the Complex Monge-Amp`ere Equation on Compact K¨ahler Manifolds

the Complex Monge-Amp`ere Equation on Compact K¨ahler Manifolds

Z BIGNIEW B ŁOCKI

A

. We prove uniqueness of weak solutions of the Dirich- let problem for the complex Monge-Amp`ere equation on com- pact K¨ahler manifolds. A qualitative version of this result implies the

stability of solutions of this equation.

1. I

Let

be a compact K¨ahler manifold of the complex dimension

with the K¨ahler form

. We say that a function

on

is admissible if it is upper semicontin- uous, locally integrable and

:

0, where

¯ and

√−

1

¯

. By [1], for bounded admissible

on

one can well define the complex Monge-Amp`ere measure

ωⁿ_ϕ= ωϕ∧ · · · ∧ ωϕ.

The main goal of this note is to show the following uniqueness result.

Theorem 1.1. Let

,

be bounded admissible functions on

such that

. Then

is constant.

For

Theorem 1.1 was proved in [2]. In this case it is equivalent to the uniqueness of the Dirichlet problem for the complex Monge-Amp`ere equation for entire plurisubharmonic functions with logarithmic growth. For arbitrary

, Calabi showed in the 1950’s that the uniqueness holds in the case when

,

are smooth and

,

0. Recently in [5] Theorem 1.1 was proved under extra assumption that

and

have a density belonging to

for some

1.

In fact, we prove the following stability result, which is a quantitative version of Theorem 1.1.

1697

Indiana University Mathematics Journal c , Vol. 52, No. 6 (2003)

Theorem 1.2. If

and

are bounded admissible on

, then

Md(ϕ− ψ) ∧ d^c(ϕ− ψ) ∧ ωⁿ⁻¹≤ C

 Z

M(ψ− ϕ)(ωⁿϕ− ωⁿψ)

21⁻ⁿ

,

where

is a positive constant depending only on

and upper bounds of

,

and the volume of

.

Theorem 1.2 and the Sobolev inequality give an

stability for the complex Monge-Amp`ere operator on compact K¨ahler manifolds (compare with [5]).

Theorem 1.3. Let

and

be bounded admissible functions on

such that

M

ϕ ωⁿ= Z

M

ψωⁿ

and

,

for some

,

. Then, if

2,

L1_(M),

where

is a positive constant depending on

and on upper bounds of

and

.

The proof of Theorem 1.2 is in a way quite elementary. Unlike in [5], it does not use the existence of solutions of the Dirichlet problem (given directly or indirectly by [6]).

Note that the uniqueness (Theorem 1.1) does not hold for unbounded solu- tions: consider for example

and for

0

ϕ([z0

:

:

:

:

log

p|z1|²+ · · · + |zn|² p|z0|²+ |z1|²+ · · · + |zn|², ψ([z₀

:

:

:

:

log

max

|z0|²+ |z1|²+ · · · + |zn|².

One can then check that

but of course

is not constant.

2. P

We will now recall some facts that will be used in the proof of Theorem 2. They

were proved mostly in [1] (see also [3], [4]). We assume that

is just a complex

manifold of dimension

. Let

be a nonnegative (and thus in particular of

order zero, or, in other words, representable by integration) closed complex

current on

(

1) and let

,

be functions defined on

which locally

can be written in the form

,

, where

,

,

are bounded and plurisubharmonic. One can then well define currents

η dd^cρ∧ T , dη∧ d^cρ∧ T ,

and they are of order zero. Moreover, we always have

0 and, if

is plurisubharmonic,

dd^cρ∧ T ≥

0

Since currents of the form

are well defined, so are

and

(but they are not necessarily of order zero).

If

1, then the Schwarz inequality gives

Mdη∧ d^cρ∧ T  ≤ Z

Mdη∧ d^cη∧ T

1/2 Z

Mdρ∧ d^cρ∧ T

1/2

.

If we take

in some chart of

, it follows that

has a locally bounded mass and therefore

.

Finally, by the Stokes theorem, if

is a real 2

1 current (with complex coe fficients) on

such that

is of order zero, and

is compact, then

Z

M

dU=

0

3. P

T

1.2

By

we will denote possibly di fferent constants depending only on the required quantities. Set

. For

0, 1,

,

1 we will prove inductively that (3.1)

Z

Mdρ∧ d^cρ∧ ωⁱϕ∧ ω^jψ∧ ω^k≤ Ca^2−k,

where

a= Z

M(ψ− ϕ)(ωⁿϕ− ωⁿψ)= Z

Mdρ∧ d^cρ∧ T , T =

n−X1 l=0

ω^{`</sup><sub>ϕ</sub>∧ ω<sup>n−</sup>ψ <sup>1−`},

and

,

are such that

1. For

1 we will then obtain the

desired estimate.

If

0, then

M

dρ∧ d^cρ∧ ωⁱϕ∧ ω^jψ≤ Z

M

dρ∧ d^cρ∧ T = a.

Assume that (3.1) holds for 0, 1,

,

1. We have

ωⁱ_ϕ∧ ω^jψ∧ ω^k= ωⁱϕ^+k∧ ω^jψ− dd^cϕ∧ α,

where

α= ωⁱϕ∧ ω^jψ∧

k−X1 l=0

ω^{`</sup><sub>ϕ</sub>∧ ω<sup>k−1−`}.

Therefore

dρ∧ d^cρ∧ ωⁱϕ∧ ω^jψ∧ ω^k≤ dρ ∧ d^cρ∧ (T − dd^cϕ∧ α)

= d(ρd^cρ∧ T − d^cϕ∧ α ∧ dρ ∧ d^cρ)

− ρdd^cρ∧ T − dρ ∧ d^cϕ∧ α ∧ dd^cρ.

This means that

Mdρ∧ d^cρ∧ ωⁱϕ∧ ω^jϕ∧ ω^k≤ a − Z

Mdρ∧ d^cϕ∧ α ∧ dd^cρ.

We have

− Z

M

dρ∧d^cϕ∧α∧dd^cρ≤ Z

M

dρ∧d^cϕ∧α∧ωϕ

+ Z

M

dρ∧d^cϕ∧α∧ωψ

.

If

is equal to

or

, the Schwarz inequality gives

Z

Mdρ∧d^cϕ∧α∧ωη

 ≤ Z

Mdρ∧d^cρ∧α∧ωη

1/2 Z

Mdϕ∧d^cϕ∧α∧ωη

1/2

.

By the inductive assumption (and since

) it remains to show that

Z

M

dϕ∧ d^cϕ∧ α ∧ ωη≤ C.

But

M

dϕ∧ d^cϕ∧ α ∧ ωη= − Z

M

ϕ∧ dd^cϕ∧ α ∧ ωη

≤ Z

M

ϕ ω∧ α ∧ ωη

 + Z

M

ϕ ωϕ∧ α ∧ ωη

≤

2

vol

The proof is complete.

Remark 3.1. Approximating

and

by smooth admissible functions and using the continuity theorems for the complex Monge-Amp`ere operator from [1], one can reduce the proof of Theorem 1.2 to smooth

and

.

Acknowledgement. The author was partially supported by KBN Grant #2 P03A 028 19.

The author would like to thank S. Kołodziej for his assistance and discussions on [2].

R

[1] E.BEDFORDand B.A.TAYLOR, A new capacity for plurisubharmonic functions, Acta Math 149 (1982), 1-41.

[2] , Uniqueness for the complex Monge-Amp`ere equation for functions of logarithmic growth, Indiana Univ. Math. J. 38 (1989), 455-469.

[3] J.-P. DEMAILLY, Monge-Amp`ere operators, Lelong numbers and intersection theory, Complex Anal- ysis and Geometry, Univ. Ser. Math. Plenum, New York, 1993, pp. 115-193.

[4] M. KLIMEK, Pluripotential Theory, Clarendon Press, 1991.

[5] S. KOŁODZIEJ, Stability of solutions to the complex Monge-Amp`ere equation on compact K¨ahler manifolds, Indiana Univ. Math. J. 52 (2003), 667-686.

[6] S.-T. YAU, On the Ricci curvature of a compact K¨ahler manifold and the complex Monge-Amp`ere equation, I Comm. Pure Appl. Math. 31 (1978), 339-411.

Jagiellonian University, Institute of Mathematics, Reymonta 4, 30-059, Krak´ow, Poland . blocki@im.uj.edu.pl

KEY WORDS AND PHRASES: complex Monge-Amp`ere equation; compact K¨ahler manifolds.

2000MATHEMATICSSUBJECTCLASSIFICATION: Primary: 32W20; secondary: 32Q15.

Received : August 1st, 2002; revised: March 26th, 2003.