A note on the preceding paper

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.

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“…Let μ be a probability measure which does not charge pluripolar sets. It follows from a generalization of Radon-Nikodym theorem [23] that…”

Section: Lemma 43mentioning

confidence: 99%

The weighted Monge–Ampère energy of quasiplurisubharmonic functions

Guedj

Zériahi

2007

Journal of Functional Analysis

225

487

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“…It follows from a generalized Radon-Nykodim Theorem [32] that there exists a positive measure ν ∈ M and a positive function f ∈ L 1 (ν) such that μ = f dν + ν s , where ν s is orthogonal to M . Observe also that every measure orthogonal to M is supported in some m-polar set since H m (h L ) ∈ M for each L K. We then deduce that ν s ≡ 0 since μ does not charge m-polar sets.…”

Section: Solving the Complex Hessian Equationmentioning

confidence: 99%

A variational approach to complex Hessian equations in Cn

2015

Journal of Mathematical Analysis and Applications

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Let Ω be an m-hyperconvex domain of C n and β be the standard Kähler form in C n . We introduce finite energy classes of m-subharmonic functions of Cegrell type, E p m (Ω), p > 0 and F m (Ω). Using a variational method we show that the degenerate complex Hessian equation (dd c ϕ) m ∧ β n−m = μ has a unique solution in E 1 m (Ω) if and only if every function in E 1 m (Ω) is integrable with respect to μ. If μ has finite total mass and does not charge m-polar sets, then the equation has a unique solution in F m (Ω).

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“…For the general case observe that since μ does not charge m-polar sets, it follows from [20] and the generalized Radon-Nikodym theorem [24] that we can write This is an analogue of Cegrell's decomposition theorem (see page 205 in [9], see also [10, Theorem 5.11]). We thus can assume that μ = H m (φ) for some bounded ω-m-sh function φ.…”

Section: Theorem 38 Let U V Be Functions In E(x ω M) and μ Be A Pmentioning

confidence: 99%

Mixed Hessian inequalities and uniqueness in the class $$\mathcal {E}(X,\omega ,m)$$ E ( X , ω , m )

Dinew

2014

Math. Z.

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We prove a general inequality for mixed Hessian measures by global arguments. Our method also yields a simplification for the case of complex Monge-Ampère equation. Exploiting this and using Kołodziej's mass concentration technique we also prove the uniqueness of the solutions to the complex Hessian equation on compact Kähler manifolds in the case of probability measures vanishing on m-polar sets.

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A note on the preceding paper

Cited by 25 publications

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The weighted Monge–Ampère energy of quasiplurisubharmonic functions

The weighted Monge–Ampère energy of quasiplurisubharmonic functions

A variational approach to complex Hessian equations in Cn

Mixed Hessian inequalities and uniqueness in the class $$\mathcal {E}(X,\omega ,m)$$ E ( X , ω , m )

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