Some global solutions to the

complex Monge-Ampère equation.

by

Sławomir Kołodziej

IMUJ PREPRINT 1997/02

Abstract. We show that perturbing in a suitable way a radial non-negative function belonging to L_{loc}^{p} one can solve the complex Monge-Ampère equation in ℂ^{n} with the right hand side equal to this new function.

Introduction.

Though much is known about solvability of the Dirichlet problem for the complex Monge-Ampère equation in a strictly pseudoconvex domain in ℂ^{n} (see [BT1], [C], [K1], [K2], [K3]) we are far from understanding which measures are Monge-Ampère measures of entire plurisubharmonic functions.

This problem has been treated in some special cases. There are fairly sharp sufficient conditions which guarantee that a measure μ having finite mass in ℂ^{n} admits a solution of

(dd^{c}u)^{n} = dμ,

which (after suitable normalization) has logarithmic growth ([K2], see also [CK]). The result is based on a generalized version of Yau’s theorem [Y].

In [J] A. Jeune proved that perturbing (the density of) the Lebesgue measure dλ in ℂ^{n} by a smooth function which, together with all its derivatives, tends to 0 fast enough at infinity one can solve the Monge-Ampère equation for this new measure. The solution is smooth and differs from the potential of the Euclidean metric by a bounded function.

In [TY] G.Tian and S.-T. Yau solve the equation in a more general setting on some special non-compact complete Kähler manifolds. In this case, as well as in [J] and [Y], the attempt to solve the Monge-Ampère equation is motivated by geometrical problem of finding a Kähler metric with preassigned Ricci curvature.

In this paper we consider any radial plurisubharmonic function v such that its Monge-Ampère measure (dd^{c}v)^{n}(z) = M(z) dλ belongs to L_{loc}^{p}(ℂ^{n}), p > 1. We wish to solve

(∗)  \begin{matrix}u ∈ P SH ∩ C(ℂ^{n}) \\ (dd^{c}u)^{n} = (M + f) dλ, \end{matrix}

Typeset by AMS-TEX

1991 Mathematics Subject Classification. primary 32F07, secondary 32F05.

Key words and phrases. plurisubharmonic function, complex Monge-Ampère operator.

for f ≥ 0, f ∈ L_{loc}^{p} when the growth of f at infinity is controled in terms of M. To state it precisely we need to introduce some auxiliary functions:

F (z) := sup_{|w|=|z|} f(w),

H(r) := ∫_{B(r)} M(z) dλ,

Φ(r) := ∫_{B(r)} F (z) dλ,

where B(r) denotes the ball {|z| < r}.

Theorem. If F ∈ L_{loc}^{p}(ℂ^{n}), p > 1, and

∫_{1}^{∞} \frac{Φ(t)}{tH^{(n−1)/n}(t)} dt < ∞.

then there exists a unique u solving the equation (∗) and such that lim_{|z|→∞}(u − v)(z) = 0.

For v(z) = |z|^{2} we are in the situation of Jeune’s result. Then the integral in the statement of Theorem is finite if, for instance, F (t) ≤ t^{−2−ϵ}, ϵ > 0, t > t_{0}. This is exactly one of the assumptions on f in [J].

Note also that it is enough to take any compactly supported f ∈ L_{loc}^{p} if H(r) ≥ const.(log^{+} r)^{\frac{n}{n−1} +ϵ},

Proof of Theorem.

Proof. By [K1],[K2] one can solve the following Dirichlet problem:

(1)  \begin{matrix}u_{j} ∈ P SH(B(j)) ∩ C(B̄(j)) \\ (dd^{c}u_{j})^{n} = (M + f) dλ \\ u_{j} = v on ∂B(j). \end{matrix}

Set U_{j}(z) := max_{|w|=|z|} u_{j}(w), j = 1, 2, .... These are radial, continuous, plurisubharmonic functions.

Proposition 1. The sequence {U_{j} − v} is uniformly bounded.

Proof. By [BT1]:

(dd^{c} max(u_{j} ∘ T_{1}, u_{j} ∘ T_{2}, ..., u_{j} ∘ T_{k}))^{n} ≥ (dd^{c}v)^{n},

where T_{l} are unitary transformations. If we take a countable, dense set of such transformations and apply the Convergence theorem [BT2] then we arrive at

(2)  (dd^{c}U_{j})^{n} ≥ (dd^{c}v)^{n}, j = 1, 2, ... .

From the Comparison Principle [BT2] one also obtains

(3)  ∫_{B(j)}(dd^{c}U_{j})^{n} ≤ ∫_{B(j)}(dd^{c}u_{j})^{n} ≤ ∫_{B(j)}(dd^{c}v)^{n} + Φ(j).

Let us denote the density of (dd^{c}U_{j})^{n} by M_{j}. Then, following [M], one can compute U_{j} when M_{j} is given. For a radial function g we shall use the notation: g̃(t) := g(t, 0, 0, ..., 0).

\begin{aligned}\frac{d}{dt}Ũ_{j}(t) & = \frac{2}{t}[2n ∫_{0}^{t} x^{2n−1}M̃_{j}(x) dx]^{1/n} \\ & = \frac{2}{t}[2n ∫_{0}^{t} x^{2n−1}(M̃(x) + h_{j}(x)) dx]^{1/n}, \end{aligned}

where h_{j} is just M̃_{j} − M̃. Use this to estimate ṽ(t) − Ũ_{j}(t).

\frac{d}{dt}((Ũ_{j} − ṽ)(t)) = \frac{2}{t}(2n)^{1/n}\{[∫_{0}^{t} x^{2n−1}(M̃(x) + h_{j}(x)) dx]^{1/n}

− [∫_{0}^{t} x^{2n−1}M̃(x) dx]^{1/n}\} ≤ \frac{const.}{t}[(H(t) + Φ(t))^{1/n} − H^{1/n}(t)]

≤ \frac{const.}{t}[\frac{Φ(t)}{nH^{(n−1)/n}(t)}].

To get the last inequality write (a − b)/(a^{1/n} − b^{1/n}) as ∑ a^{k/n}b^{(n−k−1)/n}.

Integrating both sides of the above formula and recalling that Ũ_{j}(j) = ṽ(j) one can use the assumptions to find δ > 0 such that

v − δ ≤ U_{j}, j = 1, 2, ... .

On the other hand, from the Comparison Principle and (2) we have U_{j} ≤ v, and so the proposition follows.

Corollary. The sequence {u_{j} − v} is uniformly bounded.

Proof. Denote by Û_{j} the (radial) solution to (1) with (M + f) dλ replaced by dμ̂ = (M + F) dλ. By the proposition above

v − const. ≤ Û_{j}, j = 1, 2, ... ,

and by the Comparison Principle

Û_{j} ≤ u_{j} ≤ v, j = 1, 2, ... .

The proof is completed.

To prove the existence of a (locally bounded) solution to (∗) note that the sequence {u_{j}} is (locally) decreasing. Indeed, since due to the Comparison Principle u_{j+1} ≤ v = u_{j} on ∂B(j) the inequality u_{j+1} ≤ u_{j} extends to B(j) by another application of the Comparison Principle, as the Monge-Ampère measures of both functions are equal.

Thus, using the Convergece Theorem [BT1] and Corollary above we conclude that u = lim u_{j} is the desired solution. For the equality lim_{|z|→∞}(v − u)(z) = 0 it is enough to prove that lim_{|z|→∞}(v − Û)(z) = 0, where Û := lim Û_{j} with Û_{j} defined above. First, observe that

a := inf(v − Û) = lim_{|z|→∞}(v − Û).

Indeed, otherwise, for some b > a the set {v − b < Û} would have non-empty bounded component on which the Comparison Principle would be violated. Thus Û + a ≤ v, and so by the construction of Û_{j} and the Comparison Principle applied on B(j) we get Û_{j} ≥ Û + a, which upon letting j to infinity forces a to be 0.

Uniqueness in the form from the statement also follows from the Comparison Principle and the fact that lim_{|z|→∞}(u − w)(z) = 0 if w is another solution to (∗) with lim_{|z|→∞}(v − w)(z) = 0.

It remains to show that u is continuous. For this we need a lemma from [K2, Lemma 2.3.1]. Let us recall the definition of the relative capacity introduced in [BT2]:

cap(K, Ω) = sup{∫_{K}(dd^{c}u)^{n}; u ∈ P SH(Ω), −1 < u < 0},

for compact K in Ω. This set function extends to open sets as inner capacity.

Lemma. Let Ω be a strictly pseudoconvex subset of ℂ^{n} and let w, v belong to PSH(Ω). Assume that ∅ ≠ U_{c} := {w < v − c} ⊂ Ω′ ⊂ Ω and

(4)  ∫_{U_{c}}(dd^{c}w)^{n} ≤ F (cap(U_{c}, Ω)), c ∈ (0, C),

for F (x) = Ax(log(1 + x^{−1}))^{−(n+ϵ)}, A, ϵ > 0. Then C < κ = κ(A, ||v||, ϵ, Ω′, Ω) (the estimate is independent of w and v as long as ||v|| is under control). Moreover, κ → 0 as cap(U_{0}, B(R)) → 0 when A, ϵ, Ω′, Ω are fixed.

If (dd^{c}w)^{n} belongs to L^{p}(Ω), p > 1, then the inequality (4) is satisfied (see [K1]). Now, suppose u_{j} were not locally uniformly convergent. Then for a bounded

sequence z_{j} of points in ℂ^{n} and a positive constant C we would have

(5)  u_{j}(z_{j}) > u(z_{j}) + C.

Having lim_{|z|→∞}(v − u)(z) = 0 one can fix R > 0 such that v − u < C/2 on ℂ^{n} \ B(R). Then the sets

U_{j}(c) := {u_{j} > u + c}, c ∈ (C/2, C)

are non-empty and relatively compact in B(R). Since the sequence u_{j} is decreasing one can apply [BT2] (see [K2, Corollary 1.2.10]) to conclude that

cap((U_{j}(C/2), B(R)) → 0

as j → ∞, which contradicts the lemma above.

Remark 1. Suppose we wished to solve (∗) with non-positive f. In that case, set F (z) := sup_{|w|=|z|}(−f)(w), g := F + f, and N := M − F. If (N, g) satisfies the assumption of Theorem in place of (M, f) then (∗) is solvable since M +f = N +g.

Remark 2. Theorem readily lends itself to some generalizations. For instance, one can work in spaces larger than all L_{loc}^{p}, p > 1 (see [K1], [K2]). But the real progress would be achieved by weakening the most restrictive assumption saying, roughly, that the right hand side in the Monge-Ampère equation (∗) does not differ much from the radial measure, and expressed by the finiteness of the integral in the statement.

References

[BT1] E. Bedford and B.A. Taylor, The Dirichlet problem for a complex Monge-Ampère equation, 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.

[C] U. Cegrell, Pluricomplex energy. Research Reports 14 (1996) Umeå University.

[CK] U. Cegrell and S. Kołodziej, The global Dirichlet problem for the complex Monge-Ampère equation. Research Reports 2 (1995) Umeå University, to appear in J. Geom. Analysis.

[J] A. Jeune, Prescription de la courbure de Ricci sur ℂ^{n}, n > 2. Bull.Sc.Math., 2^{e} série 116 (1992), 185-226.

[K1] S. Kołodziej, Some sufficient conditions for solvability of the Dirichlet problem for the complex Monge-Ampère equation. Ann. Polon. Math. LXV.1 (1996), 11-21.

[K2] S. Kołodziej, The complex Monge-Ampère equation. Preprint 1995/11 Instytut Matematyki Uniwersytetu Jagiellońskiego, Kraków 1995.

[K3] S. Kołodziej, The range of the complex Monge-Ampère operator II. Indiana U. Math. J. 44, 3 (1995), 765-782.

[M] D.R. Monn, Regularity of the Complex Monge-Ampère Equation for the Radially Symmetric Functions of the Unit Ball. Math. Ann. 275 (1986), 501-511.

[TY] G.Tian and S.-T. Yau, Complete Kähler manifolds with the zero Ricci curvature. I, J.Amer.Math.Soc. 3 (1990), 579-609.

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

Polish Academy of Science, Institute of Mathematics Śniadeckich 8 00-950 Warszawa

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