The weighted Monge–Ampère energy of quasiplurisubharmonic functions
The weighted Monge–Ampère energy of quasiplurisubharmonic functions
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DOI：
10.1016/j.jfa.2007.04.018
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发表时间：
2006-12
影响因子：
1.7
通讯作者：
V. Guedj;A. Zeriahi
中科院分区：
文献类型：
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作者：
V. Guedj;A. Zeriahi
研究紧致Kähler流形（X，ω）上退化的复Monge-Ampère方程.我们证明了复Monge-Ampère算子[公式：见正文]在具有有限加权Monge-Ampère能量的ω-多重次调和函数类E（X，ω）上是良好定义的.类E（X，ω）是最大的ω-psh函数类，在它上面Monge-Ampère算子被很好地定义，比较原理是有效的.它包含几个函数，其梯度不是平方可积的。我们给出了E（X，ω）上算子[公式：见正文]的值域的完整描述，以及它的一些子类。我们还研究了唯一性的性质，扩展卡拉比的结果，这种无界和退化的情况下，我们给应用程序复杂的动力学和奇异凯勒-爱因斯坦度量的存在。
We study degenerate complex Monge–Ampère equations on a compact Kähler manifold (X,ω). We show that the complex Monge–Ampère operator [Formula: see text] 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 [Formula: see text] 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.