The weighted Monge–Ampère energy of quasiplurisubharmonic functions

Abstract

We study degenerate complex Monge-Ampère equations on a compact Kähler manifold (X, ω). We show that the complex Monge-Ampère operator (ω + dd c ·) n is well defined on the class E(X, ω) of ω-plurisubharmonic functions with finite weighted Monge-Ampère energy. The class E(X, ω) is the largest class of ω-psh functions on which the Monge-Ampère operator is well defined and the comparison principle is valid. It contains several functions whose gradient is not square integrable. We give a complete description of the range of the operator (ω + dd c ·) n on E(X, ω), as well as on some of its subclasses. We also study uniqueness properties, extending Calabi's result to this unbounded and degenerate situation, and we give applications to complex dynamics and to the existence of singular Kähler-Einstein metrics.

Maximal subextensions of plurisubharmonic functions

“…However if the given function has a finite weighted Monge-Ampère energy in the sense of [GZ2], we will prove that the maximal subextension satisfies the same property.…”

“…Interesting classes have been investigated in [GZ2] and [CGZ]. We are going to introduce similar classes in the semi-global case where the complex Monge-Ampère operator is well defined and continuous under deacreasing sequences.…”

“…Let us introduce the following classes of finite weighted Monge-Ampère energy (see [Ce1], [GZ2], [BGZ]). A weight function is by definition an increasing function χ : R −→ R such that χ(t) = t is t 0 and χ(−∞) = −∞.…”

“…From the (IBP) formula, we can derive the following fundamental inequality which will be useful (see [GZ2]). …”

“…We can prove that the complex Monge-Ampère operator is well defined and continuous on decreasing sequences in the class E χ (D, ω), where χ is a convex increasing functions R −→ R (see [GZ2], [CGZ]). …”