Stability of solutions to the complex

Monge-Ampère equation on compact Kähler manifolds.

by

Sławomir Kołodziej

IMUJ PREPRINT 2001/07

Abstract. We prove a stability theorem for generalized solutions of the complex Monge-Ampère equation. The uniqueness and equicontinuity of families of solutions follow from it.

0.Introduction

We continue our study of the complex Monge-Ampère equation on compact Kähler manifolds initiated in [K2]. Let M be a compact n-dimensional Kähler manifold with the fundamental form ω given in local coordinates by

ω = \frac{i}{2} ∑_{k,j} g_{kj̄}dz^{k} ∧ dz̄^{j}.

We normalize the metric by

∫_{M} ω^{n} = 1.

In this paper a continuous function on M will be called ω-plurisubharmonic (ω-psh in short) if dd^{c}φ+ω ≥ 0, where d = ∂ +\overline{∂}, d^{c} = i(\overline{∂} −∂). The set of those functions we denote by P SH(ω). Elsewhere (e.g. [A2] [T]) such functions are often called admissible if they are also smooth and satisfy strict inequality dd^{c}φ + ω > 0.

We study ω-psh solutions φ of the complex Monge-Ampère equation

(0.1)  (ω + dd^{c}φ)^{n} = fω^{n},

Typeset by AMS-TEX

2000 Mathematics Subject Classification. primary 32U05, 53C55, secondary 32U40.

Key words and phrases. Kähler manifold, complex Monge-Ampère operator.

Partially supported by KBN Grant No. PO3A 010 19

where f ∈ L^{1}(M), f ≥ 0, ∫_{M} fω^{n} = 1 is a given function. The wedge product of positive currents on the left is defined as in [BT1].

The geometrical meaning of the equation (0.1) (with smooth positive f) is that given the volume form fω^{n} on M we seek for the Kähler metric (represented by the fundamental form ω + dd^{c}φ) which yields this volume form. More interestingly, as a short calculation shows (see [A1][A2] [T] [Y]), equation (0.1) arises when given a closed (1, 1) form τ representing the first Chern class of M we want to find a Kähler form ω′ such that τ = Ricc(ω′) (the Ricci form of ω′) and ω′ lives in the same Chern class as τ. E. Calabi conjectured that this is always possible. He also proved in [C] the uniqueness of such ω′ which is equivalent to the fact that any two solutions of (0.1) differ by a constant. The Calabi conjecture was confirmed by S.-T. Yau [Y] who proved the following theorem.

Theorem 0.1. Let f > 0, f ∈ C^{k}(M), k ≥ 3. Then there exists a solution to (0.1) belonging to Hölder class C^{k+1,α}(M) for any 0 ≤ α < 1.

The existence of generalized solutions of (0.1) was shown in [K2]. The result may be rephrased as follows (see Section 4 for details). Consider a family of functions

F(F) = {f ∈ L^{1}(M) : f ≥ 0, ∫_{M} fω^{n} = 1,
∫_{E} fω^{n} ≤ F (cap_{ω}(E)) for any Borel set E ⊂ M},

where cap_{ω} is a capacity defined in Section 1 and F (x) = \frac{Ax}{h(x^{−1/n})}, with A > 0 and h : ℝ_{+} → [1, ∞) which is nondecreasing and satisfies

∫_{1}^{∞} y^{−1}h^{−1/n}(y) dy < ∞.

Then for any f ∈ F(F) there exists a continuous solution of (0.1). Furthermore, if we normalize those solutions by max_{M} φ = 0 then there exists a constant a(F) > 0 such that φ > −a(F) for any φ solving (0.1) with f ∈ F(F).

In particular, for any p > 1 and any f ∈ L^{p}(M) such that f ≥ 0 and ∫_{M} fω^{n} = 1 the Monge-Ampère equation is solvable. Despite the fact that we deal with nonsmooth solutions the result may be useful for geometers (cf. [TZ]).

The primary objective of the present paper is to establish a stability result for those generalized solutions (Theorem 4.1). For the solutions of

(ω + dd^{c}φ)^{n} = fω^{n}, (ω + dd^{c}ψ)^{n} = gω^{n}

(suitably normalized) we find a bound on sup norm of |φ − ψ| in terms of ||f − g||_{L^{1}(M)} when f, g ∈ F(F). Such an estimate is also new for classical solutions. In the proof of Theorem 0.1 stability estimates are established between norms in C^{k,α} spaces for k ≥ 3.

As an application of the main result one obtains the following explicit estimate in the case the right hand side is bounded in L^{p} norm p > 1. For any ϵ > 0 and c_{0} > 0 there exists a constant c(ϵ, c_{0}, p) such that for all f, g with ||f||_{L^{p}} < c_{0}, ||g||_{L^{p}} < c_{0} we have

sup |φ − ψ| ≤ c(ϵ, c_{0}, p)||f − g||_{L^{p}}^{1/(n+1+ϵ)}.

Having such an estimate one can deduce the equicontinuity of solutions of (0.1) when f belongs to the set

F_{p}(c) = {f ∈ L^{p}(M) : f ≥ 0, ∫_{M} f dV = 1, ∫_{M} f^{p}dV ≤ c},

for p > 1 and c > 0. The main stability result also implies that for f ∈ F(F) the solution of the Monge-Ampère equation is unique up to an additive constant which is a generalization of Calabi’s theorem. It would be interesting to know if the uniqueness holds in general for ω-psh functions. A particular case M = ℙ^{n} was treated in [BT3].

In the proofs we use the methods of pluripotential theory. In Section 2 the comparison principle for ω-psh on compact Kähler manifolds is shown which might be of independent interest. For more backround in pluripotential theory we refer to [K][K2][K3].

Notation: We shall denote by || · ||_{p} the norm in L^{p}(M) for p ∈ [1, ∞].

For the sake of brevity we shall write

ω_{φ} = ω + dd^{c}φ.

1. Preliminaries

We work throughout on a compact Kähler manifold M with a fundamental form ω and assume

∫_{M} ω^{n} = 1.

If in an open subset of M there exists a potential function v satisfying ω = dd^{c}v then for ω-psh φ the function v + φ is a true plurisubharmonic function. Thus such properties of plurisubharmonic functions as Hartogs’ lemma or the theorem saying that the weak convergence implies convergence in L_{loc}^{p} (see e.g. [H]) hold also for ω-psh functions. The same goes for the convergence theorems, proved by Bedford and Taylor [BT2], which say that the Monge-Ampère measures associated to a convergent monotone sequence of plurisubharmonic functions converge. For a Borel set E ⊂ M one can define a capacity

cap_{ω}(E) = sup{∫_{E} ω_{φ}^{n} : φ ∈ P SH(ω), 0 ≤ φ ≤ 1}.

This is modeled on Bedford-Taylor capacity introduced in [BT2].

Let us consider two open finite coverings {V_{s}}, {V′_{s}}, s = 1, 2, ..., N of M such that V′_{s} ⊂ V_{s} and in each strictly pseudoconvex V_{s} there exists v_{s} ∈ P SH(V_{s}) with dd^{c}v_{s} = ω and v_{s} = 0 on ∂V_{s}. Given a compact set K ⊂ M define K_{s} = K ∩ \overline{V′_{s}}.

We are going to show that cap_{ω}(K) is comparable with cap′_{ω}(K) = ∑_{s} cap(K_{s}, V_{s}), where cap(K, V) denotes the Bedford-Taylor capacity. From [BT2] we know that

cap(K, V) = ∫_{K}(dd^{c}u_{K,V})^{n}

with

u_{K,V} = sup{u ∈ P SH(V) : u < 0, u ≤ −1 on K}.

For fixed s put φ_{s} = u_{K_{s},V_{s}} − v_{s}. Then φ_{s} ≥ −1 on K_{s} and φ_{s} = 0 on ∂V_{s}. One can find ψ_{s} ∈ P SH(ω) ∩ C^{∞}(M) such that ψ_{s} = 0 outside V_{s} and ψ_{s} ≤ −3δ < 0 on V′_{s} with the same δ for all s. (To see this just take any smooth ψ′_{s} which is equal to 0 outside V_{s} and negative on V′_{s} and choose ϵ > 0 so small that ψ_{s} = ϵψ′_{s} is ω-psh.) Take χ_{s} which is equal to max(δφ_{s} − δ, ψ_{s}) on V_{s} and equal to 0 elsewhere on M. Note that this function is ω-psh and equal to δφ_{s} − δ on a neighbourhood of K_{s}. Therefore

\begin{aligned}∫_{K_{s}} ω_{χ_{s}}^{n} & = ∫_{K_{s}}[δω_{φ_{s}} + (1 − δ)ω]^{n} \\ & ≥ δ^{n} ∫_{K_{s}} ω_{φ_{s}}^{n} = δ^{n} ∫_{K_{s}}(dd^{c}u_{K_{s},V_{s}})^{n} = δ^{n}cap(K_{s}, V_{s}). \end{aligned}

So

cap_{ω}(K) ≥ cap_{ω}(K_{s}) ≥ δ^{n}cap(K_{s}, V_{s})

By the definitions,

cap_{ω}(K) ≤ (C_{1} + 1)^{n} ∑_{s} cap(K_{s}, V_{s}),

where C_{1} is chosen so that v_{s} ≥ −C_{1} for any s. Indeed, let φ ∈ P SH(ω) with −1 < φ ≤ 0. Then (C_{1}+1)^{−1}(φ+v_{s}) is a competitor in the definition of cap(K_{s}, V_{s}) (see [BT2]). So

\begin{aligned}(C_{1} + 1)^{−n} ∫_{K} ω_{φ}^{n} & ≤ (C_{1} + 1)^{−n} ∑_{s} ∫_{K_{s}} ω_{φ}^{n} \\ & ≤ ∑_{s} cap(K_{s}, V_{s}) \end{aligned}

which proves the preceding inequality. Thus we finally obtain

(1.1)  \frac{δ^{n}}{N} cap′_{ω}(K) ≤ cap_{ω}(K) ≤ (C_{1} + 1)^{n}cap′_{ω}(K).

A sequence φ_{j} of functions defined in M is said to converge with respect to capacity to φ if for any t > 0 and a Borel K ⊂ M

lim_{j→∞} cap(K ∩ {|φ − φ_{j}| > t}, Ω) = 0.

The following lemma is taken from Demailly’s paper [D1] who attributes it to Richberg [R] and Green, Wu [GW] Lemma 1.1. If φ ∈ P SH(ω) and γ is a continuous (1, 1) form on M such that dd^{c}φ ≥ γ (as currents) then given δ > 0 one can find a smooth function ψ satisfying φ < ψ < φ + δ and dd^{c}ψ ≥ γ − δω on M.

Next lemma is well known for smooth forms. We need a more general version (cf [D2]).

Lemma 1.2. Suppose g ∈ L^{1}(M) and φ, ψ ∈ P SH(ω) satisfy

ω_{φ}^{n} ≥ gω^{n}, ω_{ψ}^{n} ≥ gω^{n}.

Then ω_{φ}^{k} ∧ ω_{ψ}^{n−k} ≥ gω^{n}.

Proof. The statement is local and so it is equivalent to the following one: For u, v ∈ P SH(B) ∩ C(\overline{B}) (B - a ball in ℂ^{n}) satisfying

(dd^{c}u)^{n} ≥ g dV, (dd^{c}v)^{n} ≥ g dV

we have (dd^{c}u)^{k} ∧ (dd^{c}v)^{n−k} ≥ g dV, where dV denotes the Lebesgue measure and g ∈ L^{1}(B).

For smooth u, v and g > 0 it is a well known matrix inequality which follows from concavity of the mapping A → log det^{1/n} A defined on the set of positive definite Hermitian matrices (see Corollary 7.6.9 in [HJ]). Next we shall prove it for g ∈ L^{2}(B). Let g_{j} be a sequence of smooth positive functions, positive on \overline{B} and tending in L^{2}(B) to g. Fix also two sequences f_{j}, h_{j} of smooth functions on ∂B such that f_{j} → u and h_{j} → v uniformly on ∂B. Applying [CKNS] one can find u_{j}, v_{j} ∈ P SH(B) ∩ C^{∞}(\overline{B}) solving the Dirichlet problems for the Monge-Ampère equation

\begin{aligned}(dd^{c}u_{j})^{n} & = g_{j}dV, u_{j} = f_{j} on ∂B \\ (dd^{c}v_{j})^{n} & = g_{j}dV, v_{j} = h_{j} on ∂B. \end{aligned}

By L^{2} estimates from [K1] u_{j} and v_{j} tend uniformly to u and v respectively. Hence one can apply the convergence theorem [BT2] and the statement for smooth functions to the effect

\begin{aligned}(dd^{c}u)^{k} ∧ (dd^{c}v)^{n−k} & = lim_{j→∞}(dd^{c}u_{j})^{k} ∧ (dd^{c}v_{j})^{n−k} \\ & ≥ lim_{j→∞} g_{j} dV ≥ g dV, \end{aligned}

For the general case we take an increasing sequence g_{j} ↑ g with g_{j} ∈ L^{2}(B) and repeat the above argument using [K1] to solve the suitable Dirichlet problems. Now the convergence of the approximating sequences is not uniform, but the sequences are decreasing due to the comparison principle [BT2]. So the convergence theorem still applies in this case.

2. Comparison principle

Now we shall prove the comparison principle for the Monge-Ampère operator on compact Kähler manifolds.

Theorem 2.1. If φ and ψ are ω-psh on M then for Ω = {φ < ψ} we have

∫_{Ω} ω_{ψ}^{n} ≤ ∫_{Ω} ω_{φ}^{n}.

Proof. Suppose first that φ, ψ and the boundary of Ω are smooth. Set φ_{t} = max(φ+ t, ψ), t > 0. Then close to ∂Ω we have φ_{t} = φ + t. Define the (closed) current

T_{t} = ∑_{k=1}^{n} \binom{n}{k}(dd^{c}φ_{t})^{k−1} ∧ ω^{n−k}

and set T = lim_{t→0} T_{t}. By Stokes’ theorem

\begin{aligned}∫_{Ω} ω_{φ_{t}}^{n} & = ∫_{Ω} dd^{c}φ_{t} ∧ T_{t} + ω^{n} = ∫_{∂Ω} d^{c}φ_{t} ∧ T_{t} + ∫_{Ω} ω^{n} \\ & = ∫_{∂Ω} d^{c}φ ∧ T + ∫_{Ω} ω^{n} = ∫_{Ω} ω_{φ}^{n} \end{aligned}

Since φ_{t} ↓ ψ in Ω as t → 0 we get applying the convergence theorem [BT2] that

ω_{φ_{t}}^{n} → ω_{ψ}^{n} in Ω

as measures. Hence for a test function χ in Ω with 0 ≤ χ ≤ 1 we get

∫_{Ω} χω_{ψ}^{n} = lim_{t→0} ∫_{Ω} χω_{φ_{t}}^{n} ≤ lim inf_{t→0} ∫_{Ω} ω_{φ_{t}}^{n}.

So

∫_{Ω} ω_{ψ}^{n} ≤ lim inf_{t→0} ∫_{Ω} ω_{φ_{t}}^{n} = ∫_{Ω} ω_{φ}^{n},

which completes the proof for smooth functions. Suppose now that φ and ψ are continuous and that they satisfy the extra assumption

(2.1)  dd^{c}φ ≥ (δ − 1)ω, dd^{c}ψ ≥ (δ − 1)ω

for some δ > 0. Then, applying Lemma 1.1, one can find two sequences of ω-psh functions φ_{j} and ψ_{j} converging uniformly to φ and ψ respectively. Given a compact set K ⊂ Ω we find t > 0 and a positive integer j_{0} such that K ⊂ Ω(t, j) = {φ_{j} < ψ_{j} −t} ⊂ Ω for j > j_{0} and the boundary of Ω(t, j) is smooth (using Sards’ theorem). Applying the first part of the proof and the convergence theorem we obtain

∫_{K} ω_{ψ}^{n} ≤ lim inf_{j→∞} ∫_{Ω(t,j)} ω_{ψ_{j}}^{n} ≤ lim inf_{j→∞} ∫_{Ω(t,j)} ω_{φ_{j}}^{n} ≤ ∫_{Ω} ω_{φ}^{n}.

Exhausting Ω by compact sets we get the desired inequality in this case. It remains to get rid of the extra assumption. Note that for fixed t ∈ (0, 1) and ω-psh functions φ, ψ the functions tφ and tψ satisfy (2.1) for some δ > 0. Fix a compact set K ⊂ Ω = {φ < ψ} and consider δ > 0 and t ∈ (0, 1) such that K ⊂ Ω(δ, t) = {φ < ψ − δ/t}. By the above and the convergence theorem we have

∫_{K} ω_{ψ}^{n} ≤ lim inf_{t→1} ∫_{Ω(δ,t)} ω_{tψ}^{n} ≤ lim inf_{t→1} ∫_{Ω(δ,t)} ω_{tφ}^{n} ≤ ∫_{Ω} ω_{φ}^{n}.

Again to complete the proof it is enough to consider an exhaustion sequence of compact subsets of Ω.

Remark. In the definition of ω-psh function we assumed that it is continuous (since only with such functions we deal in this paper). In the comparison principle this assumption is redundant. This can be shown using quasicontinuity of plurisubharmonic functions as in [BT2] (see also [K3]).

The following lemma is a variation of Lemma 2.3.1 in [K2]. The reason the proof must be modified is that, unlike for plurisubharmonic functions, a sum of ω-psh is not necessarily ω-psh.

Lemma 2.2. Let φ and ψ be ω-psh functions on M with 0 ≤ ψ ≤ C. Assume that U(s) := {φ − s < ψ} are nonempty for s ∈ [S, S + D]. Suppose that for some positive number A and an increasing function h : ℝ_{+} → (1, ∞) satisfying

∫_{1}^{∞}(yh^{1/n}(y))^{−1} dy < ∞,

the following inequality holds

(2.2)  ∫_{K} ω_{φ}^{n} ≤ F (cap_{ω}(K)), with F (t) = \frac{At}{h(t^{−1/n})}, A > 0,

for any compact set K. Then

D ≤ κ(a(S + D)), a(s) := cap_{ω}(U(s))

where

κ(s) = c(n)A^{1/n}(1 + C)[∫_{s^{−1/n}}^{∞} y^{−1}h^{−1/n}(y) dy + h^{−1/n}(s^{−1/n})].

Proof. Set

b(s) = ∫_{U(s)} ω_{φ}^{n}.

First we shall prove the inequality

(2.3)  t^{n}a(s) ≤ b(s + t + Ct) for 0 < t < \frac{S + D − s}{C + 1}.

Indeed, take ρ ∈ P SH(ω) with −1 ≤ ρ ≤ 0 and put V (s) := {φ − s − t − Ct < tρ + (1 − t)ψ}. One easily verifies that U(s) ⊂ V (s) ⊂ U(s + t + Ct). We can now apply the comparison principle (Theorem 2.1) to obtain

\begin{aligned}t^{n} ∫_{U(s)} ω_{ρ}^{n} & ≤ ∫_{V (s)}(tω_{ρ} + (1 − t)ω_{ψ})^{n} \\ & ≤ ∫_{V (s)} ω_{φ}^{n} ≤ ∫_{U(s+t+Ct)} ω_{φ}^{n} = b(s + t + Ct). \end{aligned}

Taking supremum over ρ we get (2.3).

Next we define an increasing sequence s_{0}, s_{1}, ..., s_{N}, setting s_{0} := S and

s_{j} := sup{s : a(s) ≤ lim_{t→s_{j−1}+} ea(t)}

for j = 1, 2, ..., N. Then this sequence is increasing and

a(s_{j}) ≥ ea(s_{j−2}).

The integer N is chosen to be the greatest one satisfying s_{N} ≤ S + D. Then

a(S + D) ≤ lim_{t→s_{N} +} ea(t)

(otherwise we would have s_{N+1} ≤ S +D.) From the last inequality, the assumptions and (2.3) it follows that for any s ∈ (s_{N}, S + D) we have

(\frac{S + D − t}{1 + C})^{n}a(s) ≤ b(S + D) ≤ F (a(S + D)) ≤ Aea(s)h^{−1}([a(S + D)]^{−1/n}).

Hence

(2.4)  S + D − s_{N} ≤ (Ae)^{1/n}(1 + C)h^{−1/n}([a(S + D)]^{−1/n})

Consider two numbers S < s′ < s < S + D such that a(s) ≤ ea(s′) and take t := \frac{s−s′}{1+C} . Therefore, by the assumptions and (2.2), we have

\begin{aligned}a(s′) & ≤ t^{−n}b(s) ≤ At^{−n}a(s)h^{−1}([a(s)]^{−1/n}) \\ & ≤ Aet^{−n}a(s′)h^{−1}([a(s)]^{−1/n}). \end{aligned}

Then proceeding like in [K2] we obtain (letting s to s_{j+1} and s′ to s_{j}) (2.5)

\begin{aligned}s_{N} − S & = ∑_{j=0}^{N−1} t_{j} \\ & ≤ 2(Ae)^{1/n}[n ∫_{[a(S+D)]^{−1/n}}^{∞}[yh^{1/n}(y)]^{−1} dy + (h(a(S + D)^{−1/n}))^{−1/n}]. \end{aligned}

It follows from (2.4) and (2.5) that

D ≤ 2n(Ae)^{1/n}(1 + C)[∫_{[a(S+D)]^{−1/n}}^{∞}[yh^{1/n}(y)]^{−1} dy + (h(a(S + D)^{−1/n}))^{−1/n}].

The proof is complete.

3. Convergence with respect to capacity

By [X] and (1.1) the convergence with respect to capacity of a sequence of ω-psh functions entails the convergence of Monge-Ampère measures associated to those functions. Below we prove that a weakly convergent sequence of solutions of (0.1) for the right hand side uniformly bounded in L^{p} norm, p > 1, is convergent with respect to capacity. Later, using this we shall show that it is in fact uniformly convergent.

Lemma 3.1. Let φ_{j} ∈ P SH(ω) and φ_{j} → φ ∈ P SH(ω) in the sense of distributions. Suppose that

ω_{φ_{j}}^{n} = f_{j}ω^{n}

where the norms ||f_{j}||_{L^{p}(ω^{n})} are uniformly bounded by C for some p > 1. Then

cap_{ω}({φ_{j} + 2s < φ}) ≤ C(||φ|| + 1)s^{−n−1}||φ − φ_{j}||_{L^{q}(ω^{n})},

where q is such that p^{−1} + q^{−1} = 1.

In particular φ_{j} → φ with respect to capacity.

Proof. Denote

E_{j}(s) = {φ_{j} + s < φ} and a_{j} := cap_{ω}(E_{j}(2s)).

Suppose 0 < φ < N − 1. Take ρ ∈ P SH(ω) with −1 ≤ ρ ≤ 0 and set V := {φ_{j} < \frac{s}{N} ρ + (1 − \frac{s}{N})φ − s\}. Then

E_{j}(2s) ⊂ V ⊂ E_{j}(s).

By comparison principle

\begin{aligned}\frac{s^{n}}{N^{n}} ∫_{E_{j}(2s)} ω_{ρ}^{n} & ≤ ∫_{V} (\frac{s}{N} ω_{ρ} + (1 − \frac{s}{N})ω_{φ})^{n} \\ & ≤ ∫_{V} ω_{φ_{j}}^{n} ≤ ∫_{E_{j}(s)} f_{j}ω^{n}. \end{aligned}

Taking supremum over ρ one gets

(3.1)  \frac{s^{n}}{N^{n}} a_{j} ≤ ∫_{E_{j}(s)} f_{j}ω^{n}.

Hence, applying Hölder’s inequality we infer

\begin{aligned}s^{n+1}N^{−n}a_{j} & ≤ ∫_{M} |φ − φ_{j}|f_{j}ω^{n} ≤ (∫_{M} |φ − φ_{j}|^{q}ω^{n})^{1/q}(∫_{M} f_{j}^{p}ω^{n})^{1/p} \\ & ≤ C(∫_{M} |φ − φ_{j}|^{q}ω^{n})^{1/q}. \end{aligned}

Thus we have obtained the estimate from the statement. It is known (see e.g. [H]) that a sequence of plurisubharmonic functions which is weakly convergent converges also in L_{loc}^{q} for any 1 ≤ q < ∞. The same is true for ω-psh functions if we apply the above statement for

v + φ_{j} → v + φ

where v is a local potential for ω. Therefore

lim_{j→∞} a_{j} ≤ lim_{j→∞} CN^{n}s^{−n−1}||φ − φ_{j}||_{L^{q}(ω^{n})} = 0.

This proves our statement.

4. Stability

In this paper we call a function F : ℝ_{+} → ℝ_{+} admissible if F has the form

F (x) = \frac{Ax}{h(x^{−1/n})},

with A > 0 and h : ℝ_{+} → [1, ∞) is nondecreasing and such that

∫_{1}^{∞} y^{−1}h^{−1/n}(y) dy < ∞.

We shall study the stability of the solutions of the complex Monge-Ampère equation when the function on the right hand side belongs to the set

F(F) = {f ∈ L^{1}(M) : f ≥ 0, ∫_{M} fω^{n} = 1,
∫_{E} fω^{n} ≤ F (cap_{ω}(E)) for any Borel set E ⊂ M},

where F is an admissible function.

It easily follows from the results of [K2] and Lemma 2.2 that the Monge-Ampère equation has a continuous solution for any f ∈ F(F). Indeed, let us construct f_{j} = t_{j}g_{j}, where g_{j} = min(f, j) and t_{j} > 0 is chosen so that ∫_{M} f_{j}ω^{n} = 1. Since f ∈ L^{1}(M) we have lim_{j→∞} t_{j} = 1 and so for j big enough f_{j} ∈ F(2F). Therefore the ω-psh solutions of

ω_{φ_{j}}^{n} = f_{j}ω^{n}, max_{M} φ_{j} = 0

are uniformly bounded (see Lemma 2.2). Those solutions exist by [K2] since f_{j} ∈ L^{2}(M). Finally, Lemma 2.2.1 from [K2] shows that φ = (lim sup φ_{j})^{∗} solves

ω_{φ}^{n} = fω^{n}.

By Lemma 2.2, given admissible F, there exists a positive constant which shall be denoted by a(F) such that for any f ∈ F(F) the ω-psh solution φ of

ω_{φ}^{n} = fω^{n}, max_{M} φ = 0,

satisfies φ ≥ −a(F). We shall denote by κ_{F} the function from Lemma 2.2 with C = a(2F). Thus, for F (x) = \frac{Ax}{h(x^{−1/n})} we set

κ_{F} (s) = c(n)A^{1/n}(1 + a(2F))[∫_{s^{−1/n}}^{∞} y^{−1}h^{−1/n}(y) dy + h^{−1/n}(s^{−1/n})].

Theorem 4.1. Let us consider two functions f and g belonging to F(F) for some admissible F and the corresponding solutions of

ω_{φ}^{n} = fω^{n}, ω_{ψ}^{n} = gω^{n}

normalized by

max_{M}(φ − ψ) = max_{M}(ψ − φ).

Set q = q(n) = (\frac{2}{3})^{1/n} and δ = δ(n) = \frac{1}{2}(q − 1)(1 − 2^{−n}). Define an incraesing function on ℝ_{+} by

γ(t) = \frac{(2a(2F))^{n}}{(a(2F) + 1)^{n}} \frac{δ}{δ + 1}κ_{F}^{−1}(t)

(where κ_{F}^{−1}(t) denotes the inverse of κ_{F} (t)). Then the inequality

||f − g||_{1} ≤ γ(t)t^{n+1}

implies

||φ − ψ||_{∞} ≤ (4a(2F) + 2)t

for t < t_{0} with t_{0} > 0 depending on γ.

Proof. Put a = a(2F). One can assume that

(4.1)  ∫_{{ψ<φ}}(f + g)ω^{n} ≤ 1

since otherwise we may interchange the roles of φ and ψ. Since lim_{t→0} γ(t) = 0 one can fix t_{0} < δ/(2δ + 1) and such that γ(t_{0})t_{0}^{n+1} < 1/3. From now on we shall work with fixed t < t_{0}. Denote by E_{k} the set {ψ < φ − kat} and put

C_{0} = ∫_{E_{2}} gω^{n}.

Then by (4.1) and the assumptions

(4.2)  \begin{aligned}∫_{E_{0}} gω^{n} & = \frac{1}{2} ∫_{E_{0}}[(f + g) + (g − f)]ω^{n} \\ & ≤ \frac{1}{2}(1 + γ(t_{0})t_{0}^{n+1}) ≤ \frac{2}{3}. \end{aligned}

Define ω-psh function ρ as the solution of

ω_{ρ}^{n} = g_{1}ω^{n}, max_{M} ρ = 0.

where g_{1} = (3/2)g on E_{0} and h is equal to a constant c_{0} ≥ 0 elsewhere, with c_{0} chosen so that ∫_{M} g_{1}ω^{n} = 1. (Observe that (4.2) implies c_{0} ≥ 0.) The solution belongs to F(2F) and so

ρ ≥ −a.

Adding the same constant to φ and ψ (which does not influence neither the hypothesis nor the assertion) one can assume that

−a ≤ φ ≤ 0.

The last two inequalities entail

E_{2} ⊂ E := {ψ < (1 − t)φ + tρ − at} ⊂ E_{0}.

Let us denote by G the set {f < (1−t)g}. From Lemma 1.2 we know that for k < n the following inequalities hold on E_{0} \ G :

ω_{φ}^{k} ∧ ω_{ρ}^{n−k} ≥ q^{n−k}(1 − t)^{k/n}gω^{n} ≥ q(1 − t)gω^{n}.

Therefore on E_{0} \ G

(4.3)  ω_{tρ+(1−t)φ}^{n} ≥ (q − \frac{q − 1}{2^{n}})(1 − t)gω^{n}.

By the assumptions we have

t ∫_{G} gω^{n} ≤ ∫_{G}(g − f)ω^{n} ≤ γ(t)t^{n+1}.

Hence

(4.4)  ∫_{G} gω^{n} ≤ γ(t)t^{n}.

The following chain of inequalities is obtained by applying, in turn, formula (4.3), the comparison principle, and formula (4.4):

(4.5)  \begin{aligned}(q − \frac{q − 1}{2^{n}})(1 − t) ∫_{E\G} gω^{n} & ≤ ∫_{E} ω_{tρ+(1−t)φ}^{n} ≤ ∫_{E} gω^{n} \\ & ≤ ∫_{E\G} gω^{n} + γ(t)t^{n}. \end{aligned}

Since, by the choice of of δ and t_{0}

(q − \frac{q − 1}{2^{n}})(1 − t) − 1 = (2δ + 1)(1 − \frac{δ}{2δ + 1}) − 1 = δ,

we infer from (4.5) that

δ ∫_{E\G} gω^{n} ≤ γ(t)t^{n}.

Therefore

δ(C_{0} − γ(t)t^{n}) ≤ δ(∫_{E_{2}} gω^{n} − ∫_{G} gω^{n}) ≤ γ(t)t^{n},

and further

C_{0} ≤ (1 + \frac{1}{δ})γ(t)t^{n}.

We also have an estimate which is analogous to (3.1) with the suitable change of notation

cap_{ω}(E_{4}) ≤ \frac{(||(1 − t)φ + tρ|| + 1)^{n}}{(2at)^{n}} ∫_{E_{2}} gω^{n}.

Observe that −a ≤ (1−t)φ+tρ ≤ 0. So coupling the last two estimates one obtains

\begin{aligned}cap_{ω}(E_{4}) & ≤ (2ta)^{−n}(a + 1)^{n}C_{0} ≤ (a + 1)^{n}(2a)^{−n}(1 + δ^{−1})γ(t) \\ & ≤ (a + 1)^{n}(2a)^{−n}(1 + δ^{−1})γ(t). \end{aligned}

Suppose E′ = {ψ < φ − (4a + 2)t} were nonempty. Then by Lemma 2.2, the above estimate and the definition of γ we would have

2t ≤ κ_{F} (cap_{ω}(E_{4})) ≤ κ_{F} ((a + 1)^{n}(2a)^{−n}(1 + δ^{−1})γ(t)) = t

which is a contradiction. Therefore E′ is empty which translates into the desired estimate

max(ψ − φ) = max(φ − ψ) ≤ (4a + 2)t.

The proof is completed.

It follows from Theorem 4.1 that for f ∈ F(F) with admissible F the solution of (0.1), normalized by max_{M} φ = 0, is unique. This ia a generalization of the result of Calabi [C].

Corollary 4.2. If φ_{1} and φ_{2} solve

ω_{φ_{1}}^{n} = fω^{n} = ω_{φ_{2}}^{n}

with f ∈ F(F) for some admissible F then φ_{1} − φ_{2} = const.

Next we shall make the estimate from Theorem 4.1 more explicit in the case when f, g belong to a bounded set in L^{p}(ω^{n}). For p > 1 we denote by F_{p}(c_{0}) the set of functions

{f ∈ L^{p}(M) : f ≥ 0, ∫_{M} fω^{n} = 1, ∫_{M} f^{p}ω^{n} ≤ c_{0}}.

An analogue of the following estimate was proved in [K4] for subsets of strictly pseudoconvex domains in ℂ^{n}.

Lemma 4.3. Let p > 1, and f ∈ L^{p}(M). Then for some c_{M} > 0 and any compact K in M

∫_{K} f dV ≤ c(p, M)||f||_{p}a(K) exp(−c_{M}a^{−1/n}(K))

where we use the notation a(K) = cap_{ω}(K) and the constant c(p, M) depends on p, and M.

Proof. Consider the covering V_{s}, the sets K_{s} and numbers δ, N as in Section 1. Then, by Lemma 4 in [K4] we have for a_{s} = cap(K_{s}, V_{s}) and m < min(4π−1, p−1):

∫_{K} f dV ≤ ∑_{s} ∫_{K_{s}} f dV ≤ c′(p, M)||f||_{p} ∑_{s} a_{s} exp(−ma_{s}^{−1/n}).

Using the equivalence of cap_{ω} and cap′_{ω} from Section 1 we therefore get

∫_{K} f dV ≤ c′(p, M)δN||f||_{p}a(K) exp(−mδ^{−1/n}a^{−1/n}(K)).

So we get the statement with c(p, M) = c′(p, M)δN and c_{M} = mδ^{−1/n}.

It follows from Lemma 4.3 that

F_{p}(c_{0}) ⊂ F(F)

with F (x) = Ac_{0} exp(−c_{M}x^{−1/n}) for some A > 0 depending on and M. We estimate κ_{F} for F given above:

κ_{F} (s) ≤ c(n)A^{1/n}(1 + a(F))(1 + \frac{n}{c_{M}}) exp(−\frac{c_{M}}{n} s^{−1/n}) ≤ c_{F} exp(−\frac{c_{M}}{n} s^{−1/n})

where the constant c_{F} depends on F. Hence

\begin{aligned}γ(t) & = (2a(2F))^{n}(a(2F) + 1)^{−n} \frac{δ}{δ + 1}κ_{F}^{−1}(t) \\ & ≥ (2a(2F))^{n}(a(2F) + 1)^{−n} \frac{δ}{δ + 1}(−\frac{n}{c_{M}} log \frac{t}{c_{F}} )^{−n} = c_{0} log^{−n} \frac{c_{F}}{t} . \end{aligned}

Thus we have obtained the following corollary to Theorem 4.2.

Corollary 4.4. Given p > 1 and c_{0} > 0 there exist positive constants α, β and t_{0} such that for f, g ∈ F_{p}(c_{0}) satisfying

||f − g||_{1} ≤ α(− log t)^{−n}t^{n+1}, t < t_{0},

one can find ω-psh solutions of the equations

ω_{φ}^{n} = fω^{n}, ω_{ψ}^{n} = gω^{n}

subject to

||φ − ψ||_{∞} ≤ βt, t < t_{0}.

In particular for any ϵ > 0 there exists c(ϵ) with

||φ − ψ||_{∞} ≤ c(ϵ)||f − g||_{1}^{1/(n+1+ϵ)}

(for suitably normalized φ and ψ).

Let us denote by P_{p}(c) the inverse image of F_{p}(c) via the Monge-Ampère operator

P_{p}(c) = {φ ∈ P SH(ω) : ω_{φ}^{n} = fω^{n}, ||f||_{p} ≤ c}.

In this family the weak convergence is equivalent to the uniform convergence.

Theorem 4.5. Suppose φ_{j} ∈ P_{p}(c) and φ_{j} → φ weakly. Then φ_{j} tends to φ uniformly.

Proof. As it was mentioned in Section 1, under our assumptions, φ_{j} → φ in L^{q}(M), where q is chosen to be conjugate to p. Applying Lemma 3.1 (and its notation) we have for given s > 0

cap_{ω}(E_{j}(2s)) → 0

as j → ∞. Then from Lemma 2.2 we deduce that E_{j}(2s) is empty for j large enough. Since, by Hartogs’ lemma, φ = (lim sup φ_{j})^{∗} the statement follows.

Corollary 4.6. P_{p}(c) is equicontinuous.

Proof. Arguing by contradiction, suppose that for some ϵ > 0, φ_{j} ∈ P_{p}(c) and for some two sequences z_{j}, w_{j} of points in M we had dist(z_{j} − w_{j}) < j^{−1} and φ_{j}(z_{j}) − φ_{j}(w_{j}) > ϵ. One can pick subsequences z_{j_{k}}, w_{j_{k}} converging to z and φ_{j_{k}} converging in L^{1} norm to φ ∈ P SH(ω). By Theorem 4.5 φ_{j_{k}} converges uniformly. From this fact and the triangle inequality we easily deduce a contradiction for k large enough:

|φ_{j_{k}}(z_{j_{k}}) − φ_{j_{k}}(w_{j_{k}})| < ϵ.

Theorem 4.5 and the corollary following it hold in wider families of solutions. For ψ : ℝ_{+} → ℝ_{+} such that ψ(x)/x increases to ∞ as x → ∞ set

L_{c}^{ψ}(M) = {g ∈ L^{1}(M) : ∫_{M} ψ(g)ω^{n} ≤ c}.

Then for any admissible F the inverse image through the Monge-Ampère operator of F(F) ∩ L_{c}^{ψ}(M) is equicontinuous. As an example: the solutions corresponding to f ∈ L_{c}^{ψ}(M) with ψ(x) = x log^{m}(1 + x), m > n are equicontinuous. To show this one has to strengthen Lemma 3.1 along the same lines as in [K4, proof of Theorem 1].

References

[A1] T. Aubin, Equations du type Monge-Ampère sur les varietes kählériennes compactes. Bull. Sci. Math. 102 (1978), 63-95.

[A2] T. Aubin, Nonlinear Analysis on Manifolds. Monge-Ampère Equations, Grundl. d. math. Wiss. 244, Springer-Verlag, Berlin-Heidelberg-New-York, 1982.

[BT1] E. Bedford and B.A. Taylor, The Dirichlet problem for the complex Monge-Ampère operator. Invent. Math. 37 (1976), 1-44.

[BT2] E. Bedford and B.A. Taylor, A new capacity for plurisubharmonic functions, Acta Math. 149 (1982), 1-40.

[BT3] E. Bedford and B.A. Taylor, Uniqueness for the complex Monge-Ampère equation

for functions of logarithmic growth, Indiana Univ. Math. J. 38 (1989), 455-469.

[CKNS] L. Caffarelli, J.J. Kohn, L. Nirenberg and J. Spruck , The Dirichlet problem for Nonlinear Second-Order Elliptic Equations. II. Complex Monge-Ampère, and Uniformly Elliptic, Equations. Comm. Pure and Appl. Math. 38 (1985), 209-252.

[C] E. Calabi, On Kähler manifolds with vanishing canonical class, Algebraic Geometry and Topology, A Symposium in Honor of S. Lefschetz, Princeton Univ. Press, (1955) 78-89.

[D1] J.-P. Demailly, Regularization of closed positive currents and intersection theory, J. Alg. Geom. 1 (1992), 361-409.

[D2] J.-P. Demailly, A numerical criterion for very ample line bundles, J. Differential

Geometry 37 (1993), 323-374.

[GW] R. E. Green, H. Wu, Embedding of open riemannian manifolds by harmonic

functions, Ann. Inst. Fourier 25 (1975), 215-235.

[H] L. Hörmander, Notions of convexity, Birkhäuser, 1996.

[HJ] R.A. Horn, C.R. Johnson, Matrix analysis, Cambridge Univ. Press, 1985.

[KL] M. Klimek, Pluripotential Theory, Oxford University Press, New York, 1991.

[K1] S. Kołodziej, Some sufficient conditions for solvability of the Dirichlet problem for the complex Monge-Ampère operator, Ann. Polon. Math. 65 (1996), 11-21.

[K2] S. Kołodziej, The complex Monge-Ampère equation, Acta Math. 180 (1998),69-117.

[K3] S. Kołodziej, Lecture notes in the complex Monge-Ampère equation, www.im.uj.edu.pl/gamma/kolodziej.

[K4] S. Kołodziej, Equicontinuity of families of plurisubharmonic functions with bounds on their Monge-Ampère masses, preprint.

[R] R. Richberg, Stetige streng pseudokonvexe Funktionen, Math. Ann. 175 (1968), 257-286.

[T] G. Tian, Canonical metrics in Kähler geometry, Lecture notes in mathematics, ETH Zürich, Birkhäuser.

[TZ] G. Tian, X. H. Zhu, Uniqueness of Kähler-Ricci solitons, Acta Math. 184 (2000), 271-305.

[X] Y. Xing, Continuity of the complex Monge-Ampère operator. Proc. of AMS 124 (1996), 457-467.

[Y] S.-T. Yau, On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation, Comm. Pure and Appl. Math. 31 (1978), 339-411.

postal address: Jagiellonian University, Institute of Mathematics Reymonta 4, 30-059 Kraków, Poland e-mail: kolodzie@ im.uj.edu.pl