zk(Γ

s
īj
) − Zī(Γskj) + ΓskrΓrīj − Γ

s
īr
Γrkj − BrkīΓ

s
rj + B

r̄
īk
Γsr̄j

}
(p0)

= Zk(Γ
s
īj
)(p0) − Zī(Γskj)(p0).

We compute that

Bs
k̄j̄
ZiZs(ϕ) = Zi(B

s
k̄j̄
Zs(ϕ)) − Zi(Bsk̄j̄)Zs(ϕ)

= Zi(∂̄
2ϕ(Zk̄,Zj̄)) − Zi(Bsk̄j̄)∂ϕ(Zs)

= ∂∂̄2ϕ(Zi,Zk̄,Zj̄) + Zi(T
s
k̄j̄
)∂ϕ(Zs), (3.9)

where we used that

∂̄
2
ϕ(Zk̄,Zj̄) = Zk̄Zj̄(ϕ) − Zj̄Zk̄(ϕ) − Bs̄k̄j̄Zs̄(ϕ)

= [Zk̄,Zj̄](ϕ) − Bs̄k̄j̄Zs̄(ϕ)

= Bs
k̄j̄
Zs(ϕ), (3.10)

∂∂̄2ϕ(Zi,Zk̄,Zj̄) = Zi(∂̄
2ϕ(Zk̄,Zj̄)) − ∂̄2ϕ([Zi,Zk̄],Zj̄) + ∂̄2ϕ([Zi,Zj̄],Zk̄)

= Zi(∂
2ϕ(Zk̄,Zj̄)) − Brs̄j̄B

s̄
ik̄
ϕr + B

r
s̄k̄
Bs̄

ij̄
ϕr

= Zi(∂
2
ϕ(Zk̄,Zj̄)).

Combining (3.8) with (3.9), we obtain that

∂̄∂∂̄ϕ(Zk̄,Zi,Zj̄) = ∂∂̄
2
ϕ(Zi,Zk̄,Zj̄) +

{
R s

jīk
− R s

kīj
− Zi(T s̄k̄j̄)

}
∂̄ϕ(Zs̄)

= ∂∂̄2ϕ(Zi,Zk̄,Zj̄) + T
r
k̄j̄
T s̄ri∂̄ϕ(Zs̄),

where we have used that from (2.15),

R s
jīk

− R s
kīj

= −Br
j̄k̄
Bs̄ri + ∇īT skj

= T r
k̄j̄
T s̄ri + Zi(T

s̄
k̄j̄
).

We compute that by applying (3.5) and (3.10),

∂̄∂∂̄ϕ(Zk̄,Zi,Zj̄) = ∂∂̄
2ϕ(Zi,Zk̄,Zj̄) + T

r
k̄j̄
T s̄ri∂̄ϕ(Zs̄)

= ∂(Bs
k̄j̄
∂ϕ(Zs))(Zi) + T

r
k̄j̄
T

s̄
ri∂̄ϕ(Zs̄)

= ∂(Bs
k̄j̄
)(Zi)∂ϕ(Zs) + B

r
k̄j̄
∂2ϕ(Zi,Zr) + T

r
k̄j̄
T s̄ri∂̄ϕ(Zs̄)

= ∂(Bs
k̄j̄
)(Zi)∂ϕ(Zs) − T rk̄j̄B

s̄
ir∂̄ϕ(Zs̄) + T

r
k̄j̄
T s̄ri∂̄ϕ(Zs̄)

= ∂(Bs
k̄j̄
)(Zi)∂ϕ(Zs),

where we have used that Bs̄ir = −T s̄ir = T s̄ri.



CUBO
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On an a priori L∞ estimate for a class of Monge-Ampère type... 249

Lemma 3.3. One has for a smooth real-valued function ϕ on M,

∂∂̄∂∂̄ϕ(Zl,Zk̄,Zi,Zj̄) = −∂2(T sk̄j̄)(Zl,Zi)∂ϕ(Zs) + ∂(T
s
k̄j̄
)(Zi)T

r̄
ls∂̄ϕ(Zr̄). (3.11)

Proof. By applying (3.5) and (3.6), we have that

∂∂̄∂∂̄ϕ(Zl,Zk̄,Zi,Zj̄) = ∂
2(Bs

k̄j̄
)(Zl,Zi)∂ϕ(Zs) + ∂(B

s
k̄j̄
)(Zi)∂

2
ϕ(Zl,Zs)

= −∂2(T s
k̄j̄
)(Zl,Zi)∂ϕ(Zs) + ∂(T

s
k̄j̄
)(Zi)T

r̄
ls∂̄ϕ(Zr̄).

In order to avoid a notational quagmire, we adopt the following ∗-convention C1 ∗ C2 between two
geometric quantities C1 and C2 with respect to a metric g:

(1) Summation over pairs of maching upper and lower indices.

(2) Contraction on upper indices with respect to the metric.

(3) Contraction on lower indices with respect to the dual metrics.

Since the point p0 was chosen arbitrary, the computations in Lemma 3.1–3.3 hold globally on an

almost Hermitian manifold M for any real-valued smooth function ϕ, which implies that we can

write (3.2), (3.7), and (3.11) globally on M as follows:

∂∂̄∂∂̄ϕ =: T1 ∗ ∂ϕ + T2 ∗ ∂̄ϕ, ∂2∂̄ϕ =: T3 ∗ ∂̄ϕ, ∂̄∂∂̄ϕ =: T4 ∗ ∂ϕ. (3.12)

4 Proof of Theorem 1.1

Let (M2n,J,ω) be a compact almost Hermitian manifold of real dimension 2n with n ≥ 2 in
this whole section. Let u be a smooth solution of (1.1). As in [5], we let Sk(λ) denote the k-th

elementary symmetric polynomial of λ ∈ Rn:

Sk(λ) :=
∑

1≤i1<···<ik≤n

λi1 · · ·λik.

For a square matrix U, we define Sα(U) := Sα(λ(U)), where λ(U) denote the eigenvalues of the

matrix U. Locally, we can write the equation (1.1) in the following form:

Sn(χu)

Sn−α(χu)
=

ψ

Cαn
, (4.1)

where Cαn :=
n!

(n−α)!α!
. We need the following generalized Newton-MacLaurin inequality.

Lemma 4.1 (cf. [1, Proposition 3], [9, Proposition 2.1]). For λ ∈ Γk := {λ ∈ Rn : Sk(λ) > 0,
∀ 1 ≤ i ≤ k} and 0 ≤ l < k ≤ n, 0 ≤ s < r, r ≤ k, s ≤ l, we have

[ Sk(λ)
Ck

n

Sl(λ)
Cln

] 1
k−l

≤
[ Sr(λ)

Cr
n

Ss(λ)
Csn

] 1
r−s

. (4.2)



250 M. Kawamura CUBO
24, 2 (2022)

In this section, the positive constant C may be changed from line to line, but it depends on the

allowed data.

Proof of Theorem 1.1. It suffices to show the following key inequality:
∫

M

|∂e−
p

2
u|2gωn ≤ Cp

∫

M

e−puωn (4.3)

for p large enough.

Lemma 4.2. Let u be a smooth admissible solution to the Monge-Ampère type equation (1,1).

Then, there are uniform constants C, p0 such that for any p ≥ p0, we have the inequality (4.3).

Proof. Without loss of generality, we may assume that

nχn−1 > (n − α)ψχn−α−1 ∧ ωα, (4.4)

and there exist uniform positive constants λ,Λ > 0 such that

λω ≤ χ ≤ Λω. (4.5)

As the local expression (4.1):

χnu

χ
n−α
u ∧ ωα

= Cαn
Sn(χu)

Sn−α(χu)
= ψ,

we locally have that

C
α
n−1

Sn−1(χu)

Sn−α−1(χu)
=

χn−1u

χ
n−α−1
u ∧ ωα

and which implies that the following inequality

nχn−1u > (n − α)ψχn−α−1u ∧ ωα (4.6)

is equivalent to
Sn−1(χu)

Sn−α−1(χu)
>

Sn(χu)

Sn−α(χu)
(4.7)

since we have locally that

n − α
n

· ψ = n − α
n

· Cαn
Sn(χu)

Sn−α(χu)
= Cαn−1

Sn(χu)

Sn−α(χu)
.

Note that we may apply Lemma 4.1 to χu since χu > 0. Applying the inequality (4.2), we have

[ Sn(χu)
Cn

n

Sn−α(χu)

C
n−α
n

] 1
α

≤
[ Sn−1(χu)

C
n−1
n

Sn−α−1(χu)

C
n−α−1
n

] 1
α

,

which can be written by

Sn(χu)

Sn−α(χu)
≤ C

n−α−1
n

C
n−1
n C

n−α
n

Sn−1(χu)

Sn−α−1(χu)

=
n − α
n(α + 1)

Sn−1(χu)

Sn−α−1(χu)

<
Sn−1(χu)

Sn−α−1(χu)
,



CUBO
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On an a priori L∞ estimate for a class of Monge-Ampère type... 251

where we used that n−α
n(α+1)

< 1. Therefore, the inequality (4.7) holds and as a consequence, we

have the inequality (4.6).

We estimate that

I :=

∫

M

e
−pu((χnu − χn) − ψ(χn−αu ∧ ωα − χn−α ∧ ωα))

=

∫

M

e−pu
(

χnu

χ
n−α
u ∧ ωα

− χ
n

χn−α ∧ ωα
)
χn−α ∧ ωα

≤ C
∫

M

e−puωn. (4.8)

On the other hand, we have that by Stokes’ theorem,

I =

∫ 1

0

∫

M

e
−pu d

dt
(χntu − ψχn−αtu ∧ ωα)dt

=

∫ 1

0

∫

M

e
−pu

√
−1∂∂̄u ∧ (nχn−1tu − (n − α)ψχn−α−1tu ∧ ωα)dt

=

∫ 1

0

∫

M

d(e−pu
√
−1∂̄u ∧ (nχn−1tu − (n − α)ψχn−α−1tu ∧ ωα))dt

−
∫ 1

0

∫

M

√
−1∂e−pu ∧ ∂̄u ∧ (nχn−1tu − (n − α)ψχn−α−1tu ∧ ωα)dt

+

∫ 1

0

∫

M

√
−1e−pu∂̄u ∧ ∂(nχn−1tu − (n − α)ψχn−α−1tu ∧ ωα)dt

= p

∫ 1

0

∫

M

e
−pu

√
−1∂u ∧ ∂̄u ∧ (nχn−1tu − (n − α)ψχn−α−1tu ∧ ωα)dt

−1
p

∫ 1

0

∫

M

√
−1∂̄e−pu ∧ ∂(nχn−1tu − (n − α)ψχn−α−1tu ∧ ωα)dt

= p

∫ 1

0

∫

M

e−pu
√
−1∂u ∧ ∂̄u ∧ (nχn−1tu − (n − α)ψχn−α−1tu ∧ ωα)dt

−1
p

∫ 1

0

∫

M

d(
√
−1e−pu∂(nχn−1tu − (n − α)ψχn−α−1tu ∧ ωα))dt

+
1

p

∫ 1

0

∫

M

e
−pu

√
−1∂̄∂(nχn−1tu − (n − α)ψχn−α−1tu ∧ ωα))dt

= p

∫ 1

0

∫

M

e
−pu

√
−1∂u ∧ ∂̄u ∧ (nχn−1tu − (n − α)ψχn−α−1tu ∧ ωα)dt

−1
p

∫ 1

0

∫

M

e−pu
√
−1∂∂̄(nχn−1tu − (n − α)ψχn−α−1tu ∧ ωα)dt, (4.9)

where we have used that d = A + ∂ + ∂̄ + Ā,

∂̄(e−pu
√
−1∂̄u ∧ (nχn−1tu − (n − α)ψχn−α−1tu ∧ ωα)) = 0,

A(e−pu
√
−1∂̄u ∧ (nχn−1tu − (n − α)ψχn−α−1tu ∧ ωα)) = 0,

Ā(e−pu
√
−1∂̄u ∧ (nχn−1tu − (n − α)ψχn−α−1tu ∧ ωα)) = 0,

∂(
√
−1e−pu∂(nχn−1tu − (n − α)ψχn−α−1tu ∧ ωα)) = 0,



252 M. Kawamura CUBO
24, 2 (2022)

A(
√
−1e−pu∂(nχn−1tu − (n − α)ψχn−α−1tu ∧ ωα)) = 0,

Ā(
√
−1e−pu∂(nχn−1tu − (n − α)ψχn−α−1tu ∧ ωα)) = 0

and from (2.9),

∂̄∂(nχn−1tu − (n − α)ψχn−α−1tu ∧ ωα) = −(∂∂̄ + AĀ + ĀA)(nχn−1tu − (n − α)ψχn−α−1tu ∧ ωα)

= −∂∂̄(nχn−1tu − (n − α)ψχn−α−1tu ∧ ωα)

since we have

A(nχn−1tu − (n − α)ψχn−α−1tu ∧ ωα) = Ā(nχn−1tu − (n − α)ψχn−α−1tu ∧ ωα) = 0.

We compute that for 0 ≤ t ≤ 1,

−1
p

∫

M

e−pu
√
−1∂∂̄(nχn−1tu − (n − α)ψχn−α−1tu ∧ ωα)

= −1
p

∫

M

e−pu
√
−1∂

(
n(n − 1)χn−2tu ∧ (∂̄χ +

√
−1t∂̄∂∂̄u) − (n − α)∂̄ψ ∧ χn−α−1tu ∧ ωα

−(n − α)(n − α − 1)ψχn−α−2tu ∧ (∂̄χ +
√
−1t∂̄∂∂̄u) ∧ ωα − α(n − α)ψχn−α−1tu ∧ ωα−1 ∧ ∂̄ω

)

= −1
p

∫

M

√
−1e−pu

{
n(n − 1)(n − 2)χn−3tu ∧ (∂χ + t

√
−1∂∂∂̄u) ∧ (∂̄χ + t

√
−1∂̄∂∂̄u)

+n(n − 1)χn−2tu ∧ (∂∂̄χ + t
√
−1∂∂̄∂∂̄u) − (n − α)∂∂̄ψ ∧ χn−α−1tu ∧ ωα

+(n − α)(n − α − 1)∂̄ψ ∧ χn−α−2tu ∧ (∂χ + t
√
−1∂∂∂̄u) ∧ ωα

+α(n − α)∂̄ψ ∧ χn−α−1tu ∧ ωα−1 ∧ ∂ω

−(n − α)(n − α − 1)∂ψ ∧ χn−α−2tu ∧ (∂̄χ + t
√
−1∂̄∂∂̄u) ∧ ωα

−(n − α)(n − α − 1)(n − α − 2)ψχn−α−3tu ∧ (∂χ + t
√
−1∂∂∂̄u) ∧ (∂̄χ + t

√
−1∂̄∂∂̄u) ∧ ωα

−(n − α)(n − α − 1)ψχn−α−2tu ∧ (∂∂̄χ + t
√
−1∂∂̄∂∂̄u) ∧ ωα

−α(n − α)(n − α − 1)ψχn−α−2tu ∧ (∂̄χ + t
√
−1∂̄∂∂̄u) ∧ ωα−1 ∧ ∂ω

−α(n − α)∂ψ ∧ χn−α−1tu ∧ ωα−1 ∧ ∂̄ω

−α(n − α)(n − α − 1)ψχn−α−2tu ∧ (∂χ + t
√
−1∂∂∂̄u) ∧ ωα−1 ∧ ∂̄ω

−α(n − α)(α − 1)ψχn−α−1tu ∧ ωα−2 ∧ ∂ω ∧ ∂̄ω

−α(n − α)ψχn−α−1tu ∧ ωα−1 ∧ ∂∂̄ω
}

≥ −C
p

∫

M

e−puχ
n−3
tu ∧ ω3 −

C

p

∫

M

e−pu
√
−1∂u ∧ ∂̄u ∧ χn−3tu ∧ ω2 − C

∫

M

e−puχ
n−2
tu ∧ ω2

−C
p

∫

M

e−pu
√
−1∂u ∧ ∂̄u ∧ χn−2tu ∧ ω

−C
p

∫

M

e−puχ
n−α−1
tu ∧ ωα+1 −

C

p

∫

M

e−puχ
n−α−2
tu ∧ ωα+2

−C
p

∫

M

e
−pu

√
−1∂u ∧ ∂̄u ∧ χn−α−2tu ∧ ωα+1 −

C

p

∫

M

e
−pu

χ
n−α−3
tu ∧ ωα+3

−
C

p

∫

M

e
−pu

√
−1∂u ∧ ∂̄u ∧ χn−α−3tu ∧ ωα+2, (4.10)



CUBO
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On an a priori L∞ estimate for a class of Monge-Ampère type... 253

where we have used that for instance, by applying (3.12),
∫

M

√
−1e−puχn−2tu ∧ t

√
−1∂∂̄∂∂̄u

=

∫

M

√
−1e−puχn−2tu ∧ t

√
−1(T1 ∗ ∂u + T2 ∗ ∂̄u)

≤ C
∫

M

e
−pu

χ
n−2
tu ∧

√
−1∂u ∧ ∂̄u ∧ ω + C

∫

M

e
−pu

χ
n−2
tu ∧ ω2, (4.11)

∫

M

∂̄ψ ∧ χn−α−2tu ∧ t
√
−1∂∂∂̄u ∧ ωα

=

∫

M

∂̄ψ ∧ χn−α−2tu ∧ t
√
−1T3 ∗ ∂̄u ∧ ωα

≤ C
∫

M

χ
n−α−2
tu ∧

√
−1∂u ∧ ∂̄u ∧ ωα+1 + C

∫

M

χ
n−α−2
tu ∧ ωα+2, (4.12)

∫

M

√
−1e−puχn−3tu ∧ ∂χ ∧ t

√
−1∂̄∂∂̄u ∧ ω

=

∫

M

√
−1e−puχn−3tu ∧ ∂χ ∧ t

√
−1T4 ∗ ∂u ∧ ω

≤ C
∫

M

e−puχ
n−3
tu ∧

√
−1∂u ∧ ∂̄u ∧ ω + C

∫

M

e−puχ
n−3
tu ∧ ω3. (4.13)

Since we have assumed that χ,χu > 0, then we have that χtu > 0 for any 0 ≤ t ≤ 1. Now we
introduce the following crucial inequalities (cf. [6]):

Lemma 4.3. For any 0 < t ≤ 1, 1 < l ≤ n, one has that
l

l − 1

∫ t

0

∫

M

e−pu
√
−1∂u∧∂̄u∧χl−1su ∧ωn−lds ≥ λ

∫ t

0

∫

M

e−pu
√
−1∂u∧∂̄u∧χl−2su ∧ωn−l+1ds, (4.14)

and for any 0 < t ≤ 1, 1 ≤ k ≤ n, one has that
k + 1

k

∫ t

0

χksu ∧ ωn−kds ≥ λ
∫ t

0

χk−1su ∧ ωn−k+1ds, (4.15)

where λ > 0 is the uniform constant in (4.5).

Proof. By using integration by parts and G̊arding’s inequality as in [6, (2.22)], we have that by

using χ ≥ λω,
∫ t

0

∫

M

e−pu
√
−1∂u ∧ ∂̄u ∧ χl−1su ∧ ωn−lds

=

∫ t

0

∫

M

e−pu
√
−1∂u ∧ ∂̄u ∧ χl−2su ∧ (χ + s

√
−1∂∂̄u) ∧ ωn−lds

≥ λ
∫ t

0

∫

M

e−pu
√
−1∂u ∧ ∂̄u ∧ χl−2su ∧ ωn−l+1ds

+
1

l − 1

∫ t

0

∫

M

e−pu
√
−1∂u ∧ ∂̄u ∧ s d

ds
χl−1su ∧ ωn−lds

≥ λ
∫ t

0

∫

M

e−pu
√
−1∂u ∧ ∂̄u ∧ χl−2su ∧ ωn−l+1ds

−
1

l − 1

∫ t

0

∫

M

e
−pu

√
−1∂u ∧ ∂̄u ∧ χl−1su ∧ ωn−lds, (4.16)



254 M. Kawamura CUBO
24, 2 (2022)

where we used that

∫ t

0

∫

M

e−pu
√
−1∂u ∧ ∂̄u ∧ s d

ds
χl−1su ∧ ωn−lds

= t

∫

M

e−pu
√
−1∂u ∧ ∂̄u ∧ χl−1tu ∧ ωn−l −

∫ t

0

∫

M

e−pu
√
−1∂u ∧ ∂̄u ∧ χl−1su ∧ ωn−lds

≥ −
∫ t

0

∫

M

e−pu
√
−1∂u ∧ ∂̄u ∧ χl−1su ∧ ωn−lds.

The inequality (4.16) gives the desired one (4.14). Next we compute that by using integration by

parts and G̊arding’s inequality as in [6, (3.7)], for 1 ≤ k ≤ n, using χ ≥ λω,
∫ t

0

χ
k
su ∧ ωn−kds =

∫ t

0

χ
k−1
su ∧ (χ + s

√
−1∂∂̄u) ∧ ωn−kds

≥ λ
∫ t

0

χ
k−1
su ∧ ωn−k+1ds +

1

k

∫ t

0

s
d

ds
(χksu ∧ ωn−k)ds

= λ

∫ t

0

χ
k−1
su ∧ ωn−k+1ds +

t

k
χ
k
tu ∧ ωn−k −

1

k

∫ t

0

χ
k
su ∧ ωn−kds

≥ λ
∫ t

0

χ
k−1
su ∧ ωn−k+1ds −

1

k

∫ t

0

χ
k
su ∧ ωn−kds,

which implies the inequality (4.15).

By applying these inequalities (4.14) and (4.15) for t = 1 to the estimate (4.10), we obtain that

−1
p

∫ 1

0

∫

M

e
−pu

√
−1∂∂̄(nχn−1tu − (n − α)ψχn−α−1tu ∧ ωα)dt

≥ −
C

p

∫ 1

0

∫

M

e
−pu

χ
n−1
tu ∧ ωdt −

C

p

∫ 1

0

∫

M

e
−pu

√
−1∂u ∧ ∂̄u ∧ χn−1tu dt. (4.17)

Combining (4.17) with (4.9), we have that

I ≥ p
∫ 1

0

∫

M

e−pu
√
−1∂u ∧ ∂̄u ∧

{(
n − C

p2

)
χ
n−1
tu − (n − α)ψχn−α−1tu ∧ ωα

}
dt

−C
p

∫ 1

0

∫

M

e
−pu

χ
n−1
tu ∧ ωdt. (4.18)

By the concavity of hyperbolic polynomials, for 0 < τ < 1, 1 ≤ k ≤ n, we have (cf. [6, (2.13)])

1

τ
S

1
k

k
(χτtu) +

(
1 −

1

τ

)
S

1
k

k
(χ) ≥ S

1
k

k
(χtu),

which gives

Sk(χτtu) ≥ τkSk(χtu).

For τ = 1
2
, k = n − 1, we obtain that

∫ 1

0

∫

M

e−puχ
n−1
tu ∧ ωdt ≤ 2n−1

∫ 1

0

∫

M

e−puχ
n−1
tu
2

∧ ωdt

= 2n
∫ 1

2

0

∫

M

e
−pu

χ
n−1
tu ∧ ωdt. (4.19)



CUBO
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On an a priori L∞ estimate for a class of Monge-Ampère type... 255

By combining (4.8), (4.18) and (4.19), we have that

p

∫ 1

0

∫

M

e−pu
√
−1∂u ∧ ∂̄u ∧

{(
n − C

p2

)
χ
n−1
tu − (n − α)ψχn−α−1tu ∧ ωα

}
dt

≤ C
∫

M

e
−pu

ω
n +

C

p

∫ 1
2

0

∫

M

e
−pu

χ
n−1
tu ∧ ωdt. (4.20)

Since we have χtu > 0 and

nχ
n−1
tu − (n − α)ψχn−α−1tu ∧ ωα > 0

for any 0 ≤ t ≤ 1, we can choose a sufficiently large p so that

nχ
n−1
tu − (n − α)ψχn−α−1tu ∧ ωα −

C

p2
χ
n−1
tu > 0.

Then we have that by the concavity of the quotient equation, for some 0 < δ < 1, we have (cf. [6,

(3.10)])

nχ
n−1
tu − (n − α)ψχn−α−1tu ∧ ωα > n

{
1 − 1

(1 + δ − tδ)α
}
χ
n−1
tu ,

hence for sufficiently large p,

∫ 1

0

∫

M

e
−pu

√
−1∂u ∧ ∂̄u ∧

{(
n − C

p2

)
χ
n−1
tu − (n − α)ψχn−α−1tu ∧ ωα

}
dt

≥
∫ 1

2

0

∫

M

e−pu
√
−1∂u ∧ ∂̄u ∧

{(
n − C

p2

)
χ
n−1
tu − (n − α)ψχn−α−1tu ∧ ωα

}
dt

≥
∫ 1

2

0

n
{
1 −

C

np2
−

1

(1 + δ − tδ)α
} ∫

M

e
−pu

√
−1∂u ∧ ∂̄u ∧ χn−1tu dt

≥ n
{
1 − C

np2
− 1

(1 + δ
2
)α

} ∫ 1
2

0

∫

M

e−pu
√
−1∂u ∧ ∂̄u ∧ χn−1tu dt. (4.21)

On the other hand, we compute by Stokes’ theorem,

1

p

∫ 1
2

0

∫

M

e
−pu

χ
n−1
tu ∧ ωdt

=
1

p

∫ 1
2

0

∫ t

0

d

ds

( ∫

M

e−puχn−1su ∧ ω
)
dsdt +

1

2p

∫

M

e−puχn−1 ∧ ω

=
n − 1
p

∫ 1
2

0

∫ t

0

∫

M

e−pu
√
−1∂∂̄u ∧ χn−2su ∧ ωdsdt +

1

2p

∫

M

e−puχn−1 ∧ ω

=
n − 1
p

∫ 1
2

0

∫ t

0

∫

M

d(e−pu
√
−1∂̄u ∧ χn−2su ∧ ω)dsdt

−n − 1
p

∫ 1
2

0

∫ t

0

∫

M

√
−1∂e−pu ∧ ∂̄u ∧ χn−2su ∧ ωdsdt

+
n − 1
p

∫ 1
2

0

∫ t

0

∫

M

e
−pu

√
−1∂̄u ∧ ∂(χn−2su ∧ ω)dsdt +

1

2p

∫

M

e
−pu

χ
n−1 ∧ ω



256 M. Kawamura CUBO
24, 2 (2022)

= (n − 1)
∫ 1

2

0

∫ t

0

∫

M

e−pu
√
−1∂u ∧ ∂̄u ∧ χn−2su ∧ ωdsdt

−n − 1
p2

∫ 1
2

0

∫ t

0

∫

M

√
−1∂̄e−pu ∧ ∂(χn−2su ∧ ω)dsdt +

1

2p

∫

M

e
−pu

χ
n−1 ∧ ω

= (n − 1)
∫ 1

2

0

∫ t

0

∫

M

e−pu
√
−1∂u ∧ ∂̄u ∧ χn−2su ∧ ωdsdt

−n − 1
p2

∫ 1
2

0

∫ t

0

∫

M

d(
√
−1e−pu∂(χn−2su ∧ ω))dsdt

+
n − 1
p2

∫ 1
2

0

∫ t

0

∫

M

e−pu
√
−1∂̄∂(χn−2su ∧ ω)dsdt +

1

2p

∫

M

e−puχn−1 ∧ ω

= (n − 1)
∫ 1

2

0

∫ t

0

∫

M

e−pu
√
−1∂u ∧ ∂̄u ∧ χn−2su ∧ ωdsdt

−n − 1
p2

∫ 1
2

0

∫ t

0

∫

M

e
−pu

√
−1∂∂̄(χn−2su ∧ ω)dsdt +

1

2p

∫

M

e
−pu

χ
n−1 ∧ ω, (4.22)

where we used that as in the computation in (4.9),

d(e−pu
√
−1∂̄u∧χn−2su ∧ω) = (∂+∂̄+A+Ā)(e−pu

√
−1∂̄u∧χn−2su ∧ω) = ∂(e−pu

√
−1∂̄u∧χn−2su ∧ω),

d(
√
−1e−pu ∧∂(χn−2su ∧ω)) = (∂+∂̄+A+Ā)(

√
−1e−pu∧∂(χn−2su ∧ω)) = ∂̄(

√
−1e−pu ∧∂(χn−2su ∧ω)),

and

∂̄∂(χn−2su ∧ ω) = −(∂∂̄ + AĀ + ĀA)(χn−2su ∧ ω) = −∂∂̄(χn−2su ∧ ω).

Applying (3.12), we estimate that as in (4.11)-(4.13) such as
∫

M

√
−1e−puχn−3su ∧ s

√
−1∂∂̄∂∂̄u ∧ ω

≤ C
∫

M

e−puχn−3su ∧
√
−1∂u ∧ ∂̄u ∧ ω2 + C

∫

M

e−puχn−3su ∧ ω3, (4.23)

∫

M

e−puχn−4su ∧ s
√
−1∂∂∂̄u ∧ ∂̄χ ∧ ω

≤ C
∫

M

e
−pu

χ
n−4
su ∧

√
−1∂u ∧ ∂̄u ∧ ω3 + C

∫

M

e
−pu

χ
n−4
su ∧ ω4, (4.24)

∫

M

e−puχn−4su ∧ ∂χ ∧ s
√
−1∂̄∂∂̄u ∧ ω2

≤ C
∫

M

e−puχn−4su ∧
√
−1∂u ∧ ∂̄u ∧ ω3 + C

∫

M

e−puχn−4su ∧ ω4. (4.25)

Then we estimate that by applying these estimates (4.23)-(4.25) and the inequalities (4.14)-(4.15),

n − 1
p2

∫ t

0

∫

M

e−pu
√
−1∂∂̄(χn−2su ∧ ω)ds

=
n − 1
p2

∫ t

0

∫

M

e−pu
√
−1∂((n − 2)χn−3su ∧ (∂̄χ + s

√
−1∂̄∂∂̄u) ∧ ω)ds

=
n − 1
p2

∫ t

0

∫

M

e−pu
√
−1

{
(n − 2)(n − 3)χn−4su ∧ (∂χ + s

√
−1∂∂∂̄u) ∧ (∂̄χ + s

√
−1∂̄∂∂̄u) ∧ ω

+(n − 2)χn−3su ∧ (∂∂̄χ + s
√
−1∂∂̄∂∂̄u) ∧ ω + (n − 2)χn−3su ∧ (∂̄χ + s

√
−1∂̄∂∂̄u) ∧ ∂ω



CUBO
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On an a priori L∞ estimate for a class of Monge-Ampère type... 257

+(n − 2)χn−3su ∧ (∂χ + s
√
−1∂∂∂̄u) ∧ ∂̄ω + χn−2su ∧ ∂∂̄ω

}
ds

≤
C

p2

∫ t

0

∫

M

e
−pu

χ
n−4
su ∧ ω4ds +

C

p2

∫ t

0

∫

M

e
−pu

χ
n−4
su ∧

√
−1∂u ∧ ∂̄u ∧ ω3ds

+
C

p2

∫ t

0

∫

M

e
−pu

χ
n−3
su ∧ ω3ds +

C

p2

∫ t

0

∫

M

e
−pu

χ
n−3
su ∧

√
−1∂u ∧ ∂̄u ∧ ω2ds

+
C

p2

∫ t

0

∫

M

e−puχn−2su ∧ ω2ds

≤ C1
p2

∫ t

0

∫

M

e−pu
√
−1∂u ∧ ∂̄u ∧ χn−2su ∧ ωds +

C2

p2

∫ t

0

∫

M

e−puχn−2su ∧ ω2ds. (4.26)

By choosing p sufficiently large such that C1
p2

< n − 1, C2
p
< λ · n−1

n
, by combining (4.22) with

(4.26), and applying (4.15) for t = 1
2
, k = n − 1 such that

∫ 1
2

0

∫

M

e
−pu

χ
n−2
tu ∧ ω2dt ≤

1

λ
· n
n − 1

∫ 1
2

0

∫

M

e
−pu

χ
n−1
tu ∧ ωdt,

we obtain that for 0 ≤ t ≤ 1
2
,

1

p

∫ 1
2

0

∫

M

e
−pu

χ
n−1
tu ∧ ωdt

≤ (n − 1)
∫ 1

2

0

∫ t

0

∫

M

e−pu
√
−1∂u ∧ ∂̄u ∧ χn−2su ∧ ωdsdt +

1

2p

∫

M

e−puχn−1 ∧ ω

+
C1

p2

∫ 1
2

0

∫ t

0

∫

M

e−pu
√
−1∂u ∧ ∂̄u ∧ χn−2su ∧ ωdsdt +

C2

p2

∫ 1
2

0

∫ t

0

∫

M

e−puχn−2su ∧ ω2dsdt

≤ (n − 1)
∫ 1

2

0

∫

M

e
−pu

√
−1∂u ∧ ∂̄u ∧ χn−2tu ∧ ωdt

+
1

2p
· λ(n − 1)

n

∫ 1
2

0

∫

M

e−puχ
n−2
tu ∧ ω2dt +

1

2p

∫

M

e−puχn−1 ∧ ω

≤ (n − 1)
∫ 1

2

0

∫

M

e
−pu

√
−1∂u ∧ ∂̄u ∧ χn−2tu ∧ ωdt

+
1

2p

∫ 1
2

0

∫

M

e−puχ
n−1
tu ∧ ωdt +

1

2p

∫

M

e−puχn−1 ∧ ω (4.27)

which implies that we have by applying (4.14) for t = 1
2
, l = n,

1

2p

∫ 1
2

0

∫

M

e
−pu

χ
n−1
tu ∧ ωdt

≤ (n − 1)
∫ 1

2

0

∫

M

e−pu
√
−1∂u ∧ ∂̄u ∧ χn−2tu ∧ ωdt +

1

2p

∫

M

e−puχn−1 ∧ ω

≤
n

λ

∫ 1
2

0

∫

M

e
−pu

√
−1∂u ∧ ∂̄u ∧ χn−1tu dt +

1

2p

∫

M

e
−pu

χ
n−1 ∧ ω. (4.28)

Therefore, by combining (4.28) with (4.20), (4.21), we obtain that

[
np

{
1 − C

np2
− 1

(1 + δ
2
)α

}
− C 2n

λ

] ∫ 1
2

0

∫

M

e−pu
√
−1∂u ∧ ∂̄u ∧ χn−1tu dt

≤ C
∫

M

e
−pu

ω
n +

C

p

∫

M

e
−pu

χ
n−1 ∧ ω ≤ C

∫

M

e
−pu

ω
n
. (4.29)



258 M. Kawamura CUBO
24, 2 (2022)

We choose p sufficiently large such that
[
n
{
1 − C

np2
− 1

(1 + δ
2
)α

}
− C 2n

λp

]
> 0.

By applying (4.14) for t = 1
2
repeatedly, we obtain

∫ 1
2

0

∫

M

e−pu
√
−1∂u ∧ ∂̄u ∧ χn−1tu dt ≥ λ

n − 1
n

∫ 1
2

0

∫

M

e−pu
√
−1∂u ∧ ∂̄u ∧ χn−2tu ∧ ωdt

≥ λ2 n − 1
n

n − 2
n − 1

∫ 1
2

0

∫

M

e−pu
√
−1∂u ∧ ∂̄u ∧ χn−3tu ∧ ω2dt

· · ·

≥ λ
n−1

n

1

2

∫

M

e−pu
√
−1∂u ∧ ∂̄u ∧ ωn−1. (4.30)

By combining (4.29) with (4.30), we finally obtain that for sufficiently large p,

p

∫

M

e−pu
√
−1∂u ∧ ∂̄u ∧ ωn−1 ≤ C

∫

M

e−puωn,

which tells us that there exists a sufficiently large p0 such that for all p ≥ p0, the desired inequality
(4.3) holds.

The rest of the proof is similar to the ones in [7, 8]. In the following, we give a brief proof for reader’s

convenience. We introduce the definition of Gauduchon metrics on almost complex manifolds.

Definition 4.4. Let (M2n,J) be an almost complex manifold. A metric g is called a Gauduchon

metric on M if g is an almost Hermitian metric whose associated real (1,1)-form ω =
√
−1gij̄ζi∧ζj̄

satisfies d∗(Jd∗ω) = 0, where d∗ is the adjoint of d with respect to g, which is equivalent to

d(Jd(ωn−1)) = 0, or ∂∂̄(ωn−1) = 0.

One has the following well-known result.

Proposition 4.5 (cf. [2, Theorem 2.1], [3]). Let (M2n,J,ω) be a compact almost Hermitian

manifold with n ≥ 2. Then there exists a smooth function σ, unique up to addition of a constant,
such that the conformal almost Hermitian metric eσω is Gauduchon.

Thanks to Proposition 4.5, there exists a smooth function σ : M → R with supM σ = 0 such that
ωG := e

σω is Gauduchon on M.

Lemma 4.6 (cf. [8, Lemma 2.3]). Let M be a compact almost complex manifold of real dimension

2n (n ≥ 2) with a Gauduchon metric ωG. If φ is a smooth nonnegative function on M with
∆Gφ ≥ −C0, where ∆G is the Laplacian operator with respect to ωG, then there exists a positive
constant C1, C2 depending only on (M,ωG) and C0 such that

∫

M

|∂φ
p+1
2 |2ωGω

n
G ≤ C1p

∫

M

φpωnG (4.31)

for any p ≥ 1, and
sup
M

φ ≤ C2 max
{ ∫

M

φωnG,1
}
. (4.32)



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On an a priori L∞ estimate for a class of Monge-Ampère type... 259

Proof. We compute for p ≥ 1, by Stokes’ theorem,
∫

M

|∂φ
p+1
2 |2ωGω

n
G = n

∫

M

√
−1∂φ

p+1
2 ∧ ∂̄φ

p+1
2 ∧ ωn−1G

=
n(p + 1)2

4

∫

M

√
−1φp−1∂φ ∧ ∂̄φ ∧ ωn−1

G

=
n(p + 1)2

4p

∫

M

√
−1∂(φp) ∧ ∂̄φ ∧ ωn−1G

=
n(p + 1)2

4p

∫

M

√
−1(∂ + ∂̄ + A + Ā)(φp∂̄φ ∧ ωn−1G )

−n(p + 1)
2

4p

∫

M

φp
√
−1∂∂̄φ ∧ ωn−1

G
+
n(p + 1)

4p

∫

M

√
−1∂̄(φp+1) ∧ ∂ωn−1

G

= −(p + 1)
2

4p

∫

M

φpn

√
−1∂∂̄φ ∧ ωn−1G

ωnG
ωnG

+
n(p + 1)

4p

∫

M

√
−1(∂ + ∂̄ + A + Ā)(φp+1∂ωn−1G )

−n(p + 1)
4p

∫

M

φp+1
√
−1∂̄∂ωn−1

G

=
(p + 1)2

4p

∫

M

φ
p(−∆Gφ)ωnG

≤ C1p
∫

M

φ
p
ω
n
G, (4.33)

where we used that (∂̄ + A + Ā)(φp∂̄φ ∧ ωn−1G ) = 0, (∂ + A + Ā)(φp+1∂ω
n−1
G ) = 0, and that

∂̄∂ω
n−1
G

= −(∂∂̄ + AĀ + ĀA)ωn−1
G

= −∂∂̄ωn−1
G

= 0

since we have Aωn−1
G

= Āωn−1
G

= 0.

We apply the Sobolev inequality: for β := n
n−1

> 1, and for any smooth function f,

( ∫

M

f
2β
ω
n
) 1

β ≤ C
( ∫

M

|∂f|2gωn +
∫

M

f
2
ω
n
)
. (4.34)

Taking ω = ωG and f = φ
q

2 , where we put q := p + 1, then for q ≥ 2, we have that
( ∫

M

φqβωnG

) 1
β ≤ Cq max

{ ∫

M

φqωnG,1
}
.

By repeatedly replacing q by qβ and iterating, after setting q = 2, then we obtain that

sup
M

φ ≤ C max
{( ∫

M

φ2ωnG

) 1
2

,1
}
≤ C max

{(
sup
M

φ
) 1

2
( ∫

M

φωnG

) 1
2

,1
}
,

which gives us the desired estimate (4.32).

By applying the inequality (4.3) and the Sobolev inequality (4.34), for any p ≥ p0, we obtain that

‖e−u‖Lpβ ≤ C
1
p p

1
p ‖e−u‖Lp,



260 M. Kawamura CUBO
24, 2 (2022)

and by the standard iteration, we have that

e−p0 infM u ≤ C
∫

M

e−p0uωn. (4.35)

We need the following lemma, whose proof goes in the same way as in the Hermitian case.

Lemma 4.7 (cf. [7, Lemma 3.2], [8, Lemma 2.2]). Let f be a smooth function on a compact almost

Hermitian manifold (M,J,ω). Write dµ := ω
n

∫
M

ωn
. If there exists a constant C1 such that

e− infM f ≤ eC1
∫

M

e−fdµ, (4.36)

then

|{f ≤ inf
M
f + C1 + 1}| ≥

e−C1

4
, (4.37)

where | · | denotes the volume of the set with respect to dµ.

We apply Lemma 4.6 to f = p0u, and then since we have the inequality (4.35), there exist uniform

constants C, δ > 0 such that

|{u ≤ inf
M
u + C}| ≥ δ. (4.38)

Now, we define φ := u − infM u. Since it satisfies that ∆Gφ = e−σ∆φ > −C, where ∆ is the
Laplacian operator with respect to ω, we may apply Lemma 4.3 to the function φ. From the

Poincaré inequality and the estimate (4.31) with p = 1, we obtain that

‖φ − φ‖L2 ≤ C
( ∫

M

|∂φ|2ωGω
n
G

) 1
2 ≤ C‖φ‖

1
2

L1
, (4.39)

where we put φ := 1∫
M

ωn
G

∫
M
φωnG.

By making use of (4.38), the set S := {φ ≤ C} satisfies that |S|G ≥ δ, where | · |G denotes the
volume of a set with respect to ωnG. Therefore, we obtain that

δφ ≤
∫

S

φωnG ≤
∫

S

(|φ − φ| + C)ωnG ≤
∫

M

|φ − φ|ωnG + C,

which gives that by applying (4.39),

‖φ‖L1 ≤ C(‖φ − φ‖L1 + 1) ≤ C(‖φ − φ‖L2 + 1) ≤ C(‖φ‖
1
2

L1
+ 1).

Hence, φ is uniformly bounded in L1, and from (4.32) and (1.2), we obtain a uniform bound of u

in the L∞ norm.

Acknowledgments

This work was supported by JSPS KAKENHI Grant Number JP21K13798.



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On an a priori L∞ estimate for a class of Monge-Ampère type... 261

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https://arxiv.org/pdf/2107.12035.pdf
https://arxiv.org/pdf/2201.05030.pdf
https://arxiv.org/pdf/2101.00380.pdf

	Introduction
	Preliminaries
	The Nijenhuis tensor of the almost complex structure
	The torsion and the curvature on almost complex manifolds

	Some results for a smooth function on almost Hermitian manifolds
	Proof of Theorem 1.1