THE DIRICHLET PROBLEM FOR COMPLEX MONGE … DIRICHLET PROBLEM FOR COMPLEX MONGE-AMPERE` EQUATIONS AND APPLICATIONS BO GUAN Abstract. We are concerned with the Dirichlet problem for

THE DIRICHLET PROBLEM FOR COMPLEX MONGE-AMPEREEQUATIONS AND APPLICATIONS

BO GUAN

Abstract. We are concerned with the Dirichlet problem for complex Monge-Ampere equations and their applications in complex geometry and analysis.2000 Mathematical Subject Classification: 35J65, 35J70, 53C21, 58J10, 58J32,32W20, 32U05, 32U35, 32Q15.Key words and phrases: Complex Monge-Ampere equations, Dirichlet problem,plurisubharmonic solutions, CKNS Theorem, a priori estimates, Kahler manifolds,Kahler potentials, Calabi-Yau Theorem, Donaldson conjecture, intrinsic norms,Chern-Levine-Nirenberg conjecture, pluricomplex Green functions.

1. Introduction

The complex Monge-Ampere equation is at the center of fully nonlinear elliptic

equation theory, and is of fundamental importance in complex geometry and analysis,

especially in Kahler geometry and pluripotential theory.

In a series of work, Bedford and Taylor (e.g. [4], [6]) established the foundation

of weak solutions based on the concept of positive currents and the Chern-Levine-

Nirenberg inequality [16]. The theory has developed enormously over the years with

contributions from many researchers; we refer the reader to, e.g., [31] and [32] for

background and references to the subject and pluripotential theory. In [48], S.-T.

Yau proved fundamental existence theorems of classical solutions to the complex

Monge-Ampere equation on compact Kahler manifolds, which are the centerpiece of

his proof of the Calabi conjecture.

The classical solvability of the Dirichlet problem was established by Caffarelli,

Kohn, Nirenberg and Spruck (CKNS) [13] for strongly pseudoconvex domains in

Cn. Later on the author [23] extended their results to general domains under the

assumption of existence of a C2 strictly plurisubharmonic subsolution.

The author was supported in part by NSF grants.Department of Mathematics, Ohio State University, Columbus, OH 43210. [email protected].

1

2 BO GUAN

In these notes we present the results of CKNS [13], with modifications by the

author [23] to general domains. As applications of the main theorem (Theorem 2.7)

we will discuss briefly some problems in complex analysis and geometry, most of which

involve solving the homogeneous complex Monge-Ampere equation in domains that

are not pseudoconvex. The notes are organized as follows.

In Section 2 we discuss briefly the basic properties of the complex Monge-Ampere

operator and the continuity method which reduces, at least in some important special

cases, the proof of existence results to deriving a priori estimates for the perspective

solutions. The materials in this section are all well known. At the end of the section

we state the theorem of CKNS (Theorem 2.7).

In Section 3 we present the C2 a priori estimates for plurisubharmonic solutions of

the Dirichlet problem in bounded smooth domains in Cn. This is roughly divided into

three steps: the gradient estimates, the boundary, and global estimates for second

derivatives. Most part of the proof in this section can be found in [13], [23] and [24]

(with minor modifications), except the estimate for double-normal derivative on the

boundary which was first given in a lecture by the author at NCTS, Taiwan, in 1999.

This section also contains some results that are needed in Section 6.

Section 4 concerns the complex Monge-Ampere equation on Kahler manifolds. This

is a broad subject with extensive research activities and far-reaching results. In this

section we restrict ourselves to a brief presentation of existence theorems of Yau [48]

(Theorems 4.1 and 4.3) and an analogue of Theorem 2.7 for the Dirichlet problem

(Theorem 4.2). As an application we discuss Donaldson’s conjectures [18] on geodesics

in the Mabuchi space of Kahler potentials and outline the proof of the existence of

(almost) C1,1 solutions in Chen [15].

In Section 5 we describe the intrinsic norms introduced by Chern, Levine and

Nirenberg [16] and the proof of the Chern-Levine-Nirenberg conjecture by Pengfei

Guan [27], [28]. As we only received [28] recently, we shall mainly focus on results

in [27]. Finally, in Section 6 we discuss the C1,1 regularity of pluricomplex Green

functions which are related to some special Dirichlet problems for the homogeneous

complex Monge-Ampere equation.

These notes are based in part on series of lectures that I gave for graduate students

at University of Science and Technology of China (UCTC) and Harbin Institute of

Technology (HIT) in the summer of 2006. I would like to thank Professors Xiuxiong

COMPLEX MONGE-AMPERE EQUATIONS 3

Chen, Xinan Ma, Yongqiang Fu, Boying Wu, and Departments of Mathematics of

both universities for their hospitality. Due to the limitation in space I have left out

a substantial part of materials that were covered in the lectures, and many more

which are important in the area and interesting to myself. (We should remark that

this article is not intended to be a survey.) I also wish to thank Joel Spruck for

stimulating discussions and his constant encouragement during the writing of these

notes, and over the years. Needless to say, I learned much of the subject from him.

(Of course I am solely responsible for any mistakes in these notes.) This article is

dedicated to Professor Guangchang Dong on the occasion of his 80th birthday, and

for his contributions to fully nonlinear partial differential equations.

2. The Dirichlet problem in Cn

The classical complex Monge-Ampere equation in Cn takes the form

(2.1) det∂2u

∂zj∂zk

= ψ.

This equation is elliptic for a C2 strictly plurisubharmonic solution u, i.e., the complex

Hessian matrix of u

HessC(u) =( ∂2u

∂zj∂zk

)is positive definite. In this case the function ψ on the right hand side of (2.1) is

necessarily positive; equation (2.1) becomes degenerate when ψ is only non-negative.

When ψ ≡ 0, (2.1) is referred as the homogeneous complex Monge-Ampere equation

which is well defined on complex manifolds and arises in many problems in complex

analysis and geometry.

We shall consider the Dirichlet problem for equation (2.1): Let Ω be a bounded

domain in Cn with smooth boundary ∂Ω, and ϕ ∈ C∞(∂Ω). We seek a plurisubhar-

monic solution u of equation (2.1) with boundary value u = ϕ on ∂Ω. The main goal

of this section is to explain the continuity method approach and how it reduces the

problem to a priori estimates for the perspective solutions.

4 BO GUAN

2.1. Some notation and basic properties. Let z = (z1, . . . , zn) be complex coor-

dinates in Cn, zj = xj +√−1yj. Recall that for a C2 function u defined on Cn,

(2.2)∂u

∂zj

=1

2

( ∂u

∂xj

−√−1

∂u

∂yj

),∂u

∂zj

=1

2

( ∂u

∂xj

+√−1

∂u

∂yj

).

When there is no confusion we shall write

uj = uzj=∂u

∂zj

uj = uzj=∂u

∂zj

ujk = uzj zk=

∂2u

∂zj∂zk

etc.

Note that

(2.3) uzj zk=

1

4

[(uxjxk

+ uyjyk) +

√−1(uxjyk

− uyjxk)].

Define

|∇u|2 =∑

j

|uj|2 =∑

j

ujuj =1

4

∑j

(u2xj

+ u2yj

),

∆u =∑

j

ujj =1

4

∑j

(uxjxj+ uyjyj

).

The following notation also appears in literature frequently:

∂u =∑

j

uzjdzj, ∂u =

∑j

uzjdzj,

du = ∂u+ ∂u, dcu =√−1(∂u− ∂u).

Therefore, d2 = ∂∂ + ∂∂ = ∂2 = ∂2 = 0 and

ddcu = 2√−1∂∂u = 2

√−1

∑j,k

uzj zkdzj ∧ dzk.

The complex Monge-Ampere operator (ddcu)n is defined as

(ddcu)n = 4nn! det(ujk) dV

where

dV =(√−1

2

)n

dz1 ∧ dz1 ∧ · · · ∧ dzn ∧ dzn

is the volume form of Cn; the definition of (ddcu)n extends to general locally bounded

plurisubharmonic functions ([6]).


We call the matrix

HessC(u) = (ujk)

the complex Hessian of u, while the real Hessian matrix of u is

HessR(u) =

[uxjxk

uxjyk

uyjxkuyjyk

].

Following the literature we shall use (ujk) to denote the inverse matrix of (ujk) when

it is invertible, i.e. ujkulk = δjl.

We next look at some important properties of the complex Monge-Ampere operator.

Lemma 2.1. Let z 7→ w be a holomorphic change of variables. Then

detuzj zk= | det(wz)|2 detuwjwk

.

It follows that the homogeneous complex Monge-Ampere equation

(2.4) det(uzj zk) = 0

is invariant under holomorphic changes of variables. This is one of the reasons the

equation is of interest in complex analysis and geometry. In particular, it is globally

well defined on complex manifolds.

Lemma 2.2. A differential operator u 7→ F (ujk) is elliptic (with respect to the real

variables x1, . . . , xn, y1, . . . , yn) if and only if the matrix( ∂F

∂uzj zk

)is positive definite. In particular, the complex Monge-Ampere operator u 7→ detujk is

elliptic in u if and only if the complex Hessian (ujk) is positive (or negative) definite.

This follows from the formula, by (2.3),

(2.5)

[∂F

∂uxjxk

∂F∂uxjyk

∂F∂uyjxk

∂F∂uyjyk

]=

1

4

∂F∂uzj zk

+ ∂F∂uzkzj

√−1

(∂F

∂uzj zk− ∂F

∂uzkzj

)−√−1

(∂F

∂uzj zk− ∂F

∂uzkzj

)∂F

∂uzj zk+ ∂F

∂uzkzj

.A function u in Cn is called (strictly) plurisubharmonic if it is (strictly) subharmonic

on every complex line (or equivalently, in each complex variable zj). A C2 function u is

strictly plurisubharmonic if and only if its complex Hessian (ujk) is positive definite.

We shall use PSH(Ω) to denote the collection of plurisubharmonic functions in a

domain Ω ⊂ Cn, while SPSH(Ω) denotes the collection of strictly plurisubharmonic

functions in Ω.

6 BO GUAN

Lemma 2.3. Let u be a strictly plurisubharmonic function and set F = log detujk.

Then

(2.6)

F jk :=∂F

∂uzj zk

= ujk,

F jk,pq :=∂2F

∂uzj zk∂uzpzq

= −ujqupk.

Consequently, log detujk is concave in ujk, i.e.

(2.7) F jk,pqξjkξpq = −ujqupkξjkξpq ≤ 0, ∀ ξjk ∈ Cn×n.

As we shall see, the concavity of log detujk is very important in the theory of

complex Monge-Ampere equations. Another fact that is often used is the concavity

of (detujk)1n . Indeed, for any increasing, concave function φ, φ((detujk)

1n ) is concave

for plurisubharmonic functions u.

We have the following maximum principle for complex Monge-Ampere operator;

see [13].

Lemma 2.4. Let Ω be a bounded domain. If u, v ∈ C2(Ω) are plurisubharmonic

functions with detujk ≥ det vjk in Ω and u ≤ v on ∂Ω, then u ≤ v in Ω.

2.2. The continuity method. Let Ω be a bounded domain in Cn with C∞ boundary

∂Ω, and consider the Dirichlet problem for complex Monge-Ampere equations

(2.8)detuzj zk

= ψ(z, u,Du) in Ω,

u = ϕ on ∂Ω

where ψ and ϕ are real-valued smooth functions, ψ > 0.

Suppose that there exists a strictly plurisubharmonic subsolution u ∈ C2,α(Ω) of

(2.8), that is,

(2.9)detuzj zk

≥ ψ(z, u,Du) in Ω,

u = ϕ on ∂Ω.

For 0 ≤ t ≤ 1, set

(2.10)detuzj zk

= ψt(z, u,Du) in Ω,

u = ϕ on ∂Ω

where

ψt(z, u,∇u) = tψ(z, u,Du) + (1− t) detuzj zk.


We wish to show that (2.10) has a strictly plurisubharmonic solution in C2,α(Ω) for

every 0 ≤ t ≤ 1. Define

S =t ∈ [0, 1] : (2.10) is solvable in C2,α(Ω) ∩ SPSH(Ω)

.

Obviously S is nonempty since 0 ∈ S with u being the unique solution of (2.10) when

t = 0. Our goal is to show that under suitable conditions S is both open and closed

in [0, 1], and therefore is equal to [0, 1]. This approach is known as the “continuity

method”.

The openness of S usually follows from the implicit function theorem for Banach

spaces, while the closedness reduces to a priori C2,α estimates.

Proposition 2.5 (Openness of S). Suppose ψu ≥ 0. Then S is open in [0, 1].

This follows from the implicit function theorem for Banach spaces and the Schauder

theory for linear elliptic equations.

Theorem 2.6 (Implicit Function Theorem). Let X, Y, Z be Banach spaces and

F : U × V → Z a continuous map and continuously differentiable with respect to

x ∈ X where U , V are open sets in X, Y respectively. Suppose that F (x0, y0) = 0 for

some x0 ∈ U , y0 ∈ V , and that the Frechet derivative

Fx(x0, y0) : X → Z

is an isomorphism. Then there exist balls Bε(x0) ⊂ U , Bδ(y0) ⊂ V and exactly one

map η : Bδ(y0) → Bε(x0) such that η(y0) = x0 and

F (η(y), y) = 0, ∀ y ∈ Bδ(y0).

The map η is continuous.

Proof of Proposition 2.5. Let

X = C2,α0 (Ω) ≡ v ∈ C2,α(Ω) : v|∂Ω = 0, Y = R, Z = Cα(Ω)

and

U = X ∩ SPSH(Ω).

Consider

F : (v, t) 7→ log detujk − logψt(z, u,Du), u = u+ v.

Suppose F (v0, t0) = 0 and let u0 = u + v0. The Frechet derivative Fv(v0, t0) is equal

to

(2.11) L = ujk0 ∂j∂k − (logψt0)Du ·D − (logψt0)u

8 BO GUAN

which is a linear elliptic operator with coefficients in Cα(Ω).

By the Schauder theory, the Dirichlet problem for the linear elliptic equation

(2.12)

Lw = f in Ω

w = 0 on ∂Ω

has a unique solution w ∈ C2,α(Ω for any f ∈ Cα(Ω). Moreover,

(2.13) |w|C2,α(Ω) ≤ C|f |Cα(Ω).

This means that L : C2,α0 (Ω) → Cα(Ω) is invertible and L−1 : Cα(Ω) → C2,α

0 (Ω) is

bounded. By Theorem 2.6, there exists δ > 0 and unique v = v(t) ∈ C2,α0 (Ω) for all

t ∈ [0, 1], |t− t0| < δ, such that F (v(t), t) = 0. This proves Proposition 2.5.

The closedness of S reduces to a priori C2,α estimates, which are uniform in t ∈[0, 1], for perspective solutions of (2.10). This can be seen as follows.

We wish to show that if tj ∈ S and tj → t0 as j → ∞ then t0 ∈ S. Let ut denote

the unique solution of (2.10) in C2,α(Ω)∩ SPSH(Ω) for t ∈ S and let vt = ut− u. We

need to show at least there is a subsequent of vtj converging in C2,α(Ω) to a function

v0 ∈ U and therefore F (v0, t0) = 0. Applying the Ascoli-Arzela Theorem, to show

that vtj has convergent subsequence in C2,α(Ω) it suffices to establish the estimate

(2.14) |utj |C2,α′(Ω) ≤ C, independent of j.

Consequently, the solvability of the Dirichlet problem (2.8) reduces to the a priori

estimate

(2.15) |u|C2,α(Ω) ≤ C

for strictly plurisubharmonic solutions.

2.3. CKNS Theorem. In the non-degenerate case (ψ > 0), a fundamental existence

theorem of classical solution for the Dirichlet problem (2.8) was proved by Caffarelli,

Kohn, Nirenberg and Spruck [13] for strongly pseudoconvex domains. It was later

extended to general domains by the author in [23] under the assumption of existence

of subsolutions, following the corresponding work [26], [22] on real Monge-Ampere

equations in general domains.

Theorem 2.7. Let Ω be a bounded smooth domain in Cn, and ϕ, ψ = ψ(z, u, p) real-

valued smooth functions, ψ(z, u, p) > 0 for all z ∈ Ω, u ∈ R and p ∈ Cn. Suppose


either ψ = ψ(z, u), i.e., it does not depend on the gradient of u, or for any M > 0

there exists CM > 0 such that

(2.16) −ψu, |ψzj|, |ψpj

| ≤ CMψ1− 1

n , ∀ (z, u, p) ∈ Ω× [−M,M ]× Cn.

The Dirichlet problem (2.8) admits a strictly plurisubharmonic solution u ∈ C∞(Ω),

provided that there exists a strictly plurisubharmonic subsolution u ∈ C2(Ω) satisfying

(2.9).

Remark 2.8. For strongly pseudoconvex domains, it was shown in [13] that condition

(2.16) implies the existence of strictly plurisubharmonic subsolutions. It is, however,

difficult to formulate for nonpseudoconvex domains general conditions which guaran-

tee the existence of subsolutions. Therefore, it is important in applications to find

natural conditions that one can use to construct subsolutions. As we shall see in Sec-

tions 4-6, in many interesting problems in geometry and analysis it is possible to find

such conditions. In the proof of Theorem 2.7, condition (2.16) is only used in deriving

the gradient estimates; see Subsection 3.2. We also note here that (2.16) should be

included in the assumptions of Theorem 1.1 in [23]. This was also pointed out to the

author by Zhuoliang Hou to whom the author wishes to express his gratitude.

Remark 2.9. Note that we do not assume ψu ≥ 0 in Theorem 2.7. Therefore we can

not apply the continuity method directly in the proof of Theorem 2.7. Nevertheless,

the proof of existence reduces to establishing the C2,α estimate (2.15). This can

be done following the argument in [14] for the real Monge-Ampere equation. In

the original CKNS theorem [13], the assumption ψu ≥ 0 was also used in deriving

estimates for the second derivatives. Here we modify their argument in order to drop

the assumption.

In the next section we shall derive the C2 estimate

(2.17) ‖u‖C2(Ω) ≤ C.

Once this is established, (2.8) becomes uniformly elliptic. We therefore obtain (2.15)

by the well known Evans-Krylov Theorem [19], [33], [34] (see also [13] and [12]).

Higher order estimates then follows from the classical Schauder theory for linear

uniform elliptic equations.

10 BO GUAN

3. A priori estimates

Let u ∈ C4(Ω) be a strictly pluri-subharmonic solution of (2.8) with u ≥ u. Our

goal is to derive the a priori bound (2.17). Without loss of generality we shall assume

ϕ ∈ C4(Ω).

3.1. The linearized operator. Rewrite equation (2.8) in the form

(3.1) log detujk = logψ(z, u,Du) ≡ f(z, u,Du) in Ω.

Let L be the linearized operator:

Lv = ujkvjk − fpj(z, u,Du)vj − fpj

(z, u,Du)vj.

We have

(3.2) Lu =n− fpj(z, u,Du)uj − fpj

(z, u,Du)uj.

and, for a function η ∈ C2(R),

(3.3) Lη(u) = η′Lu+ η′′ujkujuk.

Let

τ =n∑

j=1

(aj∂zj+ bj∂zj

)

be a first order linear operator with constant (complex) coefficients. Differentiating

equation (3.1) twice, we find

(3.4) L(uτ ) = fτ + fuuτ

and

(3.5) ujkujkτ τ − ujmulkujkτulmτ = (f)τ τ ,

where

(3.6)

(f)τ τ = fpjuτ τj + fpj

uτ τ j

+ fτ τ + fuuuτuτ + fuuτ τ + fpj pk(ujτuτ k + ujτukτ )

+ 2Refτuuτ + (fτpj

+ fupjuτ )ujτ

+ (fτ pk+ fupj

uτ )ujτ + fpjpkujτukτ

and

fτ = ajfzj+ bjfzj

, etc.


Therefore,

(3.7) Luτ τ = ujmulkujkτulmτ + (f)τ τ − fpjuτ τj − fpj

uτ τ j.

3.2. C1 estimates. We first observe that

(3.8) u ≤ u ≤ h in Ω,

where h is the harmonic extension of ϕ to Ω. Thus

maxΩ|u|+ max

∂Ω|∇u| ≤ C.

We next derive the global gradient estimate

(3.9) |∇u| ≤ C in Ω

under assumption (2.16).

Lemma 3.1. For any φ ∈ C2(Ω),

(3.10) e−φL(eφ|∇u|2) ≥ (2fu + Lφ)|∇u|2 + ∆u+ 2Re(fzk+ φk)uk.

Proof. This follows from straightforward calculations. First,

(3.11) e−φLeφ = Lφ+ ujkφjφk.

Next,

(3.12) (|∇u|2)i = (ukuk)i = ukiuk + ukuki,

(3.13) (|∇u|2)ij = (ukiuk + ukuki)j = ukijuk + ukukij + ukiukj + ukjuki.

Differentiating equation (3.1) we obtain

(3.14) uij(ukijuk + ukukij) = 2fu|∇u|2 + 2Refzkuk+ fpj

(|∇u|2)j + fpj(|∇u|2)j.

Thus, from (3.13) and (3.14),

(3.15) L(|∇u|2) = 2fu|∇u|2 + ∆u+ uijukiukj + 2Refzkuk.

We also note thatujk(|∇u|2)jφk = ujk(ululj + ululj)φk

= ujφj + uijuljulφk.

By Cauchy-Schwarz inequality,

2Reuijuljulφk| ≤ ujkuljulk + |∇u|2ujkφjφk.

12 BO GUAN

Finally, plugging these into

e−φL(eφ(u)|∇u|2) = |∇u|2(Lφ+ ujkφjφk) + 2Reujk(|∇u|2)jφk+ L(|∇u|2),

we derive (3.10).

Corollary 3.2. Under assumption (2.16), (3.9) holds.

Proof. Let φ = eλ|z|2 ≥ 1. We compute

φ−1Lφ =λ2ujkzjzk + λ( ∑

ujj − 2Refpjzj

)≥λ2ujkzjzk + λ

( ∑ujj − Aψ−

1n

)for some constant A > 0. Let µ1 ≤ . . . ≤ µn be the eigenvalues of ujk. One has

(3.16)∑

ujj =∑

µj ≥ n(µ1 · · ·µn)1n = n(detujk)

1n = nψ−

1n .

We consider two cases.

Case (i). µ1 ≤ (1 + A)1−nψ−1n . In this case, µn ≥ (1 + A)ψ−

1n and therefore,∑

ujj − Aψ−1n ≥ ψ−

1n +

n−1∑1

ujj.

Consequently, by Lemma 3.1 and assumption (2.16),

(3.17)

e−φL(eφ|∇u|2) ≥∆u− C(1 + λψ−1n )|∇u|

+((λφ− 2C)ψ−

1n + λφ

∑j<n

µj

)|∇u|2

≥(λ− 2C)|∇u|2 − Cλ|∇u|ψ−1n + (

√λ− C)|∇u|.

Here we used the fact that

λ|∇u|2∑j<n

µj + ∆u = λ|∇u|2∑j<n

µj +∑

µ−1j ≥ 2(n− 1)

√λ|∇u|.

Case (ii). µ1 ≥ (1 + A)1−nψ−1n . We have

ujkzjzk ≥ (1 + A)1−nψ−1n |z|2 ≥ (1 + A)1−nψ−

1n

if we assume |z| ≥ 1 for z ∈ Ω without loss of generality. It follows, similarly to

(3.17), that

(3.18) e−φL(eφ|∇u|2) ≥ ((1+A)1−nλ2−Aλ)|∇u|2−Cλ|∇u|ψ−1n +(

√λ−C)|∇u|.


From (3.17) and (3.18) we see that in both cases,

L(eφ|∇u|2) ≥ 0 for λ sufficiently large.

By the maximum principle,

maxΩ|∇u|2 ≤ max

Ω(eφ|∇u|2) = max

∂Ω(eφ|∇u|2) ≤ Cmax

∂Ω|∇u|2,

proving (3.9).

Applying Lemma 3.1 to φ = η(u) where η is a convex function, by (3.3) and (3.2)

we find

(3.19)

e−η(u) L(eη(u)|∇u|2) ≥ ∆u+ 2Refzkuk

+ 2fu + (n+ 2− fpjuj − fpj

uj)η′ + η′′uijuiuj|∇u|2

≥ 2fu + (n+ 2− fpjuj − fpj

uj)η′ + 2

√η′′|∇u|2 − 2|∇u|

∑|fzj

|.

In the last inequality we used

η′′uijuiuj + |∇u|−2∆u ≥ 2√η′′.

which follows from the Cauchy-Schwarz inequality.

Corollary 3.3. Suppose that η is a convex function satisfying

α|∇u| −∑

|fzj| > 0 in Ω

where

α ≡ fu + (n+ 2− fpjuj − fpj

uj)η′ +

√2η′′.

Then

(3.20) maxΩ

eη(u)|∇u| = max∂Ω

eη(u)|∇u|.

Proof. In (3.19) replace η by 2η and apply the maximum principle.

Using Corollary 3.3 we can derive gradient estimates under some different condi-

tions than (3.9).

Corollary 3.4. Suppose for any M > 0 there exists CM > 0 such that

(3.21)ψu ≥ −CMψ,

∑|ψzj

| ≤ CMψ(1 + |p|),∑

(ψpjpj + ψpj

pj) ≤ (n+ 2)ψ,

∀ (z, u, p) ∈ Ω× [−M,M ]× Cn.

Then

(3.22) maxΩ|∇u| ≤ C(1 + max

∂Ω|∇u|).

14 BO GUAN

Proof. Let η(u) = 4C2M(M + u)2. Then η′ = 8C2

M(u + M) ≥ 0 and η′′ = 8C2M .

Therefore

α ≥ fu +√

2η′′ ≥ 3CM

and conditions of Corollary 3.3 are satisfied or |∇u| ≤ 1 where η(u)|∇u| achieves its

maximum. By (3.20) we obtain (3.22).

Remark 3.5. Functions of the form

ψ(z, u, p) = g(z, u)(1 + |p|2)β, β ≤ n+ 2

2, g > 0, log g ∈ C1

satisfies condition (3.21) but for β > n it does not satisfy (2.16).

3.3. Boundary estimates for second derivatives. In this subsection we derive a

priori second derivative estimates on the boundary

(3.23) max∂Ω

|D2u| ≤ C.

We assume

(3.24) |u|+ |Du| ≤ K in Ω.

Set

(3.25) ψ ≡ min|u|+|p|≤K, z∈Ω

ψ(z, u, p) > 0, ψ ≡ max|u|+|p|≤K, z∈Ω

ψ(z, u, p).

(a) Consider a boundary point on ∂Ω. We may assume it to be the origin of Cn

and choose coordinates z1, . . . , zn such that the positive xn axis is the interior normal

direction to ∂Ω at 0. For convenience we set

t1 = x1, t2 = y1, . . . ,

t2n−3 = xn−1, t2n−2 = yn−1,

t2n−1 = yn, t2n = xn

and

t′ = (t1, . . . , t2n−1).

Near the origin ∂Ω may be represented as a graph

(3.26) xn = ρ(t′) =1

2

∑α,β<2n

Bαβtαtβ +O(|t′|3).


Since (u− u)(t′, ρ(t′)) = 0, we have

(3.27) (u− u)tαtβ(0) = −(u− u)xn(0)Bαβ, α, β < 2n.

It follows that

(3.28) |utαtβ(0)| ≤ C, α, β < 2n.

(b) To estimate utαxn(0) for α ≤ 2n, we will employ a barrier function of the form

(3.29) v = (u− u) + td− Nd2

2,

where t, N are positive constants to be determined and d is the distance function to

∂Ω. We may take δ > 0 small enough so that d is smooth in Ωδ = Ω ∩Bδ(0). Recall

that u ∈ C2(Ω) is strictly plurisubharmonic. Therefore there exists ε > 0 such that

(3.30) ujk ≥ εI on Ω.

Lemma 3.6. For N sufficiently large and t, δ sufficiently small,

Lv ≤ − ε4

(1 +

∑ukk

)in Ωδ,

v ≥ 0 on ∂Ωδ.

Proof. By (3.30) we have

ujk(ujk − ujk) ≤ n− ε∑

ukk.

It follows that

(3.31) L(u− u) ≤ C0 − ε∑

ukk.

Obviously,

|Ld| ≤ C1

(1 +

∑ukk

)for some constant C1 > 0 under control. Thus

Lv ≤ C0 + C1(t+Nd) + C1(t+Nd)− ε∑

ukk −Nujkdjdk in Ωδ.

Let λ1 ≤ · · · ≤ λn be the eigenvalues of ujk. We have∑ukk =

∑λ−1

k and

(3.32) ujkdjdk ≥1

λn

since |Dd| ≡ 1. By the inequality for arithmetic and geometric means,

ε

4

∑ukk +

N

λn

≥ nε

4(Nλ−1

1 · · ·λ−1n )

1n ≥ nε

4(ψ)1/nN

1n ≡ c1N

1n .

16 BO GUAN

We now fix t > 0 sufficiently small and N large so that C1t ≤ ε4

and c1N1/n ≥ C0+ε.

Consequently,

Lv ≤ − ε4

(1 +

∑ukk

)in Ωδ

if we require δ to satisfy C1Nδ ≤ ε4

in Ωδ.

On ∂Ω ∩Bδ(0) we have v = 0. On Ω ∩ ∂Bδ(0),

v ≥ td−Nd2 ≥ (t−Nδ)d ≥ 0

if we require, in addition, Nδ ≤ t.

Remark 3.7. Lemma 3.6 was first proved in Guan [23]. A preliminary version can

be found in the author’s thesis [21]. It is parallel to results in Guan [22] for real

Monge-Ampere equations in Rn and on general Riemannian manifolds, which improve

the corresponding results of Hoffman-Rosenberg-Spruck [30], Guan-Spruck [26] and

Guan-Li [25].

Lemma 3.8. Let h ∈ C2(Ω ∩Bδ(0)). Suppose that h satisfies

h ≤ C0|z|2 on (∂Ω) ∩Bδ(0), h(0) = 0

and

−Lh ≤ C1

(1 +

∑ujj

)in Ω ∩Bδ.

Then hν(0) ≤ C, where ν is the interior unit normal to ∂Ω, and C depends on ε−1,

C0, C1, |h|C0(Ω∩Bδ(0)) and |u|C1(Ω).

Proof. By Lemma 3.6, Av +B|z|2 − h ≥ 0 on ∂(Ω ∩Bδ(0)) and

L(Av +B|z|2 − h) ≤ 0 in Ω ∩Bδ

when A B are sufficiently large. By the maximum principle,

Av +B|z|2 − h ≥ 0 in Ω ∩Bδ.

Consequently,

Avν(0)− hν(0) = Dν(Av +B|z|2 − h)(0) ≥ 0

since Av +B|z|2 − h = 0 at the origin.

(c) We first apply Lemma 3.8 to estimate utαxn(0) for α < 2n. Define

T =∂

∂tα+ ρtα

∂

∂xn

.


We have (see [13])

L(±T (u− ϕ)− (uyn − ϕyn)2) ≤ C(1 +

∑F kk

)in Ω ∩Bδ(0).

By (3.24),

|T (u− ϕ)|+ (uyn − ϕyn)2 ≤ C in Ω ∩Bδ(0).

From the fact that u− ϕ = 0 and that T is a tangential operator on ∂Ω, we see that

T (u− ϕ) = 0 on ∂Ω ∩Bδ(0),

and

(3.33) (uyn − ϕyn)2 ≤ C|z|2, on ∂Ω ∩Bδ(0)

by (3.26) since

(u− ϕ)yn = −(u− ϕ)xnρyn , on ∂Ω ∩Bδ(0).

Applying Lemma 3.8 to h = (uyn − ϕyn)2 ± T (u− ϕ), we obtain

(3.34) |utαxn(0)| ≤ C, α < 2n.

(d) It remains to establish the estimate

(3.35) |uxnxn(0)| ≤ C.

Since we have already derived

(3.36) |utαtβ(0)|, |utαxn(0)| ≤ C, α, β < 2n,

it suffices to prove

(3.37) 0 ≤ unn(0) = uxnxn(0) + uynyn(0) ≤ C.

Expanding detujk:

detujk(0) = aunn(0) + b

where

a = detuαβ(0)|1≤α,β≤n−1and b is bounded in view of (3.36). By equation (2.8) we only have to derive an a

priori positive lower bound for a, which is equivalent to

(3.38)∑

α,β<n

uzαzβ(0)ξαξβ ≥ c0|ξ|2, ∀ ξ ∈ Cn−1

for a uniform constant c0 > 0.

Proposition 3.9. There exists c0 = c0(ψ−1, ϕ, u, ∂Ω) > 0 such that (3.38) holds.

18 BO GUAN

Proof. Let σ ∈ C4 be a defining function of Ω, i.e,

Ω = σ < 0, ∂Ω = σ = 0, Dσ|∂Ω 6= 0.

For convenience we assume |Dσ| = 1 on ∂Ω, without losing generality. Recall that

for a point p ∈ ∂Ω, the real and complex tangent spaces of ∂Ω at p are defined by

Tp∂Ω =ξ ∈ Cn : Re

∑σzj

(p)ξj = 0

and

TCp ∂Ω =

ξ ∈ Cn :

∑σzj

(p)ξj = 0,

respectively. Consider

m0 = minp∈∂Ω

minξ∈T C

p ∂Ω,|ξ|=1ujk(p)ξj ξk

and assume that m0 is attained at p = 0 and ξ = (1, 0, · · · , 0). We choose coordinates

z = (z1, · · · , zn) as before so that the positive xn axis is in the inner normal direction

to ∂Ω at 0. We need to show

(3.39) m0 = u11(0) ≥ c0 > 0.

Near the boundary of Ω, write

u− u = hσ.

We have

(u− u)xn = hxnσ + hσxn

and

(u− u)jk = hjkσ + hσjk + 2 Rehjσk.Therefore,

(u− u)xn(0) = −h(0)

and

(3.40) (u− u)αβ(0) = h(0)σαβ(0) α, β < n.

In particular,

(3.41) u11(0) = u11(0)− (u− u)xn(0)σ11(0).

We assume u11(0) ≤ 12u11(0); otherwise we are done. By (3.41),

(3.42) (u− u)xn(0)σ11(0) ≥ 1

2u11(0).


It follows from (3.24) that

(3.43) σ11(0) ≥1

2Cu11(0) ≥ c1 > 0.

Let δ > 0 be small enough so that

w ≡ (|σz1|2 + |σzn|2)12 > 0 in Ω ∩Bδ(0).

Define ζ = (ζ1 . . . , ζn): ζ1 = −σzn

w,

ζj = 0, 2 ≤ j ≤ n− 1,

ζn =σz1

wand

Φ = ϕjkζj ζk − (u− ϕ)xnσjkξj ξk − u11(0).

Note that ζ ∈ TC∂Ω on ∂Ω. We have

(3.44) Φ = ujkζj ζk − u11(0) ≥ 0 on ∂Ω ∩Bδ(0)

and Φ(0) = 0.

Write G = σjkξj ξk. We obviously have

LΦ ≤ −L(uxnG) + C(1 +

∑ujj

).

Using the fact that |L(uxn)| ≤ C, we calculate

L(uxnG) =GL(uxn) + ujk(uxnkGj + uxnjGk) + uxnLG

≥ujk(uxnkGj + uxnjGk)− C(1 +

∑ujj

).

Since

uxnk = 2unk +√−1uynk

and ujkunk = δjn, by Schwarz inequality,

ujk(uxnkGj + uxnjGk) = 2(Gn +Gn) +√−1ujk(uynkGj − uynjGk)

≥ − ujkuynjuynk − C(1 +

∑ujj

).

Next,

L(uyn − φyn)2 = 2(uyn − φyn)L(uyn) + L(φ2yn

) + 2ujkuynjuynk

≥ 2ujkuynjuynk − C(1 +

∑ujj

).

20 BO GUAN

It follows that

(3.45) L(Φ− (uyn − φyn)2) ≤ C(1 +

∑ujj

)in Ω ∩Bδ(0).

Moreover, by (3.33) and (3.44),

(uyn − φyn)2 − Φ ≤ C|z|2 on ∂Ω ∩Bδ(0).

Consequently, we may apply Lemma 3.8 to

h = (uyn − φyn)2 − Φ

to obtain Φxn(0) ≥ −C, which, by (3.43), implies

(3.46) uxnxn(0) ≤ C

σ11(0)≤ C

c1.

In view of (3.36) and (3.46) we have an a priori upper bound for all eigenvalues

of the complex Hessian ujk(0). Since detujk ≥ ψ > 0, the eigenvalues of ujk(0)must admit a positive lower bound, i.e.,

minξ∈Cn

,|ξ|=1ujk(0)ξj ξk ≥ c0.

Therefore,

m0 = minξ∈T C

0 ∂Ω,|ξ|=1ujk(0)ξj ξk ≥ min

ξ∈Cn,|ξ|=1

ujk(0)ξj ξk ≥ c0.

The proof of Proposition 3.9 is complete.

This establishes (3.23).

3.4. Global estimates for second derivatives. In this subsection we derive the

global estimate for second derivatives

(3.47) maxΩ|D2u| ≤ C

where C depends on |u|C1(Ω) and |D2u|∂Ω. We wish to include the degenerate case

ψ ≥ 0. So we shall still assume ψ > 0 in this subsection but the estimates will not

depends on the lower bound of ψ.

Instead of L, it is more convenient to consider the linearized operator L = ujk∂j∂k.

Lemma 3.10. For φ ∈ C2(Ω),

(3.48) e−φL(eφ∆u) ≥ (Lφ)∆u+ ∆(log detujk).


Proof. By (3.11) and (3.5),

(3.49) e−φL(eφ∆u) = ∆u(Lφ+ ujkφjφk) + 2Reujk(∆u)jφk+ L(∆u)

and

(3.50) L∆u = uimuljuijkulmk + ∆(log detujk).

By Cauchy-Schwarz inequality,

(3.51) 2|Reuij(∆u)jφk| ≤ (∆u)−1ujk(∆u)j(∆u)k + (∆u)ujkφjφk.

On the other hand,

(3.52) (∆u)−1uij(∆u)i(∆u)j ≤ uimuljuijkulmk.

To see this we may assume uij is diagonal. Applying Cauchy-Schwarz inequality,

uij(∆u)i(∆u)j =∑

i

uii∣∣∣ ∑

k

u1/2

kku−1/2

kkukki

∣∣∣2≤ ∆u

∑i,k

uiiukk|ukki|2

≤ ∆u∑i,j,k

uiiujjuijkuijk.

This proves (3.48).

We first present a corollary of Lemma 3.10 when ψ does not depend on the gradient

Du.

Corollary 3.11. For η ∈ C2(R),

(3.53) e−η(u)L(eη(u)∆u) ≥ (ψu + nη′ + η′′uijuiuj)∆u+ fuu|∇u|2 − 2|∇fu||∇u|+ ∆f

where

∆f =∑

fzj zj, |∇fu|2 =

∑|fuzj

|2.

Consequently, if η is a convex function and ψu + nη′ > 0 then

maxΩ

eη(u)∆u = max∂Ω

eη(u)∆u

or

∆u ≤ |∇fu|2 − η′′ − fuu)|∇u|2 −∆f

fu + nη′

wherever eη(u)∆u attains an interior (local) maximum.

22 BO GUAN

Proposition 3.12. Suppose for any M > 0 there exists constant C = C(M) such

that

(3.54)

− fuu, |fzj|, |fuzj

|,∑

fzj zj,≤ Cψ−

1n ,

− fu, |fzjpk|, |fupj

| ≤ Cψ−12n ,

|fpj|, |fpjpk

| ≤ C,

fpj pk ≥ −CI

whenever |u|+ |p| ≤M . Then

(3.55) ∆u ≤ C in Ω

where C depends on |u|C1(Ω) and |D2u|∂Ω but is independent of ψ−1 (see (3.25)).

Proof. Consider G = eφ∆u where

φ = A|∇u|2 +B|z|2.

Suppose G attains its maximum at an interior point z0 ∈ Ω. Then we have (all

calculations below are at z0)

(3.56) φj∆u+ (∆u)j = 0, ∀ j

and,

(3.57) 0 ≥ e−φLG ≥ (Lφ)∆u+ ∆(f).

By (3.15) and (3.16),

(3.58)

Lφ =AL(|∇u|2) +B∑

ujj

≥A∆u+ Aujkujlulk + 2Refpjφj

+Bψ−1n + 2Afu|∇u|2 + 2ReAfzj

uj −Bfpjzj

while from (3.6) we see that

(3.59)

∆(f) ≥ 2Refpj(∆u)j+ fpj pk

(ujlulk + ujlulk) + 2Refpjpkujlulk

+ fu∆u− C∆u∑

(|fpju|+ |fpjzk|)

− C∑

(|fpju|+ |fpjzk|)|ujk|

+ fuu|∇u|2 +∑

fzj zj− C

∑|fuzj

|.


Using (3.56) and Cauchy-Schwarz inequality, we obtain

(3.60)

(Lφ)∆u+ ∆(f) ≥ (Aujk∆u+ fpj pk− Cδjk)(ujlulk + ujlulk)

+ fu∆u+ fuu|∇u|2 +∑

fzj zj− C

∑|fuzj

|

+ 2Afu|∇u|2 +Bψ−1n − C

∑(A|fzj

|+B|fpj|)

−∑

(|fpjzk|2 + |fpju|2 + |fpjpk

|2).

By assumption (3.54) if we choose A and B sufficiently large then (3.57) and (3.60)

yield

0 ≥ A

2(∆u)2 − Cψ−

12n ∆u+ (B − CA− C)ψ−

1n − CB

≥ A− 1

2(∆u)2 + (B − CA− C)ψ−

1n − CB

which gives (3.55) when B − CA− C ≥ 0 (and, certainly, A > 1).

By (3.55) and equation (2.8) we have

(3.61) c0I ≤ ujk ≤ C0I in Ω.

It follows that

(3.62) ujk ≥ C−10 I in Ω.

We next finish the proof of (3.47), that is,

(3.63) |uxixj|, |uxiyj

|, |uyiyj| ≤ C, ∀ 1 ≤ i, j ≤ n.

Let

τ = aj∂xj+ bj∂yj

, aj, bj ∈ R,∑

(a2j + b2j) = 1,

be a first order linear differential operator of constant real coefficients. We shall show

under assumption (3.54) that

(3.64) maxΩ

uττ ≤ max∂Ω

uττ + C,

which obviously implies (3.63) since u is plurisubharmonic.

By (3.6), (3.7) and assumption (3.54), we find

(3.65)

Luττ ≥ 2fpj pkuτjuτ k + 2Refpjpk

uτjuτk

+ fuuττ + fuuu2τ + fττ − C

∑|fuzj

|

− C∑

(|fpjzk|+ |fpju|)|uτj|

≥ − C∑

|uτj|2 − Cψ−1n .

24 BO GUAN

Next, by (3.62)

(3.66)Lu2

τ = 2ujkuτjuτ k + 2uτLuτ

≥ 2C−10

∑|uτj|2 − Cψ−

1n

and

(3.67) Leλ|z|2 ≥ C−10 |z|2λ2 + λψ−

1n − Cλ ≥ λψ−

1n

if we assume |z| ≥ 1 in Ω and λ is sufficiently large. Therefore,

(3.68) L(uττ + Au2τ + eλ|z|2) ≥ (C−1

0 A− C)∑

|uτj|2 + (λ− CA− C)ψ−1n ≥ 0

when A and λ are sufficiently large. This shows (3.64) by the maximum principle.

4. Complex Monge-Ampere equations in Kahler geometry

4.1. Kahler metrics and curvature. Let (M,J) be a complex manifold and g a

Riemannian metric on M . Recall that J2 = −id. We assume that J is compatible

with g, i.e.

(4.1) g(u, v) = g(Ju, Jv), u, v ∈ TM.

The Kahler form ωg of g is defined by

(4.2) ωg(u, v) = −g(u, Jv).

The complexified tangent bundle TCM = TM × C has a natural splitting

(4.3) TCM = T 1,0M + T 0,1M

where T 1,0M and T 0,1M are the ±√−1-eigenspaces of J . Obviously, the metric g is

extended C-linearly to TCM , and

(4.4) g(u, v) = 0 if u, v ∈ T 1,0M , or u, v ∈ T 0,1M.

In local coordinates (z1, . . . , zn) this gives

(4.5) g( ∂

∂zj

,∂

∂zk

)= 0, g

( ∂

∂zj

,∂

∂zk

)= 0

since

(4.6) J∂

∂zj

=√−1

∂

∂zj

, J∂

∂zj

= −√−1

∂

∂zj

.


We write

(4.7) gjk = g( ∂

∂zj

,∂

∂zk

).

The Kahler form ωg is then given by

(4.8) ωg =

√−1

2gjkdzj ∧ dzk.

The metric g is Kahler if the Kahler form ωg is closed, i.e. dωg = 0. This is

equivalent to ∇J = 0, where ∇ denote the Levi-Civita connection of g. In local

coordinates, this is characterized by

(4.9)∂gjk

∂zl

=∂glk

∂zj

, ∀ j, k, l.

Another useful fact is that a metric g is Kahler if and only if g has local potentials,

namely, in local coordinates there exists a function ϕ such that

(4.10) gjk =∂2ϕ

∂zj∂zk

.

The Christoffel symbols in local coordinates (z1, . . . , zn) are defined by

∇ ∂∂zj

∂

∂zk

= Γljk

∂

∂zl

+ Γljk

∂

∂zl

and

∇ ∂∂zj

∂

∂zk

= Γljk

∂

∂zl

+ Γljk

∂

∂zl

.

For a Kahler manifold, using ∇J = 0 and (4.6) one can see that the only possible

non-vanishing terms are Γljk and Γl

jk= Γl

jk. Moreover,

(4.11) Γljk = glm∂gjm

∂zk

.

We recall that the Riemannian curvature tensor is defined by

R(u, v)w = ∇u∇vw −∇v∇uw −∇[u,v]w.

Since ∇J = 0,

R(u, v)Jw = JR(u, v)w.

It follows that

(4.12) g(R(u, v)Jw, Jx) = g(R(u, v)w, x) ≡ R(u, v, w, x).

26 BO GUAN

Therefore R(u, v, w, x) = 0 unless w and x are of different type. By (4.11) we have

in local coordinates,

(4.13) Rijkl ≡R( ∂

∂zi

,∂

∂zj

,∂

∂zk

,∂

∂zl

)= −

∂2gij

∂zk∂zl

+ gpq ∂gpj

∂zk

∂giq

∂zl

.

The Ricci curvature is given by

(4.14) Rkl = gijRijkl = − ∂2

∂zk∂zl

log det gij,

or equivalently, the Ricci form is

(4.15) Ricg :=

√−1

2Rkldzk ∧ dzl = −

√−1

2∂∂ log det gij.

Recall that a Kahler metric g is Einstein if there exists a real constant λ such that

(4.16) Ricg = λωg.

4.2. The complex Monge-Ampere equation in Kahler manifolds. Let (Mn, ω)

be a Kahler manifold and ψ ∈ C∞(M ×R), ψ ≥ 0. We consider the complex Monge-

Ampere equation which plays fundamental roles in Kahler geometry

(4.17) (ω(φ))n = ψ(z, φ)ωn

where

ω(φ) = ω +

√−1

2∂∂φ.

If in local coordinates,

ω =

√−1

2gjkdzj ∧ dzk,

then

(4.18) ω(φ) =

√−1

2(gjk + φjk)dzj ∧ dzk

and therefore, equation (4.17) is equivalent to

(4.19) det(gij + φij) = ψ(z, φ) det gij.

As we see in Section 2 for complex Monge-Ampere equations in Cn, equation (4.17)

is elliptic for solutions in the Mabuchi space of Kahler potentials defined as

(4.20) H = φ ∈ C∞(M) : ω(φ) > 0.

The following fundamental existence result is due to S.-T. Yau [48] and is the core

of his proof of Calabi conjecture.


Theorem 4.1 ([48]). Let M be a compact Kahler manifold (without boundary) and

ψ ∈ C∞(M × R). Suppose that ψ > 0, ψu ≥ 0, and that there exists a function

φ ∈ C∞(M) such that

(4.21)

∫M

ψ(z, φ(z))ωn = Vol (M).

Then there exists a solution φ ∈ H of equation (4.17).

SinceM is compact (without boundary), one needs certain compatibility conditions

such as (4.21) to solve equation (4.17). Indeed, any solution φ ∈ H of equation (4.17)

must satisfy (4.21) with φ in place of φ. Note that φ is not assumed to be in H.

For the complete proof of Theorem 4.1 the reader is referred to Yau’s original

paper [48]. For ψ = ψ(z) it can also be found in several articles and monographs such

as [46], [47], and [10].

We next consider the Dirichlet problem for equation (4.17). Let Ω be a compact

domain with smooth boundary in a Kahler manifold M . Given ψ ∈ C∞(Ω × R),

ψ > 0, and ϕ ∈ C∞(∂Ω), we seek solutions φ ∈ H(Ω) = φ ∈ C∞(Ω) : ω(φ) > 0 of

the Dirichlet problem

(4.22)

(ω(φ))n = ψ(z, φ)ωn in Ω

φ = ϕ on ∂Ω.

Theorem 4.2. Suppose that there exists a subsolution u ∈ C2(Ω) satisfying ω(u) > 0

and

(4.23)

ω(u)n ≥ ψ(z, u)ωn in Ω

u = ϕ on ∂Ω.

There exists a solution φ ∈ H of the Dirichlet problem (4.22).

This is an analogue of Theorem 2.7. Its proof also reduces to a priori C2 estimates.

The C0 and boundary gradient estimates follow as before from the inequalities u ≤φ ≤ h where h is the harmonic extension of ϕ. For the global gradient estimates see

Blocki citeBlocki09 and P.-F. Guan [29]. The boundary estimates in Subsection 3.3

for second derivatives are still valid in this case as the proof is purely local and there

are always local Kahler potentials for Kahler manifolds. Finally, the global estimates

for second derivatives follows from Yau [48].

Below we shall see some applications of Theorems 4.1 and 4.2.

28 BO GUAN

4.3. The Calabi-Yau theorem. Given two Kahler forms on a Kahler manifold M

(4.24) ω =

√−1

2gjkdzj ∧ dzk, ω =

√−1

2gjkdzj ∧ dzk,

by (4.15) their Ricci forms are related by the formula

(4.25) Ric(ω)− Ric(ω) =

√−1

2∂∂ log

det gij

det gij

which is a globally defined function on M . Consequently, the cohomology class

[Ric(ω)] of Ric(ω) is independent of ω and is in fact equal to πc1(M) where c1(M) is

the first Chern class of M .

The Calabi conjecture asserts that the converse is true: any (1, 1) form on a compact

Kahler manifold M representing πc1(M) must be the Ricci form of a Kahler form on

M . This conjecture was proved by S.-T. Yau [48].

Theorem 4.3 ([48]). Let (Mn, ω) be a compact Kahler manifold. Given any (1, 1)

form Ω representing πc1(M), there exists a unique Kahler form ω ∈ [ω] such that

Ric(ω) = Ω.

The proof of Theorem 4.3 reduces to solving a complex Monge-Ampere equation

of form (4.17) on M . In fact, since both Ω and Ric(ω) represent πc1(M), there exists

a smooth function f such that

(4.26) Ω− Ric(ω) =

√−1

2∂∂f.

(This follows from the ∂∂-lemma; see e.g. [47].) Suppose that Ω is the Ricci form of

ω ∈ [ω]. By the ∂∂-lemma again there exists a smooth function ϕ such that

ω = ω +

√−1

2∂∂φ.

In local coordinates we have,

(4.27) ∂∂ logdet(gij + φij)

det gij

= ∂∂f.

Since M is compact, this implies that

(4.28) logdet(gij + φij)

det gij

− f = c

is a constant, that is

(4.29) det(gij + φij) = ef+c det gij.


Integrating (4.29) we see that c satisfies

(4.30)

∫M

ef+cωn = Vol(M).

Consequently, Theorem 4.3 reduces to solving equation (4.29) under the compatibility

condition (4.30).

4.4. Geodesics in the Mabuchi space of Kahler potentials. Let (M, g) be a

compact Kahler manifold. The Mabuchi space of Kahler potentials (see (4.20))

(4.31) H = φ ∈ C∞(M) : ω(φ) > 0

is an open subset of C∞(M). Therefore, its tangent space TφH at φ ∈ H is naturally

identified to C∞(M). In [39], Mabuchi introduced a Riemannian structure onH using

the L2 inner product on TφH with respect to the volume form of ω(φ):

(4.32) 〈ξ, η〉φ =

∫M

ξη (ω(φ))n, ψ ∈ TφH.

Accordingly, one can define the length of a regular curve ϕ : [0, 1] → H by

(4.33) L(ϕ) =

∫ 1

0

〈ϕ, ϕ〉12φdt.

Here and in what follows ϕ = ∂ϕ/∂t and ϕ = ∂2ϕ/∂t2. The geodesic equation takes

the form

(4.34) ϕ− |∇ϕ|2ϕ = 0.

This means in local coordinates

(4.35) ϕ− g(ϕ)jkϕzjϕzk

= 0.

Here [g(ϕ)jk] is inverse matrix of [g(ϕ)jk] = [gjk + ϕjk].

Donaldson [18] conjectured that H is geodesically convex, i.e., any two functions

in H can be connected by a smooth geodesic, and that H is a metric space.

Conjecture 4.4 (Donaldson [18]). (i) Let ϕ0, ϕ1 ∈ H. There exists a smooth geodesic

ϕ : [0, 1] → H such that

(4.36) ϕ(0) = ϕ0, ϕ(1) = ϕ1.

(ii) The function

d(ϕ0, ϕ1) = infL(ϕ)|ϕ : [0, 1] → H, ϕ(0) = ϕ0, ϕ(1) = ϕ1, ϕ0, ϕ1 ∈ H

30 BO GUAN

defines a distance on H.

This is a reformulation of the original Donaldson conjecture in [18]. Indeed, by an

observation of Donaldson [18], Mabuchi [39] and Semmes [41], part (i) of Donaldson’s

conjecture may be reduced to a Dirichlet problem for the homogeneous complex

Monge-Ampere equation in M × A where A is the cylinder [0, 1]× S1. Let

w = zn+1 = t+√−1s

be the local coordinate of A. We may view a smooth curve ϕ in H as a function in

C∞(M× [0, 1]) and therefore a rotation-invariant function (constant in s) in C∞(M×A). We have

ϕ =∂ϕ

∂t= 2

∂ϕ

∂w= 2

∂ϕ

∂w

and

ϕ =∂2ϕ

∂t2= 4

∂2ϕ

∂w∂w.

Suppose now that ϕ solves the geodesic equation (4.35). Then

(4.37)

det

ϕ1w

(g(ϕ)jk)...

ϕnw

ϕw1 · · · ϕwn ϕww

=1

4det

ϕ1

(g(ϕ)jk)...ϕn

ϕ1 · · · ϕn ϕ

=1

4det(g(ϕ)ij) · det

g(ϕ)k1ϕk

I...

g(ϕ)knϕk

ϕ1 · · · ϕn ϕ

=

1

4det(g(ϕ)ij) · (ϕ− g(ϕ)jkϕzj

ϕzk) = 0.

That is,

(4.38)(Ω +

√−1

2∂∂ϕ

)n+1

= 0 in M × A


where

(4.39) Ω = ω +

√−1

2∂∂|zn+1|2 =

√−1

2

( ∑j,k≤n

gjkdzj ∧ dzk + dzn+1 ∧ dzn+1

)is the lift of ω to the product M × A.

Conversely, if ϕ ∈ C∞(M × A) is a rotation-invariant solution of (4.38) such that

(4.40) ϕ(·, w) ∈ H, ∀ w ∈ A,

then one sees from (4.37) that ϕ is a geodesic in H.

The original conjecture of Donaldson [18] may be stated as follows.

Conjecture 4.5 (Donaldson [18]). Let A be a compact Riemann surface with bound-

ary and ρ ∈ C∞(M × ∂A) such that ρ(·, w) ∈ H for w ∈ ∂A. Then there exists

a unique solution ϕ of the Monge-Ampere equation (4.38) satisfying (4.40) and the

boundary condition ϕ = ρ.

The uniqueness was proved by Donaldson [18] as a consequence of the maximum

principle. In [15], X.-X. Chen obtained the following existence result.

Theorem 4.6. Let A and ρ be as in Conjecture 4.5. There exists a unique (weak)

solution ϕ ∈ C1,α(M×A), 0 < α < 1, with ∆ϕ ∈ L∞(M×A) of the Dirichlet problem

(4.41)

Ω +

√−1

2∂∂ϕ ≥ 0 in M × A(

Ω +

√−1

2∂∂ϕ

)n+1

= 0 in M × A

ϕ = ρ on M × ∂A.

In what follows we outline the proof of Theorem 4.6 for A = [0, 1] × S1. Since

equation (4.38) is degenerate, we shall consider the approximation problem

(4.42)

Ω +

√−1

2∂∂ϕ ≥ 0 in M × A(

Ω +

√−1

2∂∂ϕ

)n+1

= εΩn+1 in M × A

ϕ = ρ on M × ∂A.

In order to apply Theorem 4.2 to the Dirichlet problem (4.42) we need to construct

a subsolution. This is easily done for the product space M×A when A is the annulus

32 BO GUAN

[0, 1]× S1. Let ϕ0 = ρ|t=0 and ϕ1 = ρ|t=1. Define

ϕ = (1− t)ϕ0 + tϕ1 +K(t2 − t).

By the assumption that ρ(·, w) ∈ H for w ∈ ∂A we can choose K > 0 sufficiently

large such that ϕ satisfies

(4.43)

Ω +

√−1

2∂∂ϕ > 0 in M × A(

Ω +

√−1

2∂∂ϕ

)n+1

≥ Ωn+1 in M × A

ϕ = ρ on M × ∂A.

Consequently, by Theorem 4.2 we obtain a unique solution ϕε ∈ C∞(M×A) of (4.42)

for any ε ∈ (0, 1]. Moreover, by the maximum principle,

ϕε ≥ ϕε′ ∀ 0 < ε < ε′ ≤ 1

and therefore the limit

ϕ(z) = limε→0

ϕε(z)

exists for all z ∈M ×A. To finish the proof of Theorem 4.6 we need to establish the

estimate

(4.44) |ϕε|C2(M×A) ≤ C independent of ε.

It is straightforward to verify that

|ϕε|C1(M×A) ≤ C independent of ε

and

(4.45) maxM×A

∆ϕε ≤ C + C max∂(M×A)

∆ϕε.

On the boundary ∂(M × A) = M × ∂A we have

(4.46) ϕεξη = ϕ

ξη, ∀ ξ, η ∈ Tp∂(M × A), ∀ p ∈ ∂(M × A)

since ∂(M ×A) is given by t = 0 or t = 1, and, from the estimates in Subsection 3.3,

(4.47) |ϕεξt| ≤ C, ∀ ξ ∈ Tp∂(M × A), |ξ| = 1, ∀ p ∈ ∂(M × A)

where C is independent of ε. As in Subsection 3.3, using (4.46) and the fact that ϕ

is strictly plurisubharmonic we derive from equation (4.42)

(4.48) |ϕεtt| ≤ ϕε

ww + |ϕεss| ≤ C, independent of ε on ∂(M × A).

This completes the proof of (4.44) and (4.45).


Remark 4.7. If ρ is rotation invariant then so is ϕ by the uniqueness.

5. The Chern-Levine-Nirenberg conjecture

In this section we present a brief description of the work of Pengfei Guan [27], [28]

on the Chern-Levine-Nirenberg conjecture on intrinsic norms.

Let M be a compact complex manifold and boundary ∂M = Γ0 ∪ Γ1. Define

PSH1(M) = u ∈ PSH(M) : 0 < u < 1

F = C2(M) ∩ PSH1(M), F = C0(M) ∩ PSH1(M)

and

B = u ∈ F : u|Γ0 = 0, u|Γ1 = 1.In [16], Chern, Levine, and Nirenberg introduced the following nonnegative function

on the homology group H∗(M,R): for γ ∈ H∗(M,R),

(5.1) N(γ) = supu∈F

infT∈γ

|T (dcu ∧ (ddcu)k−1)|, if dimγ = 2k − 1,

|T (du ∧ dcu ∧ (ddcu)k−1)|, if dimγ = 2k.

Similarly one can define N by replacing F by F .

It was shown in [16] that N and N are a seminorm on H∗(M,R). Namely,

(5.2)N(aγ) = |a|N(γ), a ∈ R,

N(γ1 + γ2) ≤ N(γ1) +N(γ2), dimγ1 = dimγ2.

(Same is true for N .) Moreover, N and N decrease under holomorphic mappings: if

f : M → P is a holomorphic mapping then

(5.3) N(f∗γ) ≤ N(γ), N(f∗γ) ≤ N(γ), γ ∈ H∗(M,R).

From now on we shall consider a special case:

M = Ω \m⋃

j=1

Ωj, Γ0 =m⋃

j=1

∂Ωj, Γ1 = ∂Ω

where Ω1, . . . ,Ωm and Ω are bounded smooth strongly pseudoconvex domains in Cn,

Ωi ∩ Ωj = ∅ (i 6= j) and Ωj ⊂ Ω, 1 ≤ j ≤ m. Define

Jk(u) =

∫M

du ∧ dcu ∧ (ddcu)k−1 ∧ ωn−k

34 BO GUAN

where

ω =

√−1

2

∑dzj ∧ dzj

is the Kahler form of Cn.

The Chern-Levine-Nirenberg conjecture [16]:

N(Γ1) = infu∈B

Jn(u).

In [27] P.-F. Guan proved the following modified version of the conjecture, formu-

lated by Bedford and Taylor [5].

Theorem 5.1 ([27]). Suppose that

(5.4)m⋃

j=1

Ωj is holomorphically convex in Ω.

Then

N(Γ1) = infu∈B

Jn(u).

More recently, P.-F. Guan [28] was able to prove that N(Γ1) = N(Γ1) for general

complex manifolds, therefore confirmed the original Chern-Levine-Nirenberg conjec-

ture.

The Chern-Levine-Nirenberg conjecture is associated with the Dirichlet problem

(5.5)

(ddcu)n = 0 in M

u|Γ0 = 0, u|Γ1 = 1.

Indeed if u ∈ C0(M) ∩ PSH(M) solves problem 5.5 then for all T ∈ [Γ0],∫T

du ∧ (ddcu)n−1 =

∫M

du ∧ dcu ∧ (ddcu)n−1.

A crucial ingredient in the proof of Theorem 5.1 is the following regularity result

for the Dirichlet problem (5.5).

Theorem 5.2 ([27]). Under condition (5.4), problem 5.5 admits a unique solution

u ∈ C1,1(M) ∩ PSH(M).

The C1,1 regularity is optimal as shown by an example of Bdeford and Faenass [3].

Because the homogeneous complex Monge-Ampere equation (ddcu)n = 0 is degener-

ate, the best regularity of its solution one can hope is C1,1 in general, even for smooth

data.

In what follows we outline the proof of Theorem 5.2 in [27].


Proof of Theorem 5.2 (Sketch). The first step is to construct a strictly plurisubhar-

monic function u ∈ C2(M) that satisfies

(5.6)

(ddcu)n ≥ ε0 > 0 in M

u|Γ0 = 0, u|Γ1 = 1.

Let σ1, . . . , σm, σ be smooth defining functions of Ω1, . . . ,Ωm,Ω, respectively. By

condition (5.4), there is a function ψ ∈ C∞(Ω) ∩ PSH(Ω) such thatψ < 0 in a neighborhood of

⋃mk=1 Ωk

ψ > 0 in Ω \⋃m

k=1 Uk

where

Ωk ⊂ Uk ⊂ Uk ⊂ Ω, 1 ≤ k ≤ m, U j ∩ Uk = ∅, j 6= k.

We may assume that for some λ > 0,

(5.7) ψ − λ|z|2, σ − λ|z|2, σk − λ|z|2 ∈ PSH(Ω), 1 ≤ k ≤ m.

Define

ϕk(z) = maxε2σk(z), εψ(z), z ∈ Ω.

For ε > 0 small,

ϕk =

ε2σk, in a neighborhood Vk ⊂ Uk of Ωk,

εψ, outside Vk.

Therefore, we may choose ε sufficiently small and A > 0 sufficiently large such that

the function u ∈ C0,1(Ω) defined by

u = max1 + Aσ, ϕ1, . . . , ϕm,

which is plurisubharmonic, satisfies (5.6) and

u =

1 + Aσ near ∂Ω,

ϕk in Vk.

Note that u is smooth near ∂M . Finally, by a regularization procedure one obtains a

strictly plurisubharmonic function in C∞(M), still denoted by u, satisfying (5.6); see

P.-F. Guan’s original argument in [27] for details.

Using u constructed above as a subsolution, for any 0 < ε ≤ ε0 we can apply

Theorem 2.7 to obtain a unique strictly plurisubharmonic solution uε ∈ C∞(M) of

36 BO GUAN

the Dirichlet problem

(5.8)

(ddcu)n = ε in M,

u|Γ0 = 0, u|Γ1 = 1.

By the maximum principle,

uε ≤ uε′if ε ≤ ε′.

Consequently, uε converges pointwise to a function u in M as ε tends to zero. In order

to prove that u ∈ C1,1(M) and solves (5.5) we only need to derive the estimate

(5.9) |uε|C2(M) ≤ C independent of ε.

Note that (5.9) does not follow directly from the estimates in Section 3 because

equation (5.8) becomes degenerate as ε vanishes. However, as we see in Section 3 this

only affects the estimate in (3.38). In other words, (5.9) will follow if we can recover

(3.38) for uε with c0 independent of ε.

For this we make use of the fact that uε is constant on each component of ∂M .

Indeed, consider a point, which we may assume to be the origin in Cn, on ∂M .

Under a coordinate system as in Subsection 3.3 we have, since uε is constant on the

components of ∂M ,

(5.10) uεjk(0) =

uεxn

(0)

σxn(0)σjk(0), 1 ≤ j, k ≤ n− 1

where σ = σl if 0 ∈ ∂Ωl (1 ≤ l ≤ m) and σ = σ if 0 ∈ ∂Ω. On the other hand, from

u ≤ uε ≤ h in M

where h is the harmonic function in M with h = 0 on Γ0 and h = 1 on Γ1, we see

thatuε

xn(0)

σxn(0)≥ c1

for some constant c1 > 0 independent of ε. By (5.7) we obtain

(5.11) uεjk(0)|1≤j,k≤n−1 ≥ c1λI,

therefore proving (5.9).


6. Regularity of pluricomplex Green functions

6.1. Pluricomplex Green function with logarithmic poles. Let Ω be a bounded

domain in Cn and ζ ∈ Ω. The pluricomplex Green function on Ω with a logarithmic

pole at ζ is defined as

gζ(z) = supv(z) : v ∈ PSH(Ω), v < 0 and v(z) ≤ log |z − ζ|+O(1)

.

This function is a higher dimensional analogue of the Green function in C, and plays

important roles in pluripotential theory. We refer the reader to [1], [17], [31] for

background and connections in complex analysis. In this section we present some

regularity results when Ω is strongly pseudoconvex.

A striking regularity theorem is due to Lempert [36] who proves that gζ(z) is smooth

in Ω \ ζ for smooth strictly convex domains Ω. This result has found important

applications; see for instance [42] where Semmes develops a theory of generalized

Riemann mappings.

In the strongly pseudoconvex case, however, E. Bedford and J.-P. Demailly [2]

found counterexamples which show that gζ in general does not belong to C2(Ω\ζ).In [23] the author proved gζ(z) ∈ C1,α(Ω \ ζ). Later on Blocki [8] improved it to

gζ(z) ∈ C1,1(Ω \ ζ), which is optimal as shown by counterexamples in [2].

A fundamental property of the pluri-complex Green function is that it is a week

solution of the following problem

(6.1)

u is pluri-subharmonic in Ω \ ζdet(uzj zk

) = 0 in Ω \ ζu = 0 on ∂Ω

u(z) = log |z − ζ|+O(1) as z → ζ.

Theorem 6.1 ([23], [8]). Let Ω be a smooth strongly pseudoconvex domain. There

exists a unique solution of (6.1) in C1,1(Ω \ ζ). Consequently, gζ ∈ C1,1(Ω \ ζ).

Proof. The uniqueness is a consequence of the minimum principle of Bedford-Taylor [4]

as in [37]. In what follows below we prove the existence of a solution in C1,α(Ω\ζ);for the proof of C1,1 regularity see [8].

Without loss of generality we assume ζ = 0 and B1 = B1(0) ⊂ Ω. Consider the

Dirichlet problem

(6.2)

det(vjk) = 1 in Ω,

v = − log |z| on ∂Ω.

38 BO GUAN

Since Ω is strongly pseudoconvex, by Theorem 2.7 problem (6.2) admits a unique

strictly pluri-subharmonic solution v ∈ C∞(Ω). Let u ≡ v + log |z| ∈ C∞(Ω \ 0).We see that u satisfies

(6.3)

u is strictly pluri-subharmonic in Ω \ 0det(ujk) ≥ 1 in Ω \ 0u ≤ 0 in Ω \ 0, u = 0 on ∂Ω

u(z) = log |z|+O(1) as z → 0.

For ε > 0 small (we shall assume ε ≤ 12) let Ωε = Ω \Bε and consider the problem

(6.4)

det(ujk) = ε in Ωε,

u = u on ∂Ωε.

Note that u is a subsolution of (6.4) when ε ≤ 1. We may apply Theorem 2.7 again

to obtain a unique strictly pluri-subharmonic solution uε ∈ C∞(Ωε) of the Dirichlet

problem (6.4). By the maximum principle,

(6.5) u ≤ uε ≤ uε′ ≤ log |z| in Ωε if ε′ ≤ ε.

Thus the limit

u(z) ≡ limε→0

uε(z)

exists for all z ∈ Ω \ 0. We need to show that u ∈ C1,α(Ω \ 0). By standard

elliptic theory it suffices to establish the following estimate

(6.6) |∇uε|2 + ∆uε ≤ C

|z|2in Ωε, independent of ε.

Lemma 6.2. There exists a constant C1 independent of ε such that

(6.7) |∇uε| ≤ C1 on ∂Ω and |∇uε| ≤ C1

εon ∂Bε.

Proof. Since u ≤ uε ≤ 0 in Ωε and uε = u = 0 on ∂Ωε, we have

|∇uε| = uεν ≤ uν = |∇u| on ∂Ω

where ν is the exterior unit normal vector to ∂Ω. This proves the first inequality in

(6.7). To prove the second one, for z ∈ B2 \B1 let

u(z) = uε(εz)− log ε,

u(z) = u(εz)− log ε = v(εz) + log |z.|


Note that

(6.8) det(ujk) = ε2n det(ujk) = ε2n+1.

Let h be the harmonic function on B2 \ B1 with h = log 2 on ∂B2 and h(z) = v(εz)

on ∂B1. Then u ≤ u ≤ h on B2 \ B1 by the maximum principle, since ∆u ≥ 0 in

B2 \ B1, u ≤ h on ∂B2 and u = h on ∂B1. Consequently,

|∇u| = uν ≤ hν ≤ C1 on ∂B1.

Here ν is the exterior unit normal vector to ∂B1. This implies the second inequality

in (6.7) as ∇uε(z) = 1ε∇u( z

ε).

Using Lemma 6.2 and Corollary 3.3 with η(u) = u we obtain, by (6.5)

(6.9) |∇uε| ≤ Ce−uε ≤ C

|z|on Ωε.

Lemma 6.3. There exists a constant C2 independent of ε such that

(6.10) |∇2uε| ≤ C2 on ∂Ω and |∇2uε| ≤ C2

ε2on ∂Bε.

Proof. The first estimate in (6.10) can be derived as in Section 5. We only prove the

second one here. Let u, u and h be as in the proof of Lemma 6.2. It suffices to show

that

(6.11) |∇2u| ≤ C2 on ∂B1.

For a fixed point z0 ∈ ∂B1, we may assume z0 = (0, . . . , 1), i.e., the coordinates of

z0 are xj = yj = 0, 1 ≤ j ≤ n − 1, xn = 1 and yn = 0. Since u(z) = v(εz) on ∂B1

and |∇u| ≤ C1, it is trivial to obtain a bound for the pure tangential second order

derivatives at z0

(6.12) |uxixk|, |uxiyj

|, |uyjyl| ≤ C, 1 ≤ i, k ≤ n− 1, 1 ≤ j, l ≤ n.

To estimate the mixed tangential normal derivatives we need the following analogue

of Lemma 3.6.

Lemma 6.4. Let Uδ = (B2 \B1)∩Bδ(z0) and w = (u− u) + t(|z| − 1)−N(|z| − 1)2,

where t, N are positive constants. For N sufficiently large and t, δ sufficiently small,

we have

(6.13) ujkwjk ≤ − 1

64

(1 +

∑ukk

)in Uδ

and v ≥ 0 on ∂Uδ.

40 BO GUAN

Proof. We first note that this does not follow from Lemma 3.6 since ujk is not

uniformly positive definite in ε. In order to prove (6.13) we make use of a special

property of u. Since u(z) = v(εz) + log |z| and v is plurisubharmonic, we see that

ujkujk ≥ ujk(log |z|)jk ≥1

2|z|2ujk

(δjk −

zjzk

|z|2)

in Uδ.

On the other hand,

ujk((|z| − 1)2)jk = ujk(|z|2 − 2|z|)jk

=(1− 1

|z|

) ∑ukk +

1

2|z|3ujkzjzk.

≥ 1

16ujkzjzk.

The rest is obvious.

Returning to the proof of Lemma 6.3, as in Subsection 3.3 we may derive a bound

for the mixed tangential normal derivatives at z0 with the aid of Lemma 6.4,

(6.14) |uxkxn|, |uxnyj| ≤ C, 1 ≤ k ≤ n− 1, 1 ≤ j ≤ n.

It remains to establish an estimate for the pure normal second order derivative

(6.15) |uxnxn(z0)| ≤ C.

Because of (6.12) and (6.14) it suffices to prove

(6.16) unn(z0) ≤ C.

Since u− u = 0 on ∂B1,

ujk(z0) = ujk(z

0) +1

2(u− u)xn(z0)δjk, j, k < n

and therefore

(6.17)∑j,k<n

ujk(z0)ξj ξk ≥

∑j,k<n

ujk(z0)ξj ξk = |ξ|2

for any ξ = (ξ1, . . . , ξn−1) ∈ Cn−1. Finally, solving equation (6.8) for unn we see that

(6.16) follows from (6.12), (6.14) and (6.17). This completes the proof of (6.11) and

therefore that of Lemma 6.3.

Applying Corollary 3.11, by Lemma 6.3 we obtain

Corollary 6.5. There exists a constant C3 independent of ε such that

(6.18) |∆uε| ≤ C3e−2uε

in Ωε.


6.2. Pluricomplex Green function with logarithmic pole at infinity. For a

bounded subset E of Cn, the pluricomplex Green function of E (with pole at infinity)

is defined as

VE(z) = supv(z) : v ∈ L, v|E ≤ 0

where L denotes the collection of plurisubharmonic functions v on Cn of minimal

growth, i.e.,

v(z) ≤ log |z|+O(1) as |z| → ∞.

This function, which is also known as the Siciak-Zahariuta L-extremal function, was

first introduced by Siciak [43] (in an essentially equivalent form) and Zahariuta [49]

in the study of polynomial approximation of holomorphic functions of several vari-

ables and Bernstein-Walsh type theorems. It has proved to be a useful tool in the

pluripotential theory, and found applications in a wide range of problems in complex

analysis (see e.g. [1], [31] and [40]).

For a compact set E in Cn, the continuity of VE is equivalent to the Bernstein-

Markov inequality: For any t > 0 there exists an open set U ⊃ E such that ‖p‖U ≤etd(p)‖p‖E for any polynomial p, where d(p) is the degree of p ([49], [44], [45]). Other

criteria for continuity of VE have been discovered by various authors; see [1], [31], [40]

and references therein. This has been an important issue in applications.

As in the previous subsection we are concerned with the higher regularity of the

pluricomplex Green function, and our approach is based on its close connections with

the homogeneous complex Monge-Ampere equation. Let Ω be a bounded open set of

Cn and Ωc = Cn \ Ω. We consider the exterior Dirichlet problem

(6.19)

u is plurisubharmonic in Ωc

(ddcu)n = 0 in Ωc

u = 0 on ∂Ω

u(z) = log |z|+O(1) as |z| → ∞.

If ∂Ω ∈ C1 then VΩ is continuous and is the unique weak solution of (6.19) (see,

e.g. [17] and [31]). The uniqueness of solution to (6.19) also follows from a general

theorem of Bedford and Taylor [7]: If u, v ∈ L and (ddcu)n = (ddcv)n then u − v is

constant.

In [38] Lempert studied the problem using an interesting approach when Ω is a

smooth strictly convex domain. By constructing an analogue of the Kelvin transfor-

mation for the complex Monge-Ampere operator he demonstrated that problem (6.19)

42 BO GUAN

can be reduced to the Dirichlet problem (6.1) in the previous subsection. Conse-

quently, using his earlier results in [36] Lempert was able to prove that problem (6.19)

admits a unique solution in C∞(Ωc) for strictly convex Ω. This is a remarkable result

because in general solutions to the degenerate complex Monge-Ampere equation, even

with smooth data, may fail to be in C2 ([2], [3], [20]). The method in [38] actually

applies to strictly linearly convex domains. However, whether it can be extended to

more general cases is not clear; see for example [1]. In this subsection we treat prob-

lem (6.19) directly using elliptic PDE methods for strongly pseudoconvex domains.

More generally, let Ω1, . . . ,ΩN be bounded smooth strongly pseudoconvex domains

in Cn with Ωi ∩ Ωj = ∅ (i 6= j) and we shall consider the case

(6.20) Ω =N⋃

j=1

Ωj.

Theorem 6.6 ([24]). Let Ω be of the form (6.20) and satisfy the assumption:

(6.21) Ω is holomorphically convex in Ω0

where Ω0 is a bounded smooth strongly pseudoconvex domain in Cn. Problem (6.19)

admits a unique plurisubharmonic solution u ∈ C1,1(Ωc). Moreover, u satisfies the a

priori estimates

(6.22) −C ≤ u− log |z| ≤ C in Ωc,

(6.23) |∇u|2 + ∆u ≤ C

|z|2in Ωc

Consequently, VΩ ∈ C1,1(Ωc).

The estimate in (6.23) shows certain resemblance beyond (6.22) between VΩ and

log |z| which is the pluricomplex Green function of the closed unit ball. We also note

that condition (6.21) is necessary for Theorem 6.6, and is clearly satisfied when Ω is

a smooth strongly pseudoconvex domain, that is, when N = 1 in (6.20).

To prove Theorem 6.6 we first construct a smooth strictly plurisubharmonic func-

tion defined on Ωc which vanishes on ∂Ω using condition (6.21). Without loss of

generality let us assume that

(6.24) Br0 ⊂⊂ Ω ⊂⊂ Ω0 ⊂⊂ B1

for some 0 < r0 ≤ 12, where Br and Br denote the ball of radius r and its closure,

respectively, centered at the origin in Cn.


Let σ0 : Cn → R be a smooth strictly plurisubharmonic defining function of Ω0:

(6.25) Ω0 = z ∈ Cn : σ0(z) < 0 and ∇σ0 6= 0 on ∂Ω0.

By Proposition 1.1 of [27] (see Section 5) we find a strictly plurisubharmonic function

σ ∈ C∞(Ωc) such that σ = 0 on ∂Ω and σ = 1 + σ0 in a neighborhood of Ωc0. Now

fix λ0 ∈ (0, 12) sufficiently small such that 2λ0σ < 1 in B2. For ε ∈ (0, λ0), note that

1

2log

ε2 + |z|2

1 + ε2>

log 3

2> λ0σ on ∂B2

and

(6.26) logε2 + |z|2

1 + ε2≤ 0 ≤ λ0σ in B1 \ Ω.

Using the smoothing technique of P.-F. Guan [27] we can construct a strictly plurisub-

harmonic function uε ∈ C∞(Ωc) satisfying

(6.27) uε(z) =

λ0σ in B1 \ Ω12log ε2+|z|2

1+ε2 in Cn \B2

and

(6.28) uε(z) ≥ maxλ0σ,

1

2log

ε2 + |z|2

1 + ε2

in B2 \B1.

For s > 0 set

(6.29)

ψs(z) :=1

2ndet

∂2

∂zj∂zk

log(s2 + |z|2)

=1

2ndet

(s2 + |z|2)δij − zizj

(s2 + |z|2)2

=s2

2n(s2 + |z|2)n+1.

From the construction in [27] (Proposition 1.1 and Lemma 3.2 (ii)) we see that

(6.30) detuεij ≥ ψε(z) in Ωc

if 0 < ε ≤ ε0 where

ε0 := (2λ0)n2 rn+1

0 δ120 ≤ 2λ0r

n+10 ≤ 1

2nminλ0, r0

and 0 < δ0 ≤ 1 satisfies

(6.31) detσjk ≥ δ0 in B2.

44 BO GUAN

For fixed R ≥ 2 let ΣR = BR \ Ω and consider the Dirichlet problem

(6.32)

detuij = ψε in ΣR

u = uε on ∂ΣR.

Clearly uε is a smooth strictly plurisubharmonic subsolution, so we may apply The-

orem 2.7 to conclude that there exists a unique strictly plurisubharmonic solution

uε,R ∈ C∞(ΣR) of (6.32). By the maximum principle,

(6.33) uε ≤ uε,R ≤ uε,R′< log

|z|r0

in ΣR

when R′ ≥ R ≥ 2 since these inequalities hold on ∂ΣR and

(6.34) det(log |z|)ij = 0.

It follows from (6.33) that

(6.35) uε(z) := limR→∞

uε,R(z)

exists for all z ∈ Ωc.

We shall next derive the following estimates for uε,R:

(6.36) |∇uε,R|2 + ∆uε,R ≤ C

|z|2in ΣR

where C is a positive constant independent of ε and R. For convenience we write

u = uε,R and u = uε.

Consider the function

ρ(z) = α(ε, R) log|z|r0

where

α(ε, R) =log(ε2 +R2)− log(1 + ε2)

2(logR− log r0)<

logR

logR/r0< 1,

and let h = hR ∈ C1,1(ΣR) be the unique plurisuharmonic solution of the problem

(see Theorem 5.2)

(6.37)

dethij = 0 in ΣR,

h = 0 on ∂Ω

h = logR− log r0 on ∂BR.

Since

(6.38)

u = u = h < ρ on ∂Ω

u = u = ρ < h on ∂BR,


by the maximum principle we have

(6.39) u < u < ρ and u < u < h in ΣR.

It follows that

(6.40)α(ε, R)

R= −∂ρ

∂ν< −∂u

∂ν< −∂u

∂ν=

R

ε2 +R2<

1

Ron ∂BR

where ν denotes the interior unit normal to ∂ΣR so it is interior to ∂BR but exterior

to ∂Ω, and

(6.41) 0 < c1 ≤∂u

∂ν<∂u

∂ν<∂h

∂ν≤ c2 on ∂Ω

for some uniform positive constants c1 and c2 independent of ε and R. In fact we

may choose

c1 = λ0 min∂Ω

∂σ

∂ν, c2 = max

∂Ω

∂h2

∂ν> 0

since hR ≤ h2 for all R ≥ 2.

Note that |uν | = 2|∇u| on ∂ΣR since u is constant on each component of ∂ΣR. By

(6.40) and (6.41) we have

sup∂ΣR

eu|∇u| ≤ 1

2max

c2,

1

R

(ε2 +R2

1 + ε2

) 12≤ maxc2, 1

2.

Applying Corollary 3.3 with η(u) = u, ψ(z, u) = logψε(z), we obtain

(6.42) |∇u| ≤ C1e−u in ΣR,

where

C1 = maxc2

2,1

2, sup

ΣR

eu|∇ logψε|n+ 2

.

Since

|∇ logψε| = (n+ 1)|z|ε2 + |z|2

,

from (6.39) we see that C1 is independent ε and R.

To complete the proof of (6.36) we next estimate the second derivatives of u on

∂ΣR. Let us first consider ∂BR. Suppose R ≥ 4 and let

u(ξ) = u(Rξ)− a, ξ ∈ B1 \B 12

where

a = u|∂BR=

1

2log

ε2 +R2

1 + ε2.

46 BO GUAN

By (6.34) and (6.39) we have

(6.43) − log 2 ≤ u ≤ u ≤ ρ ≤ 0 in B1 \B 12

where, with ε = ε/R,

u(ξ) = u(Rξ)− a =1

2log

ε2 + |ξ|2

1 + ε2,

ρ(ξ) = ρ(Rξ)− a = α(ε, R) log |ξ|.Next,

(6.44) uξi= Ruzi

, uξi= Ruzi

and

uξiξj= R2uzizj

.

Therefore,

(6.45) det uξiξj= ψε(ξ) :=

ε2

2n(ε2 + |ξ|2)n+1in B1 \B 1

2.

By (6.40), (6.42), (6.43) and (6.44) we have

(6.46) |∇u| ≤ C in B1 \B 12

and

(6.47) α(ε0, R) ≤ α(ε, R) < −∂u∂ν

<1

1 + ε2on ∂B1.

Since u = 0 on ∂B1, we may write

u(ξ) = (|ξ|2 − 1)Λ(ξ), in B1 \B 12

where Λ is a nonnegative function. We have

(6.48)∂u

∂ν= −2Λ on ∂B1.

For an arbitrary point p ∈ ∂B1, we choose coordinates of Cn

ξ1 = t1 + it2, . . . , ξn = t2n−1 + it2n

such that p is given by ξ1 = 1, ξ2 = . . . = ξn = 0. We have at p,

(6.49) utitj = 2Λδij, ∀ 2 ≤ i, j ≤ 2n.

Therefore, by (6.47) and (6.48),

(6.50) |utitj(p)| ≤ 1, ∀ 2 ≤ i, j ≤ 2n


and

(6.51)1

2α(ε0, R) ≤ uξj ξj

(p) = Λ ≤ 1

2, ∀ 2 ≤ j ≤ n.

Furthermore, by the mixed tangential-normal derivative estimates in Subsection 3.3

we obtain

(6.52) |ut1tj(p)| ≤ C, ∀ 2 ≤ j ≤ 2n.

Finally, solving uξ1ξ1 from Equation (6.45) and using (6.50)-(6.52) we derive

(6.53) uξ1ξ1(p) =det uξiξj

uξ2ξ2 · · · uξnξn

+∑i≥2

|uξ1ξi|2

uξiξi

≤ C.

In both (6.52) and (6.53) the constant C is independent of ε. Since |ut2t2(p)| ≤ 1 by

(6.50), we see that (6.53) implies

(6.54) |ut1t1(p)| ≤ C.

Combining this with (6.50) and (6.52), we have proved

(6.55) |utitj | ≤ C on ∂B1, ∀ 1 ≤ i, j ≤ 2n.

By rescaling we obtain a priori estimates for all (real) second derivatives of u on ∂BR:

(6.56) |D2u| ≤ C

R2on ∂BR

where C is independent of ε and R.

The estimate for second derivatives on ∂Ω is similar to that (on Γ0) in Section 5:

(6.57) |D2u| ≤ C on ∂Ω independent of ε and R.

Now we apply Corollary 3.11 with η(u) = 2u, f(z, u) = logψε(z) which satisfies

∆ logψε = −(n+ 1)( n

ε2 + |z|2− |z|2

(ε2 + |z|2)2

)≥ −n(n+ 1)

ε2 + |z|2.

Since C−1|z|−1 ≤ e−u ≤ e−u ≤ C|z|−1 in ΣR, we have

−e2u∆ logψε ≤ C in ΣR,

and, by (6.56) and (6.57),

e2u∆u ≤ C on ∂ΣR.

By Corollary 3.11,

(6.58) ∆u ≤ C2

|z|2

where C2 is independent ε and R. Combining with (6.42) this proves (6.36).

48 BO GUAN

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50 BO GUAN

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Documents

THE DIRICHLET PROBLEM FOR COMPLEX MONGE … DIRICHLET PROBLEM FOR COMPLEX MONGE-AMPERE` EQUATIONS AND APPLICATIONS BO GUAN Abstract. We are concerned with the Dirichlet problem for