Monge-Amp` ere equation on compact K¨ ahler manifolds

On uniqueness of the complex

Monge-Amp` ere equation on compact K¨ ahler manifolds

Z. B locki

REPORT No. 1, 2007/2008, spring ISSN 1103-467X

ISRN IML-R- -1-07/08- -SE+spring

MONGE-AMP` ERE EQUATION ON COMPACT K ¨ AHLER MANIFOLDS

Zbigniew BÃlocki

Abstract. We prove a partial uniqueness for solutions of the complex Monge-Am- p`ere equation on a compact K¨ ahler manifold in the class of quasiplurisubharmonic functions introduced recently by Guedj and Zeriahi.

1. Introduction

Let (X, ω) be a compact K¨ahler manifold of complex dimension n. P SH(X, ω) will denote the class of quasiplurisubharmonic functions ϕ on X satisfying ω _ϕ :=

ω + dd ^c ϕ ≥ 0. Guedj and Zeriahi [9] introduced the class E(X, ω) of functions ϕ ∈ P SH(X, ω) satisfying R

{ϕ>−∞} ω ⁿ _ϕ = R

X ω ⁿ (the measure ω _ϕ ⁿ = ω _ϕ ∧ · · · ∧ ω _ϕ is well defined on {ϕ > −∞} for any ϕ ∈ P SH(X, ω) by [1]). They showed in particular that for ϕ ∈ E(X, ω) the measure ω _ϕ ⁿ is well defined on X (with total mass R

X ω ⁿ ), vanishes on pluripolar sets, and is continuous (in the weak ^∗ topology) for decreasing sequences in E(X, ω).

One of the main results in [9], building on earlier work of Yau [11], KoÃlodziej [10], and Cegrell [6] was the existence of a solution of the Dirichlet problem

(1)

 



 

ϕ ∈ E(X, ω) ω ⁿ _ϕ = µ max X ϕ = 0,

provided that µ is a measure on X vanishing on pluripolar sets and with total mass R

X ω ⁿ (which are of course necessary conditions).

The uniqueness in (1) was posed as a problem in [9]. It had been proved in [3] for bounded ϕ, ψ (for X = P ⁿ it had been earlier done in [2] with much more complicated methods). As observed in [9] the method from [3] actually gives the fol- lowing result: if ϕ ∈ E(X, ω) and ψ ∈ E

¹

(X, ω) (where E ^p (X, ω) := {ψ ∈ E(X, ω) : R

X |ψ| ^p ω _ψ ⁿ < ∞}, p > 0) are such that ω ⁿ _ϕ = ω _ψ ⁿ then ϕ − ψ = const.

2000 Mathematics Subject Classification. 32W20, 32Q15.

Key words and phrases. Complex Monge-Amp`ere equation, K¨ ahler manifolds.

This paper was written during the author stay at the Institut Mittag-Leffler (Djursholm, Sweden). It was also partially supported by the projects N N201 3679 33 and 189/6 PR EU/2007/7 of the Polish Ministry of Science and Higher Education

Typeset by AMS-TEX

1

2 ZBIGNIEW BÃLOCKI

The goal of this note is to prove the following improvement:

Theorem 1. Assume ϕ ∈ E(X, ω) and ψ ∈ E ^p (X, ω) for some p > 1 − 2

¹⁻ⁿ

. If ω ⁿ _ϕ = ω _ψ ⁿ then ϕ − ψ = const.

Let us mention that if ϕ ∈ E(X, ω) are such that ω ⁿ _ϕ = ω ⁿ _ψ =: µ then we must have ω

_max{ϕ,ψ}

ⁿ = µ (see [9], Proposition 3.4, we will make use of this result in the proof of Theorem 1) and also ω ⁿ _{tϕ+(1−t)ψ} = µ, 0 ≤ t ≤ 1 (see [8]).

In [4] and [5] the class D of (germs of) plurisubharmonic functions was defined (it was shown that it is actually the same as the class E studied in [7]). It is the maximal subclass of the class of plurisubharmonic functions where the complex Monge-Amp`ere operator (dd ^c ) ⁿ can be defined (as a regular measure) so that it is continuous for decreasing sequences. It was shown in [4] that D = P SH ∩ W _loc

^1,2

for n = 2 and it was characterized (similarly, but in a more complicated way) for n ≥ 3 in [5].

A natural counterpart of the class D on a compact K¨ahler manifold is the class D(X, ω) consisting of those ϕ ∈ P SH(X, ω) such that locally ϕ + g ∈ D, where g is a local potential for ω (that is ω = dd ^c g). In particular, for n = 2 we get D(X, ω) = PSH(X, ω) ∩ W

^1,2

(X) (and ⊂ for arbitrary n). The measure ω _ϕ ⁿ is of course well defined for ϕ ∈ D(X, ω). By D _a (X, ω) denote the class of those ϕ ∈ D(X, ω) for which ω ⁿ _ϕ vanishes on pluripolar sets. It follows that D _a (X, ω) ⊂ E(X, ω) but by Example 2.14 in [9] we don’t have the equality in general.

By Lemma 5.14 in [7] the Dirichlet problem (1), where µ is a measure on X vani- shing on pluripolar sets and with total mass R

X ω ⁿ , always has a local solution in D _a . This is therefore perhaps natural to ask whether it has a global solution belonging to D _a (X, ω), which would be an improvement of Theorem A in [9]. However, using Theorem 1 we can show that this is not the case:

Theorem 2. Let (X, ω) be the projective space P ⁿ with the Fubini-Study metric.

There exists a measure µ on X, vanishing on pluripolar sets and with total mass R

X ω ⁿ , such that there is no ϕ ∈ E(X, ω) ∩ W

^1,2

(X) satisfying ω ⁿ _ϕ = µ.

In the proofs of Theorems 1 and 2 we will follow the notation from [9] and use various results proved in that article. We always assume that (X, ω) is a fixed K¨ahler manifold.

The author is grateful to SÃlawomir Dinew and SÃlawomir KoÃlodziej for helpful discussions on this subject.

Proofs

As the proof in dimension 2 is simpler and more transparent, we first prove Theorem 1 in this case.

Proof of Theorem 1 in dimension 2. If e ψ := max{ϕ, ψ} then e ψ ≥ ψ, e ψ ∈ E ^p (X, ω)

(by Lemma 2.3 in [9]), and by Proposition 3.4 in [9] we have ω _ψ

²_e

= ω _ψ

²

= ω _ϕ

²

. We

may thus assume that ϕ ≤ ψ ≤ −1. Then, if we set ψ ^j := max{ϕ, ψ − j}, we have

ψ − j ≤ ψ ^j ≤ ψ, ψ ^j ∈ E ^p (X, ω), and ψ ^j decreases to ϕ as j → ∞. Without loss

of generality we may thus assume that 0 ≤ ρ := ψ − ϕ ≤ C; then both ϕ and ψ

belong to E ^p (X, ω).

We now set ϕ _j := max{ϕ, −j}, ψ _j := max{ψ, −j}, ρ _j := ψ _j − ϕ _j , and h _j :=

(ϕ _j + ψ _j )/2. First, we claim that

(2) lim

j→∞

Z

X

dρ _j ∧ d ^c ρ _j ∧ ω _h

_j

= 0.

We have Z

X

ρ _j (ω

²

_ϕ

_j

− ω

²

_ψ

_j

) = −2 Z

X

ρ _j dd ^c ρ _j ∧ ω _h

_j

= 2 Z

X

dρ _j ∧ d ^c ρ _j ∧ ω _h

_j

.

On the other hand,

¯ ¯

¯ ¯ Z

X

ρ _j (ω _ϕ

²_j

− ω _ψ

²_j

)

¯ ¯

¯ ¯ =

¯ ¯

¯ ¯

¯ Z

{ϕ≤−j}

ρ _j (ω _ϕ

²_j

− ω _ψ

²_j

)

¯ ¯

¯ ¯

¯

≤ C ÃZ

{ϕ≤−j}

ω

²

_ϕ

_j

+ Z

{ψ≤−j}

ω _ψ

²_j

!

→ 0.

We thus get (2).

Set χ(t) := − √

−t, t ≤ −1. We want to show the following improvement of (2)

(3) lim

j→∞

Z

X

χ ◦ h _j dρ _j ∧ d ^c ρ _j ∧ ω _h

_j

= 0.

Similarly as above we have

(4)

¯ ¯

¯ ¯ Z

X

χ ◦ h _j ρ _j (ω _ϕ

²_j

− ω _ψ

²_j

)

¯ ¯

¯ ¯ =

¯ ¯

¯ ¯

¯ Z

{ϕ≤−j}

χ ◦ h _j ρ _j (ω

²

_ϕ

_j

− ω

²

_ψ

_j

)

¯ ¯

¯ ¯

¯

≤ C|χ(−j)|

ÃZ

{ϕ≤−j}

ω _ϕ

²_j

+ Z

{ψ≤−j}

ω

²

_ψ

_j

!

→ 0

because ϕ, ψ ∈ E ^p (X, ω) and p > 1/2. On the other hand, Z

X

χ ◦ h _j ρ _j (ω

²

_ϕ

_j

− ω

²

_ψ

_j

) = 2 Z

X

d(χ ◦ h _j ρ _j ) ∧ d ^c ρ _j ∧ ω _h

_j

. By (4) it is enough to estimate, using the Schwarz inequality,

¯ ¯

¯ ¯ Z

X

ρ _j χ ⁰ ◦ h _j dh _j ∧ d ^c ρ _j ∧ ω _h

_j

¯ ¯

¯ ¯

≤ C sZ

X

χ ⁰ ◦ h _j dρ _j ∧ d ^c ρ _j ∧ ω _h

_j

sZ

X

χ ⁰ ◦ h _j dh _j ∧ d ^c h _j ∧ ω _h

_j

.

In order to show that the last integral is bounded in j we write Z

X

χ ⁰ ◦ h j dh j ∧ d ^c h j ∧ ω h

j

= − Z

X

χ ◦ h j dd ^c h j ∧ ω h

j

≤ − Z

X

χ ◦ h _j ω

²

_h

_j

≤ −4 Z

X

χ ◦ ϕ _j ω

²

_ϕ

_j

4 ZBIGNIEW BÃLOCKI

by Lemma 2.3 in [9]. Now from (2) (note that 0 ≤ χ ⁰ ≤ 1) we thus get (3).

Proceeding as in [3] we write Z

X

dρ _j ∧ d ^c ρ _j ∧ ω = Z

X

dρ _j ∧ d ^c ρ _j ∧ ω _h

_j

− Z

X

dρ _j ∧ d ^c ρ _j ∧ dd ^c h _j ,

so by (2) it is enough to estimate the last integral. We have

− Z

X

dρ _j ∧ d ^c ρ _j ∧ dd ^c h _j = Z

X

dρ _j ∧ d ^c h _j ∧ dd ^c ρ _j = Z

X

dρ _j ∧ d ^c h _j ∧ (ω _ψ

_j

− ω _ϕ

_j

).

By the Schwarz inequality and since ω _ψ

_j

≤ 2ω _h

_j

¯ ¯

¯ ¯ Z

X

dρ _j ∧ d ^c h _j ∧ ω _ψ

_j

¯ ¯

¯ ¯

≤ 2 sZ

X

1

χ ⁰ ◦ h _j dρ _j ∧ d ^c ρ _j ∧ ω _h

_j

sZ

X

χ ⁰ ◦ h _j dh _j ∧ d ^c h _j ∧ ω _h

_j

.

The last integral is bounded in j. In our case we also have 1/χ ⁰ = 2χ and by (3) we get

j→∞ lim Z

X

dρ _j ∧ d ^c h _j ∧ ω _ψ

_j

= 0.

Similarly we show that

j→∞ lim Z

X

dρ _j ∧ d ^c h _j ∧ ω _ϕ

_j

= 0, and thus

j→∞ lim Z

X

dρ _j ∧ d ^c ρ _j ∧ ω = 0. ¤

For the proof of Theorem 1 in arbitrary dimension we will need some preparatory results.

Lemma 3. For p > 0, k = 1, . . . , n, and ϕ ∈ P SH(X, ω) ∩ L ^∞ (X) with ϕ ≤ −1 we have

(5)

Z

X

(−ϕ) ^p ω _ϕ ^n−k ∧ ω ^k ≤ Z

X

(−ϕ) ^p ω ⁿ _ϕ and

(6)

Z

X

(−ϕ) ^p−1 dϕ ∧ d ^c ϕ ∧ ω ^n−k _ϕ ∧ ω ^k−1 ≤ 1 p

Z

X

(−ϕ) ^p ω _ϕ ⁿ . Proof. Set T := ω _ϕ ^n−k ∧ ω ^k−1 . Then

Z

X

(−ϕ) ^p−1 dϕ ∧ d ^c ϕ ∧ T = − 1 p

Z

X

d((−ϕ) ^p ) ∧ d ^c ϕ ∧ T = 1 p

Z

X

(−ϕ) ^p dd ^c ϕ ∧ T.

Therefore the last integral is nonnegative and thus Z

X

(−ϕ) ^p ω ∧ T ≤ Z

X

(−ϕ) ^p ω _ϕ ∧ T, so by induction on k we get (5). We also obtain

Z

X

(−ϕ) ^p−1 dϕ ∧ d ^c ϕ ∧ T ≤ 1 p

Z

X

(−ϕ) ^p ω _ϕ ∧ T

which, by virtue of (5), gives (6). ¤

Lemma 4. For k = 0, 1, . . . , n − 1 set p _k := 1 − 2 ^−k . Assume that ϕ, ψ ∈ P SH(X, ω) ∩ L ^∞ (X) are ≤ −1 and denote ρ := ψ − ϕ, h := (ϕ + ψ)/2. Then for p ≥ p _n−1

Z

X

(−h) ^p−p

^k

dρ ∧ d ^c ρ ∧ ω _h ^n−1−k ∧ ω ^k ≤ C µZ

X

(−h) ^p dρ ∧ d ^c ρ ∧ ω _h ⁿ⁻¹

¶

₂^−k

,

where C is a positive constant depending only on n and on upper bounds for R

X (−h) ^p ω _h ⁿ and R

X (−h) ^p dρ ∧ d ^c ρ ∧ ω _h ⁿ⁻¹ .

Proof. We use induction on k. For k = 0 there is nothing to prove and we assume the estimate holds for k − 1. We may write the left-hand side as

Z

X

(−h) ^p−p

^k

dρ ∧ d ^c ρ ∧ ω h ∧ T − Z

X

(−h) ^p−p

^k

dρ ∧ d ^c ρ ∧ dd ^c h ∧ T,

where T = ω _h ^n−1−k ∧ ω ^k−1 . The first integral is now estimated by the inductive assumption (and since h ≤ −1), so it is enough to bound the second term from above. Note that for q ≥ 0 we have

−(−h) ^q dd ^c h = 1

q + 1 dd ^c ((−h) ^q+1 ) − q(−h) ^q−1 dh ∧ d ^c h ≤ 1

q + 1 dd ^c ((−h) ^q+1 ).

Therefore

− Z

X

(−h) ^p−p

^k

dρ ∧ d ^c ρ ∧ dd ^c h ∧ T

≤ 1

p − p _k + 1 Z

X

dρ ∧ d ^c ρ ∧ dd ^c ((−h) ^p−p

^k+1

) ∧ T

= − 1

p − p _k + 1 Z

X

d((−h) ^p−p

^k+1

) ∧ d ^c ρ ∧ dd ^c ρ ∧ T

= Z

X

(−h) ^p−p

^k

dh ∧ d ^c ρ ∧ (ω _ψ − ω _ϕ ) ∧ T.

Since ω _ψ ≤ 2ω _h , by the Schwarz inequality we get

¯ ¯

¯ ¯ Z

X

(−h) ^p−p

^k

dh ∧ d ^c ρ ∧ ω _ψ ∧ T

¯ ¯

¯ ¯

≤ 2 sZ

X

(−h) ^p−1 dh ∧ d ^c h ∧ ω _h ∧ T sZ

X

(−h) ^p−p

^k−1

dρ ∧ d ^c ρ ∧ ω _h ∧ T . Similarly we can deal with the term involving ω _ϕ and the required estimate follows from Lemma 3. ¤

Lemma 4 easily gives Theorem 1:

Proof of Theorem 1 for arbitrary n. Using the same notation as previously we can similarly as in the proof of (3) show that

j→∞ lim Z

X

(−h _j )

¹⁻²¹⁻ⁿ

dρ _j ∧ d ^c ρ _j ∧ ω _h ⁿ⁻¹

j

= 0.

6 ZBIGNIEW BÃLOCKI

Lemma 4 applied for k = n − 1, together with Lemma 2.3 in [9], now give

j→∞ lim Z

X

dρ _j ∧ d ^c ρ _j ∧ ω ⁿ⁻¹ = 0. ¤

For the proof of Theorem 2 we will need the following quantitative version of Example 2.14 in [9]:

Proposition 5. Assume that ψ ∈ P SH(X, ω) is negative and 0 < α < 1. Then

−(−ψ) ^α ∈ E ^p (X, ω) for p < (1 − α)/α.

Proof. Without loss of generality we may assume that ψ ≤ −1. Set ϕ := −(−ψ) ^α . We have

ω _ϕ = α(1 − α)|ψ| ^α−2 dψ ∧ d ^c ψ + α|ψ| ^α−1 ω _ψ + (1 − α|ψ| ^α−1 )ω

≤ α(1 − α)|ψ| ^α−2 dψ ∧ d ^c ψ + α|ψ| ^α−1 ω _ψ + ω and (for bounded ψ)

ω ⁿ _ϕ /C ≤

n−1 X

k=0

|ψ| ^{α−2+k(α−1)} dψ ∧ d ^c ψ ∧ ω _ψ ^k ∧ ω ^n−1−k

+ X n

l=1

|ψ| ^l(α−1) ω _ψ ^l ∧ ω ^n−l + ω ⁿ ,

where C is a positive constant depending on α and n. For a > 0 and T = ω _ψ ^k ∧ ω ^n−1−k we get

Z

X

(−ψ) ^−a−1 dψ ∧ d ^c ψ ∧ T = 1 a

Z

X

d(−ψ) ^−a ∧ d ^c ψ ∧ T

= − 1 a

Z

X

(−ψ) ^−a dd ^c ψ ∧ T

≤ 1 a

Z

X

(−ψ) ^−a ω ∧ T

≤ 1 a

Z

X

ω ∧ T

= 1 a

Z

X

ω ⁿ .

Therefore for b < 1 − α we obtain Z

X

|ψ| ^b ω ⁿ _ϕ ≤ C(n, α, b) µ

1 + Z

X

|ψ| ^b ω ⁿ

¶

and approximating arbitrary ψ by max{ψ, −j} the proposition follows. ¤ Proof of Theorem 2. For z ∈ C ⁿ set

ψ(z) := log |z

₁

| − 1

2 log(1 + |z|

²

) − 1.

It can be extended to a function from P SH(X, ω). Then e ψ := − √

−ψ / ∈ W

^1,2

(X) and by Proposition 5 e ψ ∈ E ^p (X, ω) for p < 1. Using Theorem 1 we then conclude that for any ϕ ∈ E(X, ω) satisfying ω ⁿ _ϕ = ω ⁿ

_e

ψ we have ϕ = e ψ +const / ∈ W

^1,2

(X). ¤

References

[1] E. Bedford, B.A. Taylor, Fine topology, ˇ Silov boundary, and (dd

^c

)

ⁿ

, J. Funct. Anal. 72 (1987), 225-251.

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

[3] Z. BÃlocki, Uniqueness and stability for the Monge-Amp`ere equation on compact K¨ ahler mani- folds, Indiana Univ. Math. J. 52 (2003), 1697-1702.

[4] Z. BÃlocki, On the definition of the Monge-Amp`ere operator in C

²

, Math. Ann. 328 (2004), 415-423.

[5] Z. BÃlocki, The domain of definition of the complex Monge-Amp`ere operator, Amer. J. Math.

128 (2006), 519-530.

[6] U. Cegrell, Pluricomplex energy, Acta Math. 180 (1998), 187-217.

[7] U. Cegrell, The general definition of the complex Monge-Amp`ere operator, Ann. Inst. Fourier 54 (2004), 159-179.

[8] S. Dinew, An inequality for mixed Monge-Amp`ere measures, preprint, 2007.

[9] V. Guedj, A. Zeriahi, The weighted Monge-Amp`ere energy of quasiplurisubharmonic func- tions, J. Funct. Anal. 443 (2007), 442-482.

[10] S. KoÃlodziej, The complex Monge-Amp`ere equation, Acta Math. 180 (1998), 69-117.

[11] 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

E-mail address: Zbigniew.Blocki@im.uj.edu.pl