The weighted Monge–Ampère energy of quasiplurisubharmonic functions

By Vincent Guedj, Ahmed Zeriahi. 2006

331 citations recorded by the source corpus.

Abstract We study degenerate complex Monge–Ampere equations on a compact Kahler manifold ( X , ω ) . We show that the complex Monge–Ampere operator ( ω + d d c ⋅ ) n is well defined on the class E ( X , ω ) of ω -plurisubharmonic functions with finite weighted Monge–Ampere energy. The class E ( X , ω ) is the largest class of ω -psh functions on which the Monge–Ampere 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 + d d 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 Kahler–Einstein metrics.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/700cac5a3bf9d73f32e6f378a404624cbecc85ae

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Open problems in pluripotential theory (2015, methodology): “It follows therefore from [25,42] that there exists a unique finite energy function φ ∈ E(X,ω) such that μ = MA(φ), and the problem is to show that φ is bounded below.”
Geometric pluripotential theory on Kähler manifolds (2019, methodology): “In Chapter 2, we develop some background in finite energy pluripotential theory, necessary for later developments, closely following the original treatises of Guedj–Zeriahi and collaborators [71, 72, 11, 12], that were inspired by work of Cegrell [33] in the local case.”
On the constant scalar curvature Kähler metrics (II)—Existence results (2020, background): “3 of [41] introduced the following space for any p ≥ 1: (2.”